,transcriber's ,notes3 ,underscores 80 before and after a word or phrase indicate italics in the original text4 ,small capitals have been converted to ,,solid capitals4 ,illustrations have been moved so they do not break up paragraphs4 ,typographical errors have been silently corrected4 ,,experimental ,,mechanics ,illustration ,illustration3 ,,the ,,path ,,of ,a ,,projectile ,,is ,a ,,parabola4 ,,experimental ,,mechanics ,a ,,course ,,of ,,lectures ,,delivered ,,at ,,the ,,royal ,,college ,,of ,,science ,,for ,,ireland ,,by ,,sir ,,robert ,,stawell ,,ball1 ,,ll4,d41 ,f4,r4,s4 ,,astronomer ,,royal ,,of ,,ireland ,,formerly ,,professor ,,of ,,applied ,,mathematics ,,and ,,mechanism ,,in ,,the ,,royal ,,college ,,of ,,science ,,for ,,ireland 7,,science ,,and ,,art ,,department7 ,,with ,,illustrations ,,second ,,edition ,london ,,macmillan ,,and ,,co4 ,,and ,,new ,,york #ahhh4 ,the ,right of ,translation and ,reproduction is reserved ,,richard ,,clay ,,and ,,sons1 ,,limited1 ,,london ,,and ,,bungay4 ,the ,first ,edition was printed in #ahga4 ,,preface4 ,i here present the revised edition of a course of lectures on ,experimental ,mechanics which ,i delivered in the ,royal ,college of ,science at ,dublin eighteen years ago4 ,the audience was a large evening class consisting chiefly of artisans4 ,the teacher of ,elementary ,mechanics1 whether he be in a ,board ,school1 a ,technical ,school1 a ,public ,school1 a ,science ,college1 or a ,university1 frequently desires to enforce his lessons by exhibiting working apparatus to his pupils1 and by making careful measurements in their presence4 ,he wants for this purpose apparatus of substantial proportions visible from every part of his lecture room4 ,he wants to have it of such a universal character that he can produce from it day after day combinations of an ever-varying type4 ,he wishes it to be composed of well-designed and well-made parts that shall be strong and durable1 and that will not easily get out of order4 ,he wishes those parts to be such that even persons not specially trained in manual skill shall presently learn how to combine them with good effect4 ,lastly1 he desires to economize his money in the matters of varnish1 mahogany1 and glass cases4 ,i found that ,i was able to satisfy all these requirements by a suitable adaptation of the very ingenious system of mechanical apparatus devised by the late ,professor ,willis of ,cambridge4 ,the elements of the system ,i have briefly described in an ,appendix1 and what adaptations ,i have made of it are shown in almost every page and every figure of the book4 ,in revising the present edition ,i have been aided by my friends ,mr4 ,g4 ,l4 ,cathcart1 the ,rev4 ,m4 ,h4 ,close1 and ,mr4 ,e4 ,p4 ,culverwell4 ,,robert ,s4 ,,ball4 ,,observatory1 ,,co4 ,,dublin1 #crd ,august1 #ahhh4 ,,table ,,of ,,contents4 ,,lecture ,i4 ,,the ,,composition ,,of ,,forces4 ,,page ,introduction4--,the ,definition of ,force4--,the ,measurement of ,force4--,equilibrium of ,two ,forces4--,equilibrium of ,three ,forces4--,a ,small ,force can sometimes balance ,two ,larger ,forces #a ,,lecture ,,ii4 ,,the ,,resolution ,,of ,,forces4 ,introduction4--,one ,force resolved into ,two ,forces4--,experimental ,illustrations4--,sailing4--,one ,force resolved into ,three ,forces not in the same ,plane4--,the ,jib and ,tie-rod #af ,,lecture ,,iii4 ,,parallel ,,forces4 ,introduction4--,pressure of a ,loaded ,beam on its ,supports4--,equilibrium of a ,bar supported on a ,knife-edge4--,the ,composition of ,parallel ,forces4--,parallel ,forces acting in opposite directions4--,the ,couple4--,the ,weighing ,scales #cd ,,lecture ,,iv4 ,,the ,,force ,,of ,,gravity4 ,introduction4--,specific ,gravity4--,the ,plummet and ,spirit-,level4--,the ,centre of ,gravity4--,stable and ,unstable ,equilibrium4--,property of the ,centre of ,gravity in a ,revolving ,wheel #ej ,,lecture ,v4 ,,the ,,force ,,of ,,friction4 ,the ,nature of ,friction4--,the ,mode of ,experimenting4--,friction is proportional to the pressure4--,a more accurate form of the ,law4--,the ,coefficient varies with the weights used4--,the ,angle of ,friction4--,another ,law of ,friction4--,concluding ,remarks #fe ,,lecture ,,vi4 ,,the ,,pulley4 ,introduction4--,friction between a ,rope and an ,iron ,bar4--,the ,use of the ,pulley4--,large and ,small ,pulleys4--,the ,law of ,friction in the ,pulley4--,wheels4--,energy #he ,,lecture ,,vii4 ,,the ,,pulley-,,block4 ,introduction4--,the ,single ,movable ,pulley4--,the ,three-sheave ,pulley-block4--,the ,differential ,pulley-block4--,the ,epicycloidal ,pulley-block #ii ,,lecture ,,viii4 ,,the ,,lever4 ,the ,lever of the ,first ,order4--,the ,lever of the ,second ,order4--,the ,shears4--,the ,lever of the ,third ,order #aai ,,lecture ,,ix4 ,,the ,,inclined ,,plane ,,and ,,the ,,screw4 ,the ,inclined ,plane without ,friction4--,the ,inclined ,plane with ,friction4--,the ,screw4--,the ,screw-jack4--,the ,bolt and ,nut #aca ,,lecture ,x4 ,,the ,,wheel ,,and ,,axle4 ,introduction4--,experiments upon the ,wheel and ,axle4--,friction upon the ,axle4--,the ,wheel and ,barrel4--,the ,wheel and ,pinion4--,the ,crane4--,conclusion #adi ,,lecture ,,xi4 ,,the ,,mechanical ,,properties ,,of ,,timber4 ,introduction4--,the ,general ,properties of ,timber4--,resistance to ,extension4--,resistance to ,compression4--,condition of a ,beam strained by a ,transverse ,force #afi ,,lecture ,,xii4 ,,the ,,strength ,,of ,a ,,beam4 ,a ,beam free at the ,ends and loaded in the ,middle4--,a ,beam uniformly loaded4--,a ,beam loaded in the ,middle1 whose ,ends are secured4--,a ,beam supported at one end and loaded at the other #ahh ,,lecture ,,xiii4 ,,the ,,principles ,,of ,,framework4 ,introduction4--,weight sustained by ,tie and ,strut4--,bridge with ,two ,struts4--,bridge with ,four ,struts4--,bridge with ,two ,ties4--,simple ,form of ,trussed ,bridge #bjc ,,lecture ,,xiv4 ,,the ,,mechanics ,,of ,a ,,bridge4 ,introduction4--,the ,girder4--,the ,tubular ,bridge4--,the ,suspension ,bridge #bah ,,lecture ,,xv4 ,,the ,,motion ,,of ,a ,,falling ,,body4 ,introduction4--,the ,first ,law of ,motion4--,the ,experiment of ,galileo from the ,tower of ,pisa4--,the ,space is proportional to the ,square of the ,time4--,a ,body falls #af' in the ,first ,second4--,the ,action of ,gravity is independent of the ,motion of the ,body4--,how the ,force of ,gravity is defined4--,the ,path of a ,projectile is a ,parabola #bcj ,,lecture ,,xvi4 ,,inertia4 ,inertia4--,the ,hammer4--,the ,storing of ,energy4--,the ,fly-wheel4--,the ,punching ,machine #bej ,,lecture ,,xvii4 ,,circular ,,motion4 ,the ,nature of ,circular ,motion4--,circular motion in ,liquids4--,the ,applications of ,circular ,motion4--,the ,permanent ,axes #bfg ,,lecture ,,xviii4 ,,the ,,simple ,,pendulum4 ,introduction4--,the ,circular ,pendulum4--,law connecting the ,time of ,vibration with the ,length4--,the ,force of ,gravity determined by the ,pendulum4--,the ,cycloid #bhd ,,lecture ,,xix4 ,,the ,,compound ,,pendulum ,,and ,,the ,,composition ,,of ,,vibrations4 ,the ,compound ,pendulum4--,the ,centre of ,oscillation4--,the ,centre of ,percussion4--,the ,conical ,pendulum4--,the ,composition of ,vibrations #bii ,,lecture ,,xx4 ,,the ,,mechanical ,,principles ,,of ,a ,,clock4 ,introduction4--,the ,compensating ,pendulum4--,the ,escapement4--,the ,train of ,wheels4--,the ,hands4--,the ,striking ,parts #cah ,,appendix ,i4 ,the ,method of ,graphical ,construction #cci ,the ,method of ,least ,squares #cdb ,,appendix ,,ii4 ,details of the ,willis ,apparatus used in illustrating the foregoing lectures #cde ,,index #cee ,,experimental ,,mechanics4 ,,lecture ,i4 ,,the ,,composition ,,of ,,forces4 ,introduction4--,the ,definition of ,force4--,the ,measurement of ,force4--,equilibrium of ,two ,forces4--,equilibrium of ,three ,forces4--,a ,small ,force can sometimes balance ,two ,larger ,forces4 ,,introduction4 #a4 ,i shall endeavour in this course of lectures to illustrate the elementary laws of mechanics by means of experiments4 ,in order to understand the subject treated in this manner1 you need not possess any mathematical knowledge beyond an acquaintance with the rudiments of algebra and with a few geometrical terms and principles4 ,but even to those who1 having an acquaintance with mathematics1 have by its means acquired a knowledge of mechanics1 experimental illustrations may still be useful4 ,by actually seeing the truth of results with which you are theoretically familiar1 clearer conceptions may be produced1 and perhaps new lines of thought opened up4 ,besides1 many of the mechanical principles which lie rather beyond the scope of elementary works on the subject are very susceptible of being treated experimentally2 and to the consideration of these some of the lectures of this course will be devoted4 ,many of our illustrations will be designedly drawn from very commonplace sources3 by this means ,i would try to impress upon you that mechanics is not a science that exists in books merely1 but that it is a study of those principles which are constantly in action about us4 ,our own bodies1 our houses1 our vehicles1 all the implements and tools which are in daily use--in fact all objects1 natural and artificial1 contain illustrations of mechanical principles4 ,you should acquire the habit of carefully studying the various mechanical contrivances which may chance to come before your notice4 ,examine the action of a crane raising weights1 of a canal boat descending through a lock4 ,notice the way a roof is made1 or how it is that a bridge can sustain its load4 ,even a well-constructed farm-gate1 with its posts and hinges1 will give you admirable illustrations of the mechanical principles of framework4 ,take some opportunity of examining the parts of a clock1 of a sewing-machine1 and of a lock and key2 visit a saw-mill1 and ascertain the action of all the machines you see there2 try to familiarize yourself with the principles of the tools which are to be found in any workshop4 ,a vast deal of interesting and useful knowledge is to be acquired in this way4 ,,the ,,definition ,,of ,,force4 #b4 ,it is necessary to know the answer to this question1 ,what is a force8 ,people who have not studied mechanics occasionally reply1 ,a push is a force1 a steam-engine is a force1 a horse pulling a cart is a force1 gravitation is a force1 a movement is a force1 c41 c4 ,the true definition of force is that which tends to produce or to destroy motion4 ,you may probably not fully understand this until some further explanations and illustrations shall have been given2 but1 at all events1 put any other notion of force out of your mind4 ,whenever ,i use the word ,force1 do you think of the words 8something which tends to produce or to destroy motion10 and ,i trust before the close of the lecture you will understand how admirably the definition conveys what force really is4 #c4 ,when a string is attached to this small weight1 ,i can1 by pulling the string1 move the weight along the table4 ,in this case1 there is something transmitted from my hand along the string to the weight in consequence of which the weight moves3 that something is a force4 ,i can also move the weight by pushing it with a stick1 because force is transmitted along the stick1 and makes itself known by producing motion4 ,the archer who has bent his bow and holds the arrow between his finger and thumb feels the string pulling until the impatient arrow darts off4 ,here motion has been produced by the force of elasticity in the bent bow4 ,before he released the arrow there was no motion1 yet still the bow was exerting force and tending to produce motion4 ,hence in defining force we must say 8that which tends to produce motion10 whether motion shall actually result or not4 #d4 ,but forces may also be recognized by their capability or tendency to prevent or to destroy motion4 ,before ,i release the arrow ,i am conscious of exerting a force upon it in order to counteract the pull of the string4 ,here my force is merely manifested by destroying the motion that1 if it were absent1 the bow would produce4 ,so when ,i hold a weight in my hand1 the force exerted by my hand destroys the motion that the weight would acquire were ,i to let it fall2 and if a weight greater than ,i could support were placed in my hand1 my efforts to sustain it would still be properly called force1 because they tended to destroy motion1 though unsuccessfully4 ,we see by these simple cases that a force may be recognized either by producing motion or by trying to produce it1 by destroying motion or by tending to destroy it2 and hence the propriety of the definition of force must be admitted4 ,,the ,,measurement ,,of ,,force4 #e4 ,as forces differ in magnitude1 it becomes necessary to establish some convenient means of expressing their measurements4 ,the pressure exerted by one pound weight at ,london is the standard with which we shall compare other forces4 ,the piece of iron or other substance which is attracted to the earth with this force in ,london1 is attracted to the earth with a greater force at the pole and a less force at the equator2 hence1 in order to define the standard force1 we have to mention the locality in which the pressure of the weight is exerted4 ,it is easy to conceive how the magnitude of a pushing or a pulling force may be described as equivalent to so many pounds4 ,the force which the muscles of a man's arm can exert is measured by the weight which he can lift4 ,if a weight be suspended from an india-rubber spring1 it is evident the spring will stretch so that the weight pulls the spring and the spring pulls the weight2 hence the number of pounds in the weight is the measure of the force the spring is exerting4 ,in every case the magnitude of a force can be described by the number of pounds expressing the weight to which it is equivalent4 ,there is another but much more difficult mode of measuring force occasionally used in the higher branches of mechanics 7,art4 #dig71 but the simpler method is preferable for our present purpose4 ,illustration3 ,,fig4 #a4 #f4 ,the straight line in which a force tends to move the body to which it is applied is called the direction of the force4 ,let us suppose1 for example1 that a force of #c lbs4 is applied at the point ,a1 ,fig4 #a1 tending to make ,a move in the direction ,,ab4 ,a standard line ,c of certain length is to be taken4 ,it is supposed that a line of this length represents a force of #a lb4 ,the line ,,ab is to be measured1 equal to three times ,c in length1 and an arrow-head is to be placed upon it to show the direction in which the force acts4 ,hence1 by means of a line of certain length and direction1 and having an arrow-head attached1 we are able completely to represent a force4 ,,equilibrium ,,of ,,two ,,forces4 ,illustration3 ,,fig4 #b4 #g4 ,in ,fig4 #b we have represented two equal weights to which strings are attached2 these strings1 after passing over pulleys1 are fastened by a knot ,c4 ,the knot is pulled by equal and opposite forces4 ,i mark off parts ,,cd1 ,,ce1 to indicate the forces2 and since there is no reason why ,c should move to one side more than the other1 it remains at rest4 ,hence1 we learn that two equal and directly opposed forces counteract each other1 and each may be regarded as destroying the motion which the other is striving to produce4 ,if ,i make the weights unequal by adding to one of them1 the knot is no longer at rest2 it instantly begins to move in the direction of the larger force4 #h4 ,when two equal and opposite forces act at a point1 they are said to be in equilibrium4 ,more generally this word is used with reference to any set of forces which counteract each other4 ,when a force acts upon a body1 at least one more force must be present in order that the body should remain at rest4 ,if two forces acting on a point be not opposite1 they will not be in equilibrium2 this is easily shown by pulling the knot ,c in ,fig4 #b downwards4 ,when released1 it flies back again4 ,this proves that if two forces be in equilibrium their directions must be opposite1 for otherwise they will produce motion4 ,we have already seen that the two forces must be equal4 ,a book lying on the table is at rest4 ,this book is acted upon by two forces which1 being equal and opposite1 destroy each other4 ,one of these forces is the gravitation of the earth1 which tends to draw the book downwards1 and which would1 in fact1 make the book fall if it were not sustained by an opposite force4 ,the pressure of the book on the table is often called the action1 while the resistance offered by the table is the force of reaction4 ,we here see an illustration of an important principle in nature1 which says that action and reaction are equal and opposite4 ,,equilibrium ,,of ,,three ,,forces4 ,illustration3 ,,fig4 #c4 #i4 ,we now come to the important case where three forces act on a point3 this is to be studied by the apparatus represented in ,fig4 #c4 ,it consists essentially of two pulleys ,h1 ,h1 each about #b diameter1#a which are capable of turning very freely on their axles2 the distance between these pulleys is about #e'1 and they are supported at a height of #f' by a frame1 which will easily be understood from the figure4 ,over these pulleys passes a fine cord1 #i' or #aj' long1 having a light hook at each of the ends ,e1 ,f4 ,to the centre of this cord ,d a short piece is attached1 which at its free end ,g is also furnished with a hook4 ,a number of iron weights1 #je lb41 #a lb41 #b lbs41 c41 with rings at the top1 are used2 one or more of these can easily be suspended from the hooks as occasion may require4 #a ,we shall often1 in these lectures1 represent feet or inches in the manner usual among practical men--#a' is one foot1 #a is one inch4 ,thus1 for example1 #c' #d is to be read 8three feet four inches40 ,when it is necessary to use fractions we shall always employ decimals4 ,for example1 #je is the mode of expressing a length of half an inch2 #c' #ai is to be read 8three feet one inch and nine-tenths of an inch40 #aj4 ,we commence by placing one pound on each of the hooks4 ,the cords are first seen to make a few oscillations and then to settle into a definite position4 ,if we disturb the cords and try to move them into some new position they will not remain there2 when released they will return to the places they originally occupied4 ,we now concentrate our attention on the central point ,d1 at which the three forces act4 ,let this be represented by ,o in ,fig4 #d1 and the lines ,,op1 ,,oq1 and ,,os will be the directions of the three cords4 ,on examining these positions we find that the three angles ,p ,o ,s1 ,q ,o ,s1 ,p ,o ,q1 are all equal4 ,this may very easily be proved by holding behind the cords a piece of cardboard on which three lines meeting at a point and making equal angles have been drawn2 it will then be seen that the cords coincide with the three lines on the cardboard4 ,illustration3 ,,fig4 #d4 #aa4 ,a little reflection would have led us to anticipate this result4 ,for the three cords being each stretched by a tension of a pound1 it is obvious that the three forces pulling at ,o are all equal4 ,as ,o is at rest1 it seems obvious that the three forces must make the angles equal1 for suppose that one of the angles1 ,p ,o ,q for instance1 was less than either of the others1 experiment shows that the forces ,o ,p and ,,oq would be too strong to be counteracted by ,o ,s4 ,the three angles must therefore be equal1 and then the forces are arranged symmetrically4 #ab4 ,the forces being each #a lb41 mark off along the three lines in ,fig4 #d 7which represent their directions7 three equal parts ,o ,p1 ,o ,q1 ,o ,s1 and place the arrowheads to show the direction in which each force is acting2 the forces are then completely represented both in position and in magnitude4 ,since these forces make equilibrium1 each of them may be considered to be counteracted by the other two4 ,for example1 ,o ,s is annulled by ,o ,q and ,o ,p4 ,but ,o ,s could be balanced by a force ,o ,r equal and opposite to it4 ,hence ,,or is capable of producing by itself the same effect as the forces ,o ,p and ,,oq taken together4 ,therefore ,o ,r is equivalent to ,o ,p and ,,oq4 ,here we learn the important truth that two forces not in the same direction can be replaced by a single force4 ,the process is called the composition of forces1 and the single force is called the resultant of the two forces4 ,o ,r is only one pound1 yet it is equivalent to the forces ,o ,p and ,o ,q together1 each of which is also one pound4 ,this is because the forces ,o ,p and ,o ,q partly counteract each other4 #ac4 ,draw the lines ,p ,r and ,q ,r2 then the angles ,p ,o ,r and ,q ,o ,r are equal1 because they are the supplements of the equal angles ,p ,o ,s and ,q ,o ,s2 and since the angles ,p ,o ,r and ,q ,o ,r together make up one-third of four right angles1 it follows that each of them is two-thirds of one right angle1 and therefore equal to the angle of an equilateral triangle4 ,also ,o ,p being equal to ,o ,q and ,o ,r common1 the triangles ,o ,p ,r and ,o ,q ,r must be equilateral4 ,therefore the angle ,p ,r ,o is equal to the angle ,r ,o ,q2 thus ,p ,r is parallel to ,o ,q2 similarly ,q ,r is parallel to ,o ,p2 that is1 ,o ,p ,r ,q is a parallelogram4 ,here we first perceive the great law that the resultant of two forces acting at a point is the diagonal of a parallelogram1 of which they are the two sides4 #ad4 ,this remarkable geometrical figure is called the parallelogram of forces4 ,stated in its general form1 the property we have discovered asserts that two forces acting at a point have a resultant1 and that this resultant is represented both in magnitude and in direction by the diagonal of the parallelogram1 of which two adjacent sides are the lines which represent the forces4 ,illustration3 ,,fig4 #e4 #ae4 ,the parallelogram of forces may be illustrated in various ways by means of the apparatus of ,fig4 #c4 ,attach1 for example1 to the middle hook ,g #ae lb41 and place #a lb4 on each of the remaining hooks ,e1 ,f4 ,here the three weights are not equal1 and symmetry will not enable us1 as it did in the previous case1 to foresee the condition which the cords will assume2 but they will be observed to settle in a definite position1 to which they will invariably return if withdrawn from it4 ,let ,o ,p1 ,o ,q 7,fig4 #e7 be the directions of the cords2 ,o ,p and ,o ,q being each of the length which corresponds to #a lb41 while ,o ,s corresponds to #ae lb4 ,here1 as before1 ,o ,p and ,o ,q together may be considered to counteract ,o ,s4 ,but ,o ,s could have been counteracted by an equal and opposite force ,o ,r4 ,hence ,o ,r may be regarded as the single force equivalent to ,o ,p and ,o ,q1 that is1 as their resultant2 and thus it is proved experimentally that these forces have a resultant4 ,we can further verify that the resultant is the diagonal of the parallelogram of which the equal forces are the sides4 ,construct a parallelogram on a piece of cardboard having its four sides equal1 and one of the diagonals half as long again as one of the sides4 ,this may be done very easily by first drawing one of the two triangles into which the diagonal divides the parallelogram4 ,the diagonal is to be produced beyond the parallelogram in the direction ,o ,s4 ,when the cardboard is placed close against the cords1 the two cords will lie in the directions ,o ,p1 ,o ,q1 while the produced diagonal will be in the vertical ,o ,s4 ,thus the application of the parallelogram of force is verified4 ,illustration3 ,,fig4 #f4 #af4 ,the same experiment shows that two unequal forces may be compounded into one resultant4 ,for in ,fig4 #e the two forces ,o ,p and ,o ,s may be considered to be counterbalanced by the force ,o ,q2 in other words1 ,o ,q must be equal and opposite to a force which is the resultant of ,o ,p and ,o ,s4 #ag4 ,let us place on the central hook ,g a weight of #e lbs41 and weights of #c lbs4 on the hook ,e and #d lbs4 on ,f4 ,this is actually the case shown in ,fig4 #c4 ,the weights being unequal1 we cannot immediately infer anything with reference to the position of the cords1 but still we find1 as before1 that the cords assume a definite position1 to which they return when temporarily displaced4 ,let ,fig4 #f represent the positions of the cords4 ,no two of the angles are in this case equal4 ,still each of the forces is counterbalanced by the other two4 ,each is therefore equal and opposite to the resultant of the other two4 ,construct the parallelogram on cardboard1 as can be easily done by forming the triangle ,o ,p ,r1 whose sides are #c1 #d1 and #e1 and then drawing ,o ,q and ,r ,q parallel to ,r ,p and ,o ,p4 ,produce the diagonal ,o ,r to ,s4 ,this parallelogram being placed behind the cords1 you see that the directions of the cords coincide with its sides and diagonal1 thus verifying the parallelogram of forces in a case where all the forces are of different magnitudes4 #ah4 ,it is easy1 by the application of a set square1 to prove that in this case the cords attached to the #c lb4 and #d lb4 weights are at right angles to each other4 ,we could have inferred1 from the parallelogram of force1 that this must be the case1 for the sides of the triangle ,o ,p ,r are #c1 #d1 and #e respectively1 and since the square of #e is #be1 and the squares of #c and of #d are #i and #af respectively1 it follows that the square of one side of this triangle is equal to the sum of the squares of the two opposite sides1 and therefore this is a right-angled triangle 7,euclid1 i4 #dh74 ,hence1 since ,p ,r is parallel to ,o ,q1 the angle ,p ,o ,q must also be a right angle4 ,a ,,small ,,force ,,sometimes ,,balances ,,two ,,larger ,,forces4 #ai4 ,cases might be multiplied indefinitely by placing various amounts of weight on the hooks1 constructing the parallelogram on cardboard1 and comparing it with the cords as before4 ,we shall1 however1 confine ourselves to one more illustration1 which is capable of very remarkable applications4 ,attach #a lb4 to each of the hooks ,e and ,f2 the cord joining them remains straight until drawn down by placing a weight on the centre hook4 ,a very small weight will suffice to do this4 ,let us put on half-a-pound2 the position the cords then assume is indicated in ,fig4 #g4 ,as before1 each force is equal and opposite to the resultant of the other two4 ,hence a force of half-a-pound is the resultant of two forces each of #a lb4 ,the apparent paradox is explained by noticing that the forces of #a lb4 are very nearly opposite1 and therefore to a large extent counteract each other4 ,constructing the cardboard parallelogram we may easily verify that the principle of the parallelogram of forces holds in this case also4 ,illustration3 ,,fig4 #g4 #bj4 ,no matter how small be the weight we suspend from the middle of a horizontal cord1 you see that the cord is deflected3 and no matter how great a tension were applied1 it would be impossible to straighten the cord4 ,the cord could break1 but it could not again become horizontal4 ,look at a telegraph wire2 it is never in a straight line between two consecutive poles1 and its curved form is more evident the greater be the distance between the poles4 ,but in putting up a telegraph wire great straining force is used1 by means of special machines for the purpose2 yet the wires cannot be straightened3 because the weight of the heavy wire itself acts as a force pulling it downwards4 ,just as the cord in our experiments cannot be straight when any force1 however small1 is pulling it downwards at the centre1 so it is impossible by any exertion of force to straighten the long wire4 ,some further illustrations of this principle will be given in our next lecture1 and with one application of it the present will be concluded4 #ba4 ,one of the most important practical problems in mechanics is to make a small force overcome a greater4 ,there are a number of ways in which this may be accomplished for different purposes1 and to the consideration of them several lectures of this course will be devoted4 ,perhaps1 however1 there is no arrangement more simple than that which is furnished by the principles we have been considering4 ,we shall employ it to raise a #bh lb4 weight by means of a #b lb4 weight4 ,i do not say that this particular application is of much practical use4 ,i show it to you rather as a remarkable deduction from the parallelogram of forces than as a useful machine4 ,illustration3 ,,fig4 #h4 ,a rope is attached at one end of an upright1 ,a 7,fig4 #h71 and passes over a pulley ,b at the same vertical height about #af' distant4 ,a weight of #bh lbs4 is fastened to the free end of the rope1 and the supports must be heavily weighted or otherwise secured from moving4 ,the rope ,,ab is apparently straight and horizontal1 in consequence of its weight being inappreciable in comparison with the strain 7#bh lbs47 to which it is subjected2 this position is indicated in the figure by the dotted line ,,ab4 ,we now suspend from ,c at the middle of the rope a weight of #b lbs4 ,instantly the rope moves to the position represented in the figure4 ,but this it cannot do without at the same moment raising slightly the #bh lbs41 for1 since two sides of a triangle1 ,,cb1 ,,ca1 are greater than the third side1 ,,ab1 more of the rope must lie between the supports when it is bent down by the #b lb4 weight than when it was straight4 ,but this can only have taken place by shortening the rope between the pulley ,b and the #bh lb4 weight1 for the rope is firmly secured at the other end4 ,the effect on the heavy weight is so small that it is hardly visible to you from a distance4 ,we can1 however1 easily show by an electrical arrangement that the big weight has been raised by the little one4 #bb4 ,when an electric current passes through this alarum you hear the bell ring1 and the moment ,i stop the current the bell stops4 ,i have fastened one piece of brass to the #bh lb4 weight1 and another to the support close above it1 but unless the weight be raised a little the two will not be in contact2 the electricity is intended to pass from one of these pieces of brass to the other1 but it cannot pass unless they are touching4 ,when the rope is straight the two pieces of brass are separated1 the current does not pass1 and our alarum is dumb2 but the moment ,i hang on the #b lb4 weight to the middle of the rope it raises the weight a little1 brings the pieces of brass in contact1 and now you all hear the alarum4 ,on removing the #b lbs4 the current is interrupted and the noise ceases4 #bc4 ,i am sure you must all have noticed that the #b lb4 weight descended through a distance of many inches1 easily visible to all the room2 that is to say1 the small weight moved through a very considerable distance1 while in so doing it only raised the larger one a very small distance4 ,this is a point of the very greatest importance2 ,i therefore take the first opportunity of calling your attention to it4 ,,lecture ,,ii4 ,,the ,,resolution ,,of ,,forces4 ,introduction4--,one ,force resolved into ,two ,forces4--,experimental ,illustrations4--,sailing4--,one ,force resolved into ,three ,forces not in the same ,plane4--,the ,jib and ,tie-rod4 ,,introduction4 ,illustration3 ,,fig4 #i4 #bd4 ,as the last lecture was principally concerned with discussing how one force could replace two forces1 so in the present we shall examine the converse question1 ,how may two forces replace one force8 ,since the diagonal of a parallelogram represents a single force equivalent to those represented by the sides1 it is obvious that one force may be resolved into two others1 provided it be the diagonal of the parallelogram formed by them4 #be4 ,we shall frequently employ in the present lecture1 and in some of those that follow1 the spring balance1 which is represented in ,fig4 #i3 the weight is attached to the hook1 and when the balance is suspended by the ring1 a pointer indicates the number of pounds on a scale4 ,this balance is very convenient for showing the strain along a cord2 for this purpose the balance is held by the ring while the cord is attached to the hook4 ,it will be noticed that the balance has two rings and two corresponding hooks4 ,the hook and ring at the top and bottom will weigh up to #cjj lbs41 corresponding to the scale which is seen4 ,the hook and ring at the side correspond to another scale on the other face of the plate3 this second scale weighs up to about #ej lbs41 consequently for a weight under #ej lbs4 the side hook and ring are employed1 as they give a more accurate result than would be obtained by the top and bottom hook and ring1 which are intended for larger weights4 ,these ingenious and useful balances are sufficiently accurate1 and can easily be tested by raising known weights4 ,besides the instrument thus described1 we shall sometimes use one of a smaller size1 and we shall be able with this aid to trace the existence and magnitude of forces in a most convenient manner4 ,,one ,,force ,,resolved ,,into ,,two ,,forces4 #bf4 ,we shall first illustrate how a single force may be resolved into a pair of forces2 for this purpose we shall use the arrangement shown in ,fig4 #aj 7see next page74 ,the ends of a cord are fastened to two small spring balances2 to the centre ,e of this cord a weight of #d lbs4 is attached4 ,at ,a and ,b are pegs from which the balances can be suspended4 ,let the distances ,,ae1 ,,be be each #ab1 and the distance ,,ab #af4 ,when the cord is thus placed1 and the weight allowed to hang freely1 each of the cords ,,ea1 ,,eb is strained by an amount of force that is shown to be very nearly #c lbs4 by the balances4 ,but the weight of #d lbs4 is the only weight acting2 hence it must be equivalent to two forces of very nearly #c lbs4 each along the directions ,,ae and ,,be4 ,here the two forces to which #d lbs4 is equivalent are each of them less than #d lbs41 though taken together they exceed it4 ,illustration3 ,,fig4 #aj4 #bg4 ,but remove the cords from ,,ab and hang them on ,,cd1 the length ,,cd being #a' #aj1 then the forces shown along ,,fc and ,d are each #e lbs42 here1 therefore1 one force of #d lbs4 is equivalent to two forces each of #e lbs4 ,in the last lecture 7,art4 #ai7 we saw that one force could balance two greater forces2 here we see the analogous case of one force being changed into two greater forces4 ,further1 we learn that the number of pairs of forces into which one force may be decomposed is unlimited1 for with every different distance between the pegs different forces will be indicated by the balances4 ,whenever the weight is suspended from a point half-way between the balances1 the forces along the cords are equal2 but by placing the weight nearer one balance than the other1 a greater force will be indicated on that balance to which the weight is nearest4 ,,experimental ,,illustrations4 ,illustration3 ,,fig4 #aa4 #bh4 ,the resolution or decomposition of one force into two forces each greater than itself is capable of being illustrated in a variety of ways1 two of which will be here explained4 ,in ,fig4 #aa an arrangement for this purpose is shown4 ,a piece of stout twine ,,ab1 able to support from #bj lbs4 to #cj lbs41 is fastened at one end ,a to a fixed support1 and at the other end ,b to the eye of a wire-strainer4 ,a wire-strainer consists of an iron rod1 with an eye at one end and a screw and a nut at the other2 it is used for tightening wires in wire fencing2 and is employed in this case for the purpose of stretching the cord4 ,this being done1 ,i take a piece of ordinary sewing-thread1 which is of course weaker than the stout twine4 ,i tie the thread to the middle of the cord at ,c1 catch the other end in my fingers1 and pull2 something must break--something has broken3 but what has broken8 ,not the slight thread1 it is still whole2 it is the cord which has snapped4 ,now this illustrates the point on which we have been dwelling4 ,the force which ,i transmitted along the thread was insufficient to break it2 the thread transferred the force to the cord1 but under such circumstances that the force was greatly magnified1 and the consequence was that this magnified force was able to break the cord before the original force could break the thread4 ,we can also see why it was necessary to stretch the cord4 ,in ,fig4 #aj the strains along the cords are greater when the cords are attached at ,c and ,d than when they are attached at ,a and ,b2 that is to say1 the more the cord is stretched towards a straight line1 the greater are the forces into which the applied force is resolved4 #bi4 ,we give a second example1 in illustration of the same principle4 ,in ,fig4 #ab is shown a chain #h' long1 one end of which ,b is attached to a wire-strainer1 while the other end is fastened to a small piece of pine ,a1 which is #je square in section1 and #e long between the two upright irons by which it is supported4 ,by means of the nut of the wire-strainer ,i straighten the chain as ,i did the string of ,fig4 #aa1 and for the same reason4 ,i then put a piece of twine round the chain and pull it gently4 ,the strain brought to bear on the wood is so great that it breaks across4 ,here1 the small force of a few pounds1 transmitted to the chain by pulling the siring1 is magnified to upwards of a hundredweight1 for less than this would not break the wood4 ,the explanation is precisely the same as when the string was broken by the thread4 ,illustration3 ,,fig4 #ab4 ,,sailing4 #cj4 ,the action of the wind upon the sails of a vessel affords a very instructive and useful example of the decomposition of forces4 ,by the parallelogram of forces we are able to explain how it is that a vessel is able even to sail against the wind4 ,a force is that which tends to produce motion1 and motion generally takes place in the line of the force4 ,in the case of the action of wind on a vessel through the medium of the sails1 we have motion produced which is not necessarily in the direction of the wind1 and which may be to a certain extent opposed to it4 ,this apparent paradox requires some elucidation4 ,illustration3 ,,fig4 #ac4 #ca4 ,let us first suppose the wind to be blowing in a direction shown by the arrows of ,fig4 #ac1 perpendicular to the line ,,ab in which the ship's course lies4 ,in what direction must the sail be set8 ,it is clear that the sail must not be placed along the line ,,ab1 for then the only effect of the wind would be to blow the vessel sideways2 nor could the sail be placed with its edge to the wind1 that is1 along the line ,o ,w1 for then the wind would merely glide along the sail without producing a propelling force4 ,let1 then1 the sail be placed between the two positions1 as in the direction ,p ,q4 ,the line ,o ,w represents the magnitude of the force of the wind pressing on the sail4 ,we shall suppose for simplicity that the sail extends on both sides of ,o4 ,through ,o draw ,o ,r perpendicular to ,p ,q1 and from ,w let fall the perpendicular ,w ,x on ,p ,q1 and ,w ,r on ,o ,r4 ,by the principle of the parallelogram of forces1 the force ,o ,w may be decomposed into the two forces ,o ,x and ,o ,r1 since these are the sides of the parallelogram of which ,o ,w1 the force of the wind1 is the diagonal4 ,we may then leave ,o ,w out of consideration1 and imagine the force of the wind to be replaced by the pair of forces ,o ,x and ,o ,r2 but the force ,o ,x cannot produce an effect1 it merely represents a force which glides along the surface of the sail1 not one which pushes against it2 so far as this component goes1 the sail has its edge towards it1 and therefore the force produces no effect4 ,on the other hand1 the sail is perpendicular to the force ,o ,r1 and this is therefore the efficient component4 ,the force of the wind is thus measured by ,o ,r1 both in magnitude and direction3 this force represents the actual pressure on the mast produced by the sail1 and from the mast communicated to the ship4 ,still ,o ,r is not in the direction in which the ship is sailing3 we must again decompose the force in order to find its useful effect4 ,this is done by drawing through ,r the lines ,r ,l and ,r ,m parallel to ,o ,a and ,o ,w1 thus forming the parallelogram ,o ,m ,r ,l4 ,hence1 by the parallelogram of forces1 the force ,o ,r is equivalent to the two forces ,o ,l and ,o ,m4 ,the effect of ,o ,l upon the vessel is to propel it in a direction perpendicular to that in which it is sailing4 ,we must1 therefore1 endeavour to counteract this force as far as possible4 ,this is accomplished by the keel1 and the form of the ship is so designed as to present the greatest possible resistance to being pushed sideways through the water3 the deeper the keel the more completely is the effect of ,o ,l annulled4 ,still ,o ,l would in all cases produce some leeway were it not for the rudder1 which1 by turning the head of the vessel a little towards the wind1 makes her sail in a direction sufficiently to windward to counteract the small effect of ,o ,l in driving her to leeward4 ,thus ,o ,l is disposed of1 and the only force remaining is ,o ,m1 which acts directly to push the vessel in the required direction4 ,here1 then1 we see how the wind1 aided by the resistance of the water1 is able to make the vessel move in a direction perpendicular to that in which the wind blows4 ,we have seen that the sail must be set somewhere between the direction of the wind and that of the ship's motion4 ,it can be proved that when the direction of the sail supposed to be flat and vertical1 is such as to bisect the angle ,w ,o ,b1 the magnitude of the force ,o ,m is greater than when the sail has any other position4 #cb4 ,the same principles show how a vessel is able to sail against the wind3 she cannot1 of course1 sail straight against it1 but she can sail within half a right angle of it1 or perhaps even less4 ,this can be seen from ,fig4 #ad4 ,the small arrows represent the wind1 as before4 ,let ,o ,w be the line parallel to them1 which measures the force of the wind1 and let the sail be placed along the line ,p ,q2 ,o ,w is decomposed into ,o ,x and ,o ,y1 ,o ,x merely glides along the sail1 and ,o ,y is the effective force4 ,this is decomposed into ,o ,l and ,o ,m2 ,o ,l is counteracted1 as already explained1 and ,o ,m is the force that propels the vessel onwards4 ,hence we see that there is a force acting to push the vessel onwards1 even though the movement be partly against the wind4 ,illustration3 ,,fig4 #ad4 ,it will be noticed in this case that the force ,o ,l acting to leewards exceeds ,o ,m pushing onwards4 ,hence it is that vessels with a very deep keel1 and therefore opposing very great resistance to moving leewards1 can sail more closely to the wind than others not so constructed2 a vessel should be formed so that she shall move as freely as possible in the direction of her length1 for which reason she is sharpened at the bow1 and otherwise shaped for gliding through the water easily2 this is in order that ,o ,m may have to overcome as little resistance as possible4 ,if the sail were flat and vertical it should bisect the angle ,a ,,ow for the wind to act in the most efficient manner4 ,since1 then1 a vessel can sail towards the wind1 it follows that1 by taking a zigzag course1 she can proceed from one port to another1 even though the wind be blowing from the place to which she would go towards the place from which she comes4 ,this well known manuvre is called 8tacking40 ,you will understand that in a sailing-vessel the rudder has a more important part to play than in a steamer3 in the latter it is only useful for changing the direction of the vessel's motion1 while in the former it is not only necessary for changing the direction1 but must also be used to keep the vessel to her course by counteracting the effect of leeway4 ,,one ,,force ,,resolved ,,into ,,three ,,forces ,,not ,,in ,,the ,,same ,,plane4 ,illustration3 ,,fig4 #ae4 #cc4 ,up to the present we have only been considering forces which lie in the same plane1 but in nature we meet with forces acting in all directions1 and therefore we must not be satisfied with confining our inquiries to the simpler case4 ,we proceed to show1 in two different ways1 how a force can be decomposed into three forces not in the same plane1 though passing through the same point4 ,the first mode of doing so is as follows4 ,to three points ,a1 ,b1 ,c 7,fig4 #ae7 three spring balances are attached2 ,a1 ,b1 ,c are not in the same straight line1 though they are at the same vertical height3 to the spring balances cords are attached1 which unite in a point ,o1 from which a weight ,w is suspended4 ,this weight is supported by the three cords1 and the strains along these cords are indicated by the spring balances4 ,the greatest strain is on the shortest cord and the least strain on the longest4 ,here the force ,w lbs4 produces three forces which1 taken together1 exceed its own amount4 ,if ,i add an equal weight ,w1 ,i find1 as we might have anticipated1 that the strains indicated by the scales are precisely double what they were before4 ,thus we see that the proportion of the force to each of the components into which it is decomposed does not depend on the actual magnitude of the force1 but on the relative direction of the force and its components4 ,illustration3 ,,fig4 #af4 #cd4 ,another mode of showing the decomposition of one force into three forces not in the same plane is represented in ,fig4 #af4 ,the tripod is formed of three strips of pine1 #d' #je #je1 secured by a piece of wire running through each at the top2 one end of this wire hangs down1 and carries a hook to which is attached a weight of #bh lbs4 ,this weight is supported by the wire1 but the strain on the wire must be borne by the three wooden rods3 hence there is a force acting downwards through the wooden rods4 ,we cannot render this manifest by a contrivance like the spring scales1 because it is a push instead of a pull4 ,however1 by raising one of the legs ,i at once become aware that there is a force acting downwards through it4 ,the weight is1 then1 decomposed into three forces1 which act downwards through the legs2 these three forces are not in a plane1 and the three forces taken together are larger than the weight4 #ce4 ,the tripod is often used for supporting weights2 it is convenient on account of its portability1 and it is very steady4 ,you may judge of its strength by the model represented in the figure1 for though the legs are very slight1 yet they support very securely a considerable weight4 ,the pulleys by means of which gigantic weights are raised are often supported by colossal tripods4 ,they possess stability and steadiness in addition to great strength4 #cf4 ,an important point may be brought out by contrasting the arrangements of ,figs4 #ae and #af4 ,in the one case three cords are used1 and in the other three rods4 ,three rods would have answered for both1 but three cords would not have done for the tripod4 ,in one the cords are strained1 and the tendency of the strain is to break the cords1 but in the other the nature of the force down the rods is entirely different2 it does not tend to pull the rod asunder1 it is trying to crush the rod1 and had the weight been large enough the rods would bend and break4 ,i hold one end of a pencil in each hand and then try to pull the pencil asunder2 the pencil is in the condition of the cords of ,fig4 #ae2 but if instead of pulling ,i push my hands together1 the pencil is like the rods in ,fig4 #af4 #cg4 ,this distinction is of great importance in mechanics4 ,a rod or cord in a state of tension is called a 8tie02 while a rod in a state of compression is called a 8strut40 ,since a rod can resist both tension and compression it can serve either as a tie or as a strut1 but a cord or chain can only act as a tie4 ,a pillar is always a strut1 as the superincumbent load makes it to be in a state of compression4 ,these distinctions will be very frequently used during this course of lectures1 and it is necessary that they be thoroughly understood4 ,,the ,,jib ,,and ,,tie ,,rod4 #ch4 ,as an illustration of the nature of the 8tie0 and 8strut10 and also for the purpose of giving a useful example of the decomposition of forces1 ,i use the apparatus of ,fig4 #ag 7see next page74 ,it represents the principle of the framework in the common lifting crane1 and has numerous applications in practical mechanics4 ,a rod of wood ,b ,c #c' #f long and #a #a section is capable of turning round its support at the bottom ,b by means of a joint or hinge3 this rod is called the 8jib02 it is held at its upper end by a tie ,a ,c #c' long1 which is attached to the support above the joint4 ,a ,b is one foot long4 ,from the point ,c a wire descends1 having a hook at the end on which a weight can be hung4 ,the tie is attached to the spring balance1 the index of which shows the strain4 ,the spring balance is secured by a wire-strainer1 by turning the nut of which the length of the wire can be shortened or lengthened as occasion requires4 ,this is necessary1 because when different weights are suspended from the hook the spring is stretched more or less1 and the screw is then employed to keep the entire length of the tie at #c'4 ,the remainder of the tie consists of copper wire4 #ci4 ,suppose a weight of #bj lbs4 be suspended from the hook ,w1 it endeavours to pull the top of the jib downwards2 but the tie holds it back1 consequently the tie is put into a state of tension1 as indeed its name signifies1 and the magnitude of that tension is shown to be #fj lbs4 by the spring balance4 ,here we find again what we have already so often referred to2 namely1 one force developing another force that is greater than itself1 for the strain along the tie is three times as great as the strain in the vertical wire by which it was produced4 ,illustration3 ,,fig4 #ag4 #dj4 ,what is the condition of the jib8 ,it is evidently being pushed downwards on its joint at ,b2 it is therefore in a state of compression2 it is a strut4 ,this will be evident if we think for a moment how absurd it would be to endeavour to replace the jib by a string or chain3 the whole arrangement would collapse4 ,the weight of #bj lbs4 is therefore decomposed by this contrivance into two other forces1 one of which is resisted by a tie and the other by a strut4 ,illustration3 ,,fig4 #ah4 #da4 ,we have no means of showing the magnitude of the strain along the strut1 but we shall prove that it can be computed by means of the parallelogram of force2 this will also explain how it is that the tie is strained by a force three times that of the weight which is used4 ,through ,c 7,fig4 #ah7 draw ,c ,p parallel to the tie ,a ,b1 and ,p ,q parallel to the strut ,c ,b then ,b ,p is the diagonal of the parallelogram whose sides are each equal to ,b ,c and ,b ,q4 ,if therefore we consider the force of #bj lbs4 to be represented by ,b ,p1 the two forces into which it is decomposed will be shown by ,b ,q and ,b ,c2 but ,a ,b is equal to ,b ,q1 since each of them is equal to ,c ,p2 also ,b ,p is equal to ,a ,c4 ,hence the weight of #bj lbs4 being represented by ,a ,c1 the strain along the tie will be represented by the length ,a ,b1 and that along the strut by the length ,b ,c4 ,remembering that ,a ,b is #c' long1 ,c ,b #c' #f1 and ,a ,c #a'1 it follows that the strain along the tie is #fj lbs41 and along the strut #gj lbs41 when the weight of #bj lbs4 is suspended from the hook4 #db4 ,in every other case the strains along the tie and strut can be determined1 when the suspended weight is known1 by their proportionality to the sides of the triangle formed by the tie1 the jib1 and the upright post1 respectively4 #dc4 ,in this contrivance you will recognize1 no doubt1 the framework of the common lifting crane1 but that very essential portion of the crane which provides for the raising and lowering is not shown here4 ,to this we shall return again in a subsequent lecture 7,art4 #ccb74 ,you will of course understand that the tie rod we have been considering is entirely different from the chain for raising the load4 #dd4 ,it is easy to see of what importance to the engineer the information acquired by means of the decomposition of forces may become4 ,thus in the simple case with which we are at present engaged1 suppose an engineer were required to erect a frame which was to sustain a weight of #aj tons1 let us see how he would be enabled to determine the strength of the tie and jib4 ,it is of importance in designing any structure not to make any part unnecessarily strong1 as doing so involves a waste of valuable material1 but it is of still more vital importance to make every part strong enough to avoid the risk of accident1 not only under ordinary circumstances1 but also under the exceptionally great shocks and strains to which every machine is liable4 #de4 ,according to the numerical proportions we have employed for illustration1 the strain along the tie rod would be #cj tons when the load was #aj tons1 and therefore the tie must at least be strong enough to bear a pull of #cj tons2 but it is customary1 in good engineering practice1 to make the machine of about ten times the strength that would just be sufficient to sustain the ordinary load4 ,hence the crank must be so strong that the tie would not break with a tension less than #cjj tons1 which would be produced when the crane was lifting #ajj tons4 ,so great a margin of safety is necessary on account of the jerks and other occasional great strains that arise in the raising and the lowering of heavy weights4 ,for a crane intended to raise #aj tons1 the engineer must therefore design a tie rod which not less than #cjj tons would tear asunder4 ,it has been proved by actual trial that a rod of wrought iron of average quality1 one square inch in section1 can just withstand a pull of twenty tons4 ,hence fifteen such rods1 or one rod the section of which was equal to fifteen square inches1 would be just able to resist #cjj tons2 and this is therefore the proper area of section for the tie rod of the crane we have been considering4 #df4 ,in the same way we ascertain the actual thrust down the jib2 it amounts to #ce tons1 and the jib should be ten times as strong as a strut which would collapse under a strain of #ce tons4 #dg4 ,it is easy to see from the figure that the tie rod is pulling the upright1 and tending1 in fact1 to make it snap off near ,b4 ,it is therefore necessary that the upright support ,a ,b 7,fig4 #ag7 be secured very firmly4 ,,lecture ,,iii4 ,,parallel ,,forces4 ,introduction4--,pressure of a ,loaded ,beam on its ,supports4--,equilibrium of a ,bar supported on a ,knife-edge4--,the ,composition of ,parallel ,forces4--,parallel ,forces acting in opposite directions4--,the ,couple4--,the ,weighing ,scales4 ,,introduction4 #dh4 ,the parallelogram of forces enables us to find the resultant of two forces which intersect3 but since parallel forces do not intersect1 the construction does not avail to determine the resultant of two parallel forces4 ,we can1 however1 find this resultant very simply by other means4 ,illustration3 ,,fig4 #ai4 #di4 ,fig4 #ai represents a wooden rod #d' long1 sustained by resting on two supports ,a and ,b1 and having the length ,a ,b divided into #ad equal parts4 ,let a weight of #ad lbs4 be hung on the rod at its middle point ,c2 this weight must be borne by the supports1 and it is evident that they will bear the load in equal shares1 for since the weight is at the middle of the rod there is no reason why one end should be differently circumstanced from the other4 ,hence the total pressure on each of the supports will be #g lbs41 together with half the weight of the wooden bar4 #ej4 ,if the weight of #ad lbs4 be placed1 not at the centre of the bar1 but at some other point such as ,d1 it is not then so easy to see in what proportion the weight is distributed between the supports4 ,we can easily understand that the support near the weight must bear more than the remote one1 but how much more8 ,when we are able to answer this question1 we shall see that it will lead us to a knowledge of the composition of parallel forces4 ,,pressure ,,of ,a ,,loaded ,,beam ,,on ,,its ,,supports4 #ea4 ,to study this question we shall employ the apparatus shown in ,fig4 #bj4 ,an iron bar #e' #f long1 weighing #aj lbs41 rests in the hooks of the spring balances ,a1 ,c1 in the manner shown in the figure4 ,these hooks are exactly five feet apart1 so that the bar projects #c beyond each end4 ,the space between the hooks is divided into twenty equal portions1 each of course #c long4 ,the bar is sufficiently strong to bear the weight ,b of #bj lbs4 suspended from it by an ,s hook1 without appreciable deflection4 ,before the weight of #bj lbs4 is suspended1 the spring balances each show a strain of #e lbs4 ,we would expect this1 for it is evident that the whole weight of the bar amounting to #aj lbs4 should be borne equally by the two supports4 #eb4 ,when ,i place the weight in the middle1 #aj divisions from each end1 ,i find the balances each indicate #ae lbs4 ,but #e lbs4 is due to the weight of the bar4 ,hence the #bj lbs4 is divided equally1 as we have already stated that it should be4 ,but let the #bj lbs4 be moved to any other position1 suppose #d divisions from the right1 and #af from the left2 then the right-hand scale reads #ba lbs41 and the left-hand reads #i lbs4 ,to get rid of the weight of the bar itself1 we must subtract #e lbs4 from each4 ,we learn therefore that the #bj lb4 weight pulls the right-hand spring balance with a strain of #af lbs41 and the left with a strain of #d lbs4 ,observe this closely2 you see ,i have made the number of divisions in the bar equal to the number of pounds weight suspended from it1 and here we find that when the weight is #af divisions from the left1 the strain of #af lbs4 is shown on the right4 ,at the same time the weight is #d divisions from the right1 and #d lbs4 is the strain shown to the left4 ,illustration3 ,,fig4 #bj4 #ec4 ,i will state the law of the distribution of the load a little more generally1 and we shall find that the bar will prove the law to be true in all cases4 ,divide the bar into as many equal parts as there are pounds in the load1 then the pressure in pounds on one end is the number of divisions that the load is distant from the other4 #ed4 ,for example1 suppose ,i place the load #b divisions from one end3 ,i read by the scale at that end #bc lbs42 subtracting #e lbs4 for the weight of the bar1 the pressure due to the load is shown to be #ah lbs41 but the weight is then exactly #ah divisions distant from the other end4 ,we can easily verify this rule whatever be the position which the load occupies4 #ee4 ,if the load be placed between two marks1 instead of being1 as we have hitherto supposed1 exactly at one1 the partition of the load is also determined by the law4 ,were it1 for example1 #ce divisions from one end1 the strain on the other would be #ce lbs42 and in like manner for other cases4 #ef4 ,we have thus proved by actual experiment this useful and instructive law of nature2 the same result could have been inferred by reasoning from the parallelogram of force1 but the purely experimental proof is more in accordance with our scheme4 ,the doctrine of the composition of parallel forces is one of the most fundamental parts of mechanics1 and we shall have many occasions to employ it in this as well as in subsequent lectures4 #eg4 ,returning now to ,fig4 #ai1 with which we commenced1 the law we have discovered will enable us to find how the weight is distributed4 ,we divide the length of the bar between the supports into #ad equal parts because the weight is #ad lbs42 if1 then1 the weight be at ,d1 #aj divisions from one end ,a1 and #d from the other ,b1 the pressure at the corresponding ends will be #d and #aj4 ,if the weight were #be divisions from one end1 and therefore #aae from the other1 the shares in which this load would be supported at the ends are #aae lbs4 and #be lbs4 ,the actual pressure sustained by each end is1 however1 about #f ounces greater if the weight of the wooden bar itself be taken into account4 #eh4 ,let us suspend a second weight from another point of the bar4 ,we must then calculate the pressures at the ends which each weight separately would produce1 and those at the same end are to be added together1 and to half the weight of the bar1 to find the total pressure4 ,thus1 if one weight of #bj lbs4 were in the middle1 and another of #ad lbs4 at a distance of #aa divisions from one end1 the middle weight would produce #aj lbs4 at each end and the #ad lbs4 would produce #c lbs4 and #aa lbs41 and remembering the weight of the bar1 the total pressures produced would be #ac lbs4 #f oz4 and #ba lbs4 #f oz4 ,the same principles will evidently apply to the case of several weights3 and the application of the rule becomes especially easy when all the weights are equal1 for then the same divisions will serve for calculating the effect of each weight4 #ei4 ,the principles involved in these calculations are of so much importance that we shall further examine them by a different method1 which has many useful applications4 ,,equilibrium ,,of ,a ,,bar ,,supported ,,on ,a ,,knife-,,edge4 #fj4 ,the weight of the bar has hitherto somewhat complicated our calculations2 the results would appear more simply if we could avoid this weight2 but since we want a strong bar1 its weight is not so small that we could afford to overlook it altogether4 ,by means of the arrangement of ,fig4 #ba1 we can counterpoise the weight of the bar4 ,to the centre of ,a ,b a cord is attached1 which1 passing over a fixed pulley ,d1 carries a hook at the other end4 ,the bar1 being a pine rod1 #d feet long and #a inch square1 weighs about #ab ounces2 consequently1 if a weight of twelve ounces be suspended from the hook1 the bar will be counterpoised1 and will remain at whatever height it is placed4 ,illustration3 ,,fig4 #ba4 #fa4 ,a ,b is divided by lines drawn along it at distances of #a apart2 there are thus #dh of these divisions4 ,the weights employed are furnished with rings large enough to enable them to be slipped on the bar and thus placed in any desired position4 #fb4 ,underneath the bar lies an important portion of the arrangement2 namely1 the knife-edge ,c4 ,this is a blunt edge of steel firmly fastened to the support which carries it4 ,this support can be moved along underneath the bar so that the knife-edge can be placed under any of the divisions required4 ,the bar being counterpoised1 though still unloaded with weights1 may be brought down till it just touches the knife-edge2 it will then remain horizontal1 and will retain this position whether the knife-edge be at either end of the bar or in any intermediate position4 ,i shall hang weights at the extremities of the rod1 and we shall find that there is for each pair of weights just one position at which1 if the knife-edge be placed1 it will sustain the rod horizontally4 ,we shall then examine the relations between these distances and the weights that have been attached1 and we shall trace the connection between the results of this method and those of the arrangement that we last used4 #fc4 ,supposing that #f lbs4 be hung at each end of the rod1 we might easily foresee that the knife-edge should be placed in the middle1 and we find our anticipations verified4 ,when the edge is exactly at the middle1 the rod remains horizontal2 but if it be moved1 even through a very small distance1 to either side1 the rod instantly descends on the other4 ,the knife-edge is #bd inches distant from each end2 and if ,i multiply this number by the number of pounds in the weight1 in this case #f1 ,i find #add for the product1 and this product is the same for both ends of the bar4 ,the importance of this remark will be seen directly4 #fd4 ,if ,i remove one of the #f lb4 weights and replace it by #b lbs41 leaving the other weight and the knife-edge unaltered1 the bar instantly descends on the side of the heavy weight2 but1 by slipping the knife-edge along the bar1 ,i find that when ,i have moved it to within a distance of #ab inches from the #f lbs41 and therefore #cf inches from the #b lbs41 the bar will remain horizontal4 ,the edge must be put carefully at the right place2 a quarter of an inch to one side or the other would upset the bar4 ,the whole load borne by the knife-edge is of course #h lbs41 being the sum of the weights4 ,if we multiply #b1 the number of pounds at one end1 by #cf1 the distance of that end from the knife-edge1 we obtain the product #gb2 and we find precisely the same product by multiplying #f1 the number of pounds in the other weight1 by #ab1 its distance from the knife-edge4 ,to express this result concisely we shall introduce the word moment1 a term of frequent use in mechanics4 ,the #b lb4 weight produces a force tending to pull its end of the bar downwards by making the bar turn round the knife-edge4 ,the magnitude of this force1 multiplied into its distance from the knife-edge1 is called the moment of the force4 ,we can express the result at which we have arrived by saying that1 when the knife-edge has been so placed that the bar remains horizontal1 the moments of the forces about the knife-edge are equal4 #fe4 ,we may further illustrate this law by suspending weights of #g lbs4 and #e lbs4 respectively from the ends of the bar2 it is found that the knife-edge must then be placed #bj inches from the larger weight1 and1 therefore1 #bh inches from the smaller1 but #e #bh #adj1 and #g #bj #adj1 thus again verifying the law of equality of the moments4 ,from the equality of the moments we can also deduce the law for the distribution of the load given in ,art4 #ec4 ,thus1 taking the figures in the last experiment1 we have loads of #g lbs4 and #e lbs4 respectively4 ,these produce a pressure of #g #e #ab lbs4 on the knife-edge4 ,this edge presses on the bar with an equal and opposite reaction4 ,to ascertain the distribution of this pressure on the ends of the beam1 we divide the whole beam into #ab equal parts of #d inches each1 and the #g lb4 weight is #e of these parts1 i4e41 #bj inches distant from the support4 ,hence the edge should be #bj inches from the greater weight1 which is the condition also implied by the equality of the moments4 ,,the ,,composition ,,of ,,parallel ,,forces4 #ff4 ,having now examined the subject experimentally1 we proceed to investigate what may be learned from the results we have proved4 ,illustration3 ,,fig4 #bb4 ,the weight of the bar being allowed for in the way we have explained1 by subtracting one-half of it from each of the strains indicated by the spring balance 7,,fig4 #bj71 we may omit it from consideration4 ,as the balances are pulled downwards by the bar when it is loaded1 so they will react to pull the bar upwards4 ,this will be evident if we think of a weight--say #ad lbs4--suspended from one of these balances3 it hangs at rest2 therefore its weight1 which is constantly urging it downwards1 must be counteracted by an equal force pulling it upwards4 ,the balance of course shows #ad lbs42 thus the spring exerts in an upward pull a force which is precisely equal to that by which it is itself pulled downwards4 #fg4 ,hence the springs are exerting forces at the ends of the bar in pulling them upwards1 and the scales indicate their magnitudes4 ,the bar is thus subject to three forces1 viz43 the suspended weight of #bj lbs41 which acts vertically downwards1 and the two other forces which act vertically upwards1 and the united action of the three make equilibrium4 #fh4 ,let lines be drawn1 representing the forces in the manner already explained4 ,we have then three parallel forces ,,ap1 ,,bq1 ,,cr acting on a rod in equilibrium 7,fig4 #bb74 ,the two forces ,,ap and ,,bq may be considered as balanced by the force ,,cr in the position shown in the figure1 but the force ,,cr would be balanced by the equal and opposite force ,,cs1 represented by the dotted line4 ,hence this last force is equivalent to ,,ap and ,,bq4 ,in other words1 it must be their resultant4 ,here then we learn that a pair of parallel forces1 acting in the same direction1 can be compounded into a single resultant4 #fi4 ,we also see that the magnitude of the resultant is equal to the sum of the magnitudes of the forces1 and further we find the position of the resultant by the following rule4 ,add the two forces together2 divide the distance between them into as many equal parts as are contained in the sum1 measure off from the greater of these two forces as many parts as there are pounds in the smaller force1 and that is the point required4 ,this rule is very easily inferred from that which we were taught by the experiments in ,art4 #ea4 ,,parallel ,,forces ,,acting ,,in ,,opposite ,,directions4 #gj4 ,since the forces ,,ap1 ,,bq1 ,,cr 7,fig4 #bb7 are in equilibrium1 it follows that we may look on ,,bq as balancing in the position which it occupies the two forces of ,,ap and ,,cr in their positions4 ,this may remind us of the numerous instances we have already met with1 where one force balanced two greater forces3 in the present case ,,ap and ,,cr are acting in opposite directions1 and the force ,,bq which balances them is equal to their difference4 ,a force ,,bt equal and opposite to ,,bq must then be the resultant of ,,cr and ,,ap1 since it is able to produce the same effect4 ,notice that in this case the resultant of the two forces is not between them1 but that it lies on the side of the larger4 ,when the forces act in the same direction1 the resultant is always between them4 #ga4 ,the actual position which the resultant of two opposite parallel forces occupies is to be found by the following rule4 ,divide the distance between the forces into as many equal parts as there are pounds in their difference1 then measure outwards from the point of application of the larger force as many of these parts as there are pounds in the smaller force2 the point thus found determines the position of the resultant4 ,thus1 if the forces be #ad and #bj1 the difference between them is #f1 and therefore the distance between their directions is divided into six parts2 from the point of application of the force of #bj1 #ad parts are measured outwards1 and thus the position of the resultant is determined4 ,hence we have the means of compounding two parallel forces in general4 ,,the ,,couple4 #gb4 ,in one case1 however1 two parallel forces have no resultant2 this occurs when the two forces are equal1 and in opposite directions4 ,a pair of forces of this kind is called a couple2 there is no single force which could balance a couple1--it can only be counterbalanced by another couple acting in an opposite manner4 ,this remarkable case1 may be studied by the arrangement of ,fig4 #bc4 ,illustration3 ,,fig4 #bc4 ,a wooden rod1 ,a ,b #dh #je #je1 has strings attached to it at points ,a and ,d1 one foot distant4 ,the string at ,d passes over a fixed pulley ,e1 and at the end ,p a hook is attached for the purpose of receiving weights1 while a similar hook descends from ,a2 the weight of the rod itself1 which only amounts to three ounces1 may be neglected1 as it is very small compared with the weights which will be used4 #gc4 ,supposing #b lbs4 to be placed at ,p1 and #a lb4 at ,q1 we have two parallel forces acting in opposite directions2 and since their difference is #a lb41 it follows from our rule that the point ,f1 where ,d ,f is equal to ,a ,d1 is the point where the resultant is applied4 ,you see this is easily verified1 for by placing my finger over the rod at ,f it remains horizontal and in equilibrium2 whereas1 when ,i move my finger to one side or the other1 equilibrium is impossible4 ,if ,i move it nearer to ,b1 the end ,a ascends4 ,if ,i move it towards ,a1 the end ,b ascends4 #gd4 ,to study the case when the two forces are equal1 a load of #b lbs4 may be placed on each of the hooks ,p and ,q4 ,it will then be found that the finger cannot be placed in any position along the rod so as to keep it in equilibrium2 that is to say1 no single force can counteract the two forces which form the couple4 ,let ,o be the point midway between ,a and ,d4 ,the forces evidently tend to raise ,,ob and turn the part ,o ,a downwards2 but if ,i try to restrain ,o ,b by holding my finger above1 as at the point ,x1 instantly the rod begins to turn round ,x and the part from ,a to ,x descends4 ,i find similarly that any attempt to prevent the motion by holding my finger underneath is equally unsuccessful4 ,but if at the same time ,i press the rod downwards at one point1 and upwards at another with suitable force1 ,i can succeed in producing equilibrium2 in this case the two pressures form a couple2 and it is this couple which neutralizes the couple produced by the weights4 ,we learn1 then1 the important result that a couple can be balanced by a couple1 and by a couple only4 #ge4 ,the moment of a couple is the product of one of the two equal forces into their perpendicular distance4 ,two couples tending to turn the body to which they are applied in the same direction will be equivalent if their moments are equal4 ,two couples which tend to turn the body in opposite directions will be in equilibrium if their moments are equal4 ,we can also compound two couples in the same or in opposite directions into a single couple of which the moment is respectively either the sum or the difference of the original moments4 ,,the ,,weighing ,,scales4 #gf4 ,another apparatus by which the nature of parallel forces may be investigated is shown in ,fig4 #bd2 it consists of a slight frame of wood ,a ,b ,c1 #d' long4 ,at ,e1 a pair of steel knife-edges is clamped to the frame4 ,the knife-edges rest on two pieces of steel1 one of which is shown at ,o ,f4 ,when the knife-edges are suitably placed the frame is balanced1 so that a small piece of paper laid at ,a will cause that side to descend4 ,illustration3 ,,fig4 #bd4 #gg4 ,we suspend two small hooks from the points ,a and ,b3 these are made of fine wire1 so that their weight may be left out of consideration4 ,with this apparatus we can in the first place verify the principle of equality of moments3 for example1 if ,i place the hook ,a at a distance of #i from the centre ,o and load it with #a lb41 ,i find that when ,b is laden with #je lb4 it must be at a distance of #ah from ,o in order to counterbalance ,a2 the moment in the one case is #i #a1 in the other #ah #je1 and these are obviously equal4 #gh4 ,let a weight of #a lb4 be placed on each of the hooks1 the frame will only be in equilibrium when the hooks are at precisely the same distance from the centre4 ,a familiar application of this principle is found in the ordinary weighing scales2 the frame1 which in this case is called a beam1 is sustained by two knife-edges1 smaller1 however1 than those represented in the figure4 ,the pans ,p1 ,p are suspended from the extremities of the beam1 and should be at equal distances from its centre4 ,these scale-pans must be of equal weight1 and then1 when equal weights are placed in them1 the beam will remain horizontal4 ,if the weight in one slightly exceed that in the other1 the pan containing the heavier weight will of course descend4 #gi4 ,that a pair of scales should weigh accurately1 it is necessary that the weights be correct2 but even with correct weights1 a balance of defective construction will give an inaccurate result4 ,the error frequently arises from some inequality in the lengths of the arms of the beam4 ,when this is the case1 the two weights which really balance are not equal4 ,supposing1 for instance1 that with an imperfect balance ,i endeavour to weigh a pound of shot4 ,if ,i put the weight on the short side1 then the quantity of shot balanced is less than #a lb42 while if the #a lb4 weight be placed at the long side1 it will require more than #a lb4 of shot to produce equilibrium4 ,the mode of testing a pair of scales is then evident4 ,let weights be placed in the pans which balance each other2 if the weights be interchanged and the balance still remains horizontal1 it is correct4 #hj4 ,suppose1 for example1 that the two arms be #aj inches and #aa inches long1 then1 if #a lb4 weight be placed in the pan of the #aj-;inch end1 its moment is #aj2 and if _/ of #a lb4 be placed in the pan belonging to the #aa-;inch end1 its moment is also #aj3 hence #a lb4 at the short end balances _/ of #a lb4 at the long end2 and therefore1 if the shopkeeper placed his weight in the short arm1 his customers would lose _/ part of each pound for which they paid2 on the other hand1 if the shopkeeper placed his #a lb4 weight on the long arm1 then not less than _/ lb4 would be required in the pan belonging to the short arm4 ,hence in this case the customer would get _/ lb4 too much4 ,it follows1 that if a shopman placed his weights and his goods alternately in the one scale and in the other he would be a loser on the whole2 for1 though every second customer gets _/ lb4 less than he ought1 yet the others get _/ lb4 more than they have paid for4 ,,lecture ,,iv4 ,,the ,,force ,,of ,,gravity4 ,introduction4--,specific ,gravity4--,the ,plummet and ,spirit-,level4--,the ,centre of ,gravity4--,stable and ,unstable ,equilibrium4--,property of the ,centre of ,gravity in a ,revolving ,wheel4 ,,introduction4 #ha4 ,in the last three lectures we considered forces in the abstract2 we saw how they are to be represented by straight lines1 how compounded together and how decomposed into others2 we have explained what is meant by forces being in equilibrium1 and we have shown instances where the forces lie in the same plane or in different planes1 and where they intersect or are parallel to each other4 ,these subjects are the elements of mechanics2 they form the framework which in this and subsequent lectures we shall try to present in a more attractive garb4 ,we shall commence by studying the most remarkable force in nature1 a force constantly in action1 and one to which all bodies are subject1 a force which distance cannot annihilate1 and one the properties of which have led to the most sublime discoveries of human intellect4 ,this is the force of gravity4 #hb4 ,if ,i drop a stone from my hand1 it falls to the ground4 ,that which produces motion is a force3 hence the stone must have been acted upon by a force which drew it to the ground4 ,on every part of the earth's surface experience shows that a body tends to fall4 ,this fact proves that there is an attractive force in the earth tending to draw all bodies towards it4 ,illustration3 ,,fig4 #be4 #hc4 ,let ,a ,b ,c ,d 7,fig4 #be7 be points from which stones are let fall1 and let the circle represent the section of the earth2 let ,p ,q ,r ,s be the points at the surface of the earth upon which the stones will drop when allowed to do so4 ,the four stones will move in the directions of the arrows3 from ,a to ,p the stone moves in an opposite direction to the motion from ,c to ,r2 from ,b to ,q it moves from right to left1 while from ,d to ,s it moves from left to right4 ,the movements are in different directions2 but if ,i produce these directions1 as indicated by the dotted lines1 they each pass through the centre ,o4 #hd4 ,hence each stone in falling moves towards the centre of the earth1 and this is consequently the direction of the force4 ,we therefore assert that the earth has an attraction for the stone1 in consequence of which it tries to get as near the earth's centre as possible1 and this attraction is called the force of gravitation4 #he4 ,we are so excessively familiar with the phenomenon of seeing bodies fall that it does not excite our astonishment or arouse our curiosity4 ,a clap of thunder1 which every one notices1 because much less frequent1 is not really more remarkable4 ,we often look with attention at the attraction of a piece of iron by a magnet1 and justly so1 for the phenomenon is very interesting1 and yet the falling of a stone is produced by a far grander and more important force than the force of magnetism4 #hf4 ,it is gravity which causes the weight of bodies4 ,i hold a piece of lead in my hand3 gravity tends to pull it downwards1 thus producing a pressure on my hand which ,i call weight4 ,gravity acts with slightly varying intensity at various parts of the earth's surface4 ,this is due to two distinct causes1 one of which may be mentioned here1 while the other will be subsequently referred to4 ,the earth is not perfectly spherical2 it is flattened a little at the poles2 consequently a body at the pole is nearer the general mass of the earth than a body at the equator2 therefore the body at the pole is more attracted1 and seems heavier4 ,a mass which weighs #bjj lbs4 at the equator would weigh one pound more at the pole3 about one-third of this increase is due to the cause here pointed out4 7,see ,lecture ,,xvii47 #hg4 ,gravity is a force which attracts every particle of matter2 it acts not merely on those parts of a body which lie on the surface1 but it equally affects those in the interior4 ,this is proved by observing that a body has the same weight1 however its shape be altered3 for example1 suppose ,i take a ball of putty which weighs #a lb41 ,i shall find that its weight remains unchanged when the ball is flattened into a thin plate1 though in the latter case the surface1 and therefore the number of superficial particles1 is larger than it was in the former4 ,,specific ,,gravity4 #hh4 ,gravity produces different effects upon different substances4 ,this is commonly expressed by saying that some substances are heavier than others2 for example1 ,i have here a piece of wood and a piece of lead of equal bulk4 ,the lead is drawn to the earth with a greater force than the wood4 ,substances are usually termed heavy when they sink in water1 and light when they float upon it4 ,but a body sinks in water if it weighs more than an equal bulk of water1 and floats if it weigh less4 ,hence it is natural to take water as a standard with which the weights of other substances may be compared4 #hi4 ,i take a certain volume1 say a cubic inch of cast iron such as this ,i hold in my hand1 and which has been accurately shaped for the purpose4 ,this cube is heavier than one cubic inch of water1 but ,i shall find that a certain quantity of water is equal to it in weight2 that is to say1 a certain number of cubic inches of water1 and it may be fractional parts of a cubic inch1 are precisely of the same weight4 ,this number is called the specific gravity of cast iron4 #ij4 ,it would be impossible to counterpoise water with the iron without holding the water in a vessel1 and the weight of the vessel must then be allowed for4 ,i adopt the following plan4 ,i have here a number of inch cubes of wood 7,fig4 #bf71 which would each be lighter than a cubic inch of water1 but ,i have weighted the wooden cubes by placing grains of shot into holes bored into the wood4 ,the weight of each cube has thus been accurately adjusted to be equal to that of a cubic inch of water4 ,this may be tested by actual weighing4 ,i weigh one of the cubes and find it to be #beb grains1 which is well known to be the weight of a cubic inch of water4 ,illustration3 ,,fig4 #bf4 #ia4 ,but the cubes may be shown to be identical in weight with the same bulk of water by a simpler method4 ,one of them placed in water should have no tendency to sink1 since it is not heavier than water1 nor on the other hand1 since it is not lighter1 should it have any tendency to float4 ,it should then remain in the water in whatever position it may be placed4 ,it is difficult to prepare one of these cubes so accurately that this result should be attained1 and it is impossible to ensure its continuance for any time owing to changes of temperature and the absorption of water by the wood4 ,we can1 however1 by a slight modification1 prove that one of these cubes is at all events nearly equal in weight to the same bulk of water4 ,in ,fig4 #bf is shown a tall glass jar filled with a fluid in appearance like plain water1 but it is really composed in the following manner4 ,i first poured into the jar a very weak solution of salt and water1 which partially filled it2 ,i then poured gently upon this a little pure water1 and finally filled up the jar with water containing a little spirits of wine3 the salt and water is a little heavier than pure water1 while the spirit and water is a little lighter4 ,i take one of the cubes and drop it gently into the glass2 it falls through the spirit and water1 and after making a few oscillations settles itself at rest in the stratum shown in the figure4 ,this shows that our prepared cube is a little heavier than spirit and water1 and a little lighter than salt and water1 and hence we infer that it must at all events be very near the weight of pure water which lies between the two4 ,we have also a number of half cubes1 quarter cubes1 and half-quarter cubes1 which have been similarly prepared to be of equal weight with an equal bulk of water4 #ib4 ,we shall now be able to measure the specific gravity of a substance4 ,in one pan of the scales ,i place the inch cube of cast iron1 and ,i find that #g of the wooden cubes1 which we may call cubes of water1 will balance it4 ,we therefore say that the specific gravity of iron is #g4 ,the exact number found by more accurate methods is #gb4 ,it is often convenient to remember that #bc cubic inches of cast iron weigh #f lbs41 and that therefore one cubic inch weighs very nearly lb4 #ic4 ,i have also cubes of brass1 lead1 and ivory2 by counterpoising them with the cubes of water1 we can easily find their specific gravities2 they are shown together with that of cast iron in the following table3-- ,substance4 ,specific ,gravity4 ,cast ,iron #gb ,brass #ha ,lead #aac ,ivory #ah #id4 ,the mode here adopted of finding specific gravities is entirely different from the far more accurate methods which are commonly used1 but the explanation of the latter involve more difficult principles than those we have been considering4 ,our method rather offers an explanation of the nature of specific gravity than a good means of determining it1 though1 as we have seen1 it gives a result sufficiently near the truth for many purposes4 ,,the ,,plummet ,,and ,,spirit-,,level4 #ie4 ,the tendency of the earth to draw all bodies towards it is well illustrated by the useful 8line and plummet40 ,this consists merely of a string to one end of which a leaden weight is attached4 ,the string when at rest hangs vertically2 if the weight be drawn to one side1 it will1 when released1 swing backwards and forwards1 until it finally settles again in the vertical2 the reason is that the weight always tries to get as near the earth as it can1 and this is accomplished when the string hangs vertically downwards4 #if4 ,the surface of water in equilibrium is a horizontal plane2 that is also a consequence of gravity4 ,all the particles of water try to get as near the earth as possible1 and therefore if any portion of the water were higher than the rest1 it would immediately spread1 as by doing so it could get lower4 #ig4 ,hence the surface of a fluid at rest enables us to find a perfectly horizontal plane1 while the plummet gives us a perfectly vertical line3 both these consequences of gravity are of the utmost practical importance4 #ih4 ,the spirit-level is another common and very useful instrument which depends on gravity4 ,it consists of a glass tube slightly curved1 with its convex surface upwards1 and attached to a stand with a flat base4 ,this tube is nearly filled with spirit1 but a bubble of air is allowed to remain4 ,the tube is permanently adjusted so that1 when the plate is laid on a perfectly horizontal surface1 the bubble will stand in the middle3 accordingly the position of the bubble gives a means of ascertaining whether a surface is level4 ,,the ,,centre ,,of ,,gravity4 ,illustration3 ,,fig4 #bg4 #ii4 ,we proceed to an experiment which will give an insight into a curious property of gravity4 ,i have here a plate of sheet iron2 it has the irregular shape shown in ,fig4 #bg4 ,five small holes ,a ,b ,c ,d ,e are punched at different positions on the margin4 ,attached to the framework is a small pin from which ,i can suspend the iron plate by one of its holes ,a3 the plate is not supported in any other way2 it hangs freely from the pin1 around which it can be easily turned4 ,i find that there is one position1 and one only1 in which the plate will rest2 if ,i withdraw it from that position it returns there after a few oscillations4 ,in order to mark this position1 ,i suspend a line and plummet from the pin1 having rubbed the line with chalk4 ,i allow the line to come to rest in front of the plate4 ,i then flip the string against the plate1 and thus produce a chalked mark3 this of course traces out a vertical line ,a ,p on the plate4 ,i now remove the plummet and suspend the plate from another of its holes ,b1 and repeat the process1 thus drawing a second chalked line ,b ,p across the plate1 and so on with the other holes3 ,i thus obtain five lines across the plate1 represented by dotted lines in the figure4 ,it is a very remarkable circumstance that these five lines all intersect in the same point ,p2 and if additional holes were bored in the plate1 whether in the margin or not1 and the chalk line drawn from each of them in the manner described1 they would one and all pass through the same point4 ,this remarkable point is called the centre of gravity of the plate1 and the result at which we have arrived may be expressed by saying that the vertical line from the point of suspension always passes through the centre of gravity4 #ajj4 ,at the centre of gravity ,p a hole has been bored1 and when ,i place the supporting pin through this hole you see that the plate will rest indifferently in all positions3 this is a curious property of the centre of gravity4 ,the centre of gravity may in this respect be contrasted with another hole ,q1 which is only an inch distant3 when ,i support the plate by this hole1 it has only one position of rest1 viz4 when the centre of gravity ,p is vertically beneath ,q4 ,thus the centre of gravity differs remarkably from any other point in the plate4 #aja4 ,we may conceive the force of gravity on the plate to act as a force applied at ,p4 ,it will then be easily seen why this point remains vertically underneath the point of suspension when the body is at rest4 ,if ,i attached a string to the plate and pulled it1 the plate would evidently place itself so that the direction of the string would pass through the point of suspension2 in like manner gravity so places the plate that the direction of its force passes through the point of suspension4 #ajb4 ,whatever be the form of the plate it always contains one point possessing these remarkable properties1 and we may state in general that in every body1 no matter what be its shape1 there is a point called the centre of gravity1 such that if the body be suspended from this point it will remain in equilibrium indifferently in any position1 and that if the body be suspended from any other point1 then it will be in equilibrium when the centre of gravity is directly underneath the point of suspension4 ,in general1 it will be impossible to support a body exactly at its centre of gravity1 as this point is within the mass of the body1 and it may also sometimes happen that the centre of gravity does not lie in the substance of the body at all1 as for example in a ring1 in which case the centre of gravity is at the centre of the ring4 ,we need not1 however1 dwell on these exceptional cases1 as sufficient illustrations of the truth of the laws mentioned will present themselves subsequently4 ,,stable ,,and ,,unstable ,,equilibrium4 ,illustration3 ,,fig4 #bh4 #ajc4 ,an iron rod ,a ,b1 capable of revolving round an axis passing through its centre ,p1 is shown in ,fig4 #bh4 ,the centre of gravity lies at the centre ,b1 and consequently1 as is easily seen1 the rod will remain at rest in whatever position it be placed4 ,but let a weight ,r be attached to the rod by means of a binding screw4 ,the centre of gravity of the whole is no longer at the centre of the rod2 it has moved to a point ,s nearer the weight2 we may easily ascertain its position by removing the rod from its axle and then ascertaining the point about which it will balance4 ,this may be done by placing the bar on a knife-edge1 and moving it to and fro until the right position be secured2 mark this position on the rod1 and return it to its axle1 the weight being still attached4 ,we do not now find that the rod will balance in every position4 ,you see it will balance if the point ,s be directly underneath the axis1 but not if it lie to one side or the other4 ,but if ,s be directly over the axis1 as in the figure1 the rod is in a curious condition4 ,it will1 when carefully placed1 remain at rest2 but if it receive the slightest displacement1 it will tumble over4 ,the rod is in equilibrium in this position1 but it is what is called unstable equilibrium4 ,if the centre of gravity be vertically below the point of suspension1 the rod will return again if moved away3 this position is therefore called one of stable equilibrium4 ,it is very important to notice the distinction between these two kinds of equilibrium4 #ajd4 ,another way of stating the case is as follows4 ,a body is in stable equilibrium when its centre of gravity is at the lowest point3 unstable when it is at the highest4 ,this may be very simply illustrated by an ellipse1 which ,i hold in my hand4 ,the centre of gravity of this figure is at its centre4 ,the ellipse1 when resting on its side1 is in a position of stable equilibrium and its centre of gravity is then clearly at its lowest point4 ,but ,i can also balance the ellipse on its narrow end1 though if ,i do so the smallest touch suffices to overturn it4 ,the ellipse is then in unstable equilibrium2 in this case1 obviously1 the centre of gravity is at the highest point4 ,illustration3 ,,fig4 #bi4 #aje4 ,i have here a sphere1 the centre of gravity of which is at its centre2 in whatever way the sphere is placed on a plane1 its centre is at the same height1 and therefore cannot be said to have any highest or lowest point2 in such a case as this the equilibrium is neutral4 ,if the body be displaced1 it will not return to its old position1 as it would have done had that been a position of stable equilibrium1 nor will it deviate further therefrom as if the equilibrium had been unstable3 it will simply remain in the new position to which it is brought4 #ajf4 ,i try to balance an iron ring upon the end of a stick ,h1 ,fig4 #bi1 but ,i cannot easily succeed in doing so4 ,this is because its centre of gravity ,s is above the point of support2 but if ,i place the stick at ,f1 the ring is in stable equilibrium1 for now the centre of gravity is below the point of support4 ,,property ,,of ,,the ,,centre ,,of ,,gravity ,,in ,a ,,revolving ,,wheel4 #ajg4 ,there are other curious consequences which follow from the properties of the centre of gravity1 and we shall conclude by illustrating one of the most remarkable1 which is at the same time of the utmost importance in machinery4 ,illustration3 ,,fig4 #cj4 #ajh4 ,it is generally necessary that a machine should work as steadily as possible1 and that undue vibration and shaking of the framework should be avoided3 this is particularly the case when any parts of the machine rotate with great velocity1 as1 if these be heavy1 inconvenient vibration will be produced when the proper adjustments are not made4 ,the connection between this and the centre of gravity will be understood by reference to the apparatus represented in the accompanying figure 7,fig4 #cj74 ,we have here an arrangement consisting of a large cog wheel ,c working into a small one ,b1 whereby1 when the handle ,h is turned1 a velocity of rotation can be given to the iron disk ,d1 which weighs #ad lbs1 and is #ah in diameter4 ,this disk being uniform1 and being attached to the axis at its centre1 it follows that its centre of gravity is also the centre of rotation4 ,the wheels are attached to a stand1 which1 though massive1 is still unconnected with the floor4 ,by turning the handle ,i can rotate the disk very rapidly1 even as much as twelve times in a second4 ,still the stand remains quite steady1 and even the shutter bell attached to it at ,e is silent4 #aji4 ,through one of the holes in the disk ,d ,i fasten a small iron bolt and a few washers1 altogether weighing about #a lb42 that is1 only one-fourteenth of the weight of the disk4 ,when ,i turn the handle slowly1 the machine works as smoothly as before2 but as ,i increase the speed up to one revolution every two seconds1 the bell begins to ring violently1 and when ,i increase it still more1 the stand quite shakes about on the floor4 ,what is the reason of this8 ,by adding the bolt1 ,i slightly altered the position of the centre of gravity of the disk1 but ,i made no change of the axis about which the disk rotated1 and consequently the disk was not on this occasion turning round its centre of gravity3 this it was which caused the vibration4 ,it is absolutely necessary that the centre of gravity of any heavy piece1 rotating rapidly about an axis1 should lie in the axis of rotation4 ,the amount of vibration produced by a high velocity may be very considerable1 even when a very small mass is the originating cause4 #aaj4 ,in order that the machine may work smoothly again1 it is not necessary to remove the bolt from the hole4 ,if by any means ,i bring back the centre of gravity to the axis1 the same end will be attained4 ,this is very simply effected by placing a second bolt of the same size at the opposite side of the disk1 the two being at equal distances from the axis2 on turning the handle1 the machine is seen to work as smoothly as it did in the first instance4 #aaa4 ,the most common rotating pieces in machines are wheels of various kinds1 and in these the centre of gravity is evidently identical with the centre of rotation2 but if from any cause a wheel1 which is to turn rapidly1 has an extra weight attached to one part1 this weight must be counterpoised by one or more on other portions of the wheel1 in order to keep the centre of gravity of the whole in its proper place4 ,thus it is that the driving wheels of a locomotive are always weighted so as to counteract the effect of the crank and restore the centre of gravity to the axis of rotation4 ,the cause of the vibration will be understood after the lecture on centrifugal force 7,lect4 ,,xvii474 ,,lecture ,v4 ,,the ,,force ,,of ,,friction4 ,the ,nature of ,friction4--,the ,mode of ,experimenting4--,friction is proportional to the pressure4--,a more accurate form of the ,law4--,the ,coefficient varies with the weights used4--,the ,angle of ,friction4--,another ,law of ,friction4--,concluding ,remarks4 ,,the ,,nature ,,of ,,friction4 #aab4 ,a discussion of the force of friction is a necessary preliminary to the study of the mechanical powers which we shall presently commence4 ,friction renders the inquiry into the mechanical powers more difficult than it would be if this force were absent2 but its effects are too important to be overlooked4 ,illustration3 ,,fig4 #ca4 #aac4 ,the nature of friction may be understood by ,fig4 #ca1 which represents a section of the top of a table of wood or any other substance levelled so that ,c ,d is horizontal2 on the table rests a block ,a of wood or any other substance4 ,to ,a a cord is attached1 which1 after passing over a pulley ,p1 is stretched by a suspended weight ,b4 ,if the magnitude of ,b exceeds a certain limit1 then ,a is pulled along the table and ,b descends2 but if ,b be smaller than this limit1 both ,a and ,b remain at rest4 ,when ,b is not heavy enough to produce motion it is supported by the tension of the cord1 which is itself neutralized by the friction produced by a certain coherence between ,a and the table4 ,friction is by this experiment proved to be a force1 because it prevents the motion of ,b4 ,indeed friction is generally manifested as a force by destroying motion1 though sometimes indirectly producing it4 #aad4 ,the true source of the force lies in the inevitable roughness of all known surfaces1 no matter how they may have been wrought4 ,the minute asperities on one surface are detained in corresponding hollows in the other1 and consequently force must be exerted to make one surface slide upon the other4 ,by care in polishing the surfaces the amount of friction may be diminished2 but it can only be decreased to a certain limit1 beyond which no amount of polishing seems to produce much difference4 #aae4 ,the law of friction under different conditions must be inquired into1 in order that we may make allowance when its effect is of importance4 ,the discussion of the experiments is sometimes a little difficult1 and the truths arrived at are principally numerical1 but we shall find that some interesting laws of nature will appear4 ,,the ,,mode ,,of ,,experimenting4 #aaf4 ,friction is present between every pair of surfaces which are in contact3 there is friction between two pieces of wood1 and between a piece of wood and a piece of iron2 and the amount of the force depends upon the character of both surfaces4 ,we shall only experiment upon the friction of wood upon wood1 as more will be learned by a careful study of a special case than by a less minute examination of a number of pairs of different substances4 #aag4 ,the apparatus used is shown in ,fig4 #cb4 ,a plank of pine #f' #aa #b is planed on its upper surface1 levelled by a spirit-level1 and firmly secured to the framework at a height of about #d' from the ground4 ,on it is a pine slide #i #i1 the grain of which is crosswise to that of the plank2 upon the slide the load ,a is placed4 ,a rope is attached to the slide1 which passes over a very freely mounted cast iron pulley ,c1 #ad diameter1 and carries at the other end a hook weighing one pound1 from which weights ,b can be suspended4 #aah4 ,the mode of experimenting consists in placing a certain load upon ,a1 and then ascertaining what weight applied to ,b will draw the loaded slide along the plane4 ,as several trials are generally necessary to determine the power1 a rope is attached to the back of the slide1 and passes over the two pulleys ,d2 this makes it easy for the experimenter1 when applying the weights at ,b1 to draw back the slide to the end of the plane by pulling the ring ,e3 this rope is of course left quite slack during the process of the experiment1 since the slide must not be retarded4 ,the loads placed upon ,a during the series of experiments ranged between one stone and eight stone4 ,in the loads stated the weight of the slide itself1 which was less than #a lb41 is always included4 ,a variety of small weights were provided for the hook ,b2 they consisted of #ja1 #je1 #a1 #b1 #g1 and #ad lbs4 ,there is some friction to be overcome in the pulley ,c1 but as the pulley is comparatively large its friction is small1 though it was always allowed for4 ,illustration3 ,,fig4 #cb4 #aai4 ,an example of the experiments made is thus described4 ,a weight of #ef lbs4 is placed upon the slide1 and it is found on trial that #bi lbs4 on ,b 7including the weight of the hook itself7 is sufficient to start the slide2 this weight is placed upon the hook pound by pound1 care being taken to make each addition gently4 #abj4 ,experiments were made in this way with various weights upon ,a1 and the results are recorded in ,table ,i4 ,,table ,i4--,,friction4 ,smooth horizontal surface of pine #gb #aa2 slide also of pine #i #i2 grain crosswise2 slide is not started2 force acting on slide is gradually increased until motion commences4 ---------------------------------------- --------------------- ,number of ,load on slide,force necessary,force necessary ,mean ,experiment4 in lbs41 to move slide4 to move slide4 values4 including weight #ast ,series4 #bnd ,series4 of slide4 ---------------------------------------- --------------------- #a #ad #e #h #fe #b #bh #ae #af #aee #c #db #bj #ae #age #d #ef #bi #bd #bfe #e #gj #cc #ca #cbj #f #hd #dc #cc #chj #g #ih #db #ch #djj #h #aab #ej #cc #dae ---------------------------------------- --------------------- ,in the first column a number is given to each experiment for convenience of reference4 ,in the second column the load on the slide is stated in lbs4 ,in the third column is found the force necessary to overcome the friction4 ,the fourth column records a second series of experiments performed in the same manner as the first series2 while the last column shows the mean of the two frictions4 #aba4 ,the first remark to be made upon this table is1 that the results do not appear satisfactory or concordant4 ,thus from #f and #g of the #ast series it would appear that the friction of #hd lbs4 was #dc lbs41 while that of #ih lbs4 was #db lbs41 so that here the greater weight appears to have the less friction1 which is evidently contrary to the whole tenor of the results1 as a glance will show4 ,moreover the frictions in the #ast and the #bnd series do not agree1 being generally greater in the former than in the latter1 the discordance being especially noticeable in experiment #h1 where the results were #ej lbs4 and #cc lbs4 ,in the final column of means these irregularities are lessened1 for this column shows that the friction increases with the weight1 but it is sufficient to observe that as the difference of the #ast and the #bnd is #i lbs41 and that of the #bnd and the #crd is only #b lbs41 the discovery of any law from these results is hopeless4 #abb4 ,but is friction so capricious that it is amenable to no better law than these experiments appear to indicate8 ,we must look a little more closely into the matter4 ,when two pieces of wood have remained in contact and at rest for some time1 a second force besides friction resists their separation3 the wood is compressible1 the surfaces become closely approximated1 and the coherence due to this cause must be overcome before motion commences4 ,the initial coherence is uncertain2 it depends probably on a multitude of minute circumstances which it is impossible to estimate1 and its presence has vitiated the results which we have found so unsatisfactory4 #abc4 ,we can remove these irregularities by starting the slide at the commencement4 ,this may be conveniently effected by the screw shown at ,f in ,fig4 #cb2 a string attached to its end is fastened to the slide1 and by giving the handle of the screw a few turns the slide begins to creep4 ,a body once set in motion will continue to move with the same velocity unless acted upon by a force2 hence the weight at ,b just overcomes the friction when the slide moves uniformly after receiving a start3 this velocity was in one case of average speed measured to be #af inches per minute4 #abd4 ,indeed in no case can the slide commence to move unless the force exceed the friction4 ,the amount of this excess is indeterminate4 ,it is certainly greater between wooden surfaces than between less compressible surfaces like those of metals or glass4 ,in the latter case1 when the force exceeds the friction by a small amount1 the slide starts off with an excessively slow motion2 with wood the force must exceed the friction by a larger amount before the slide commences to move1 but the motion is then comparatively rapid4 #abe4 ,if the power be too small1 the load either does not continue moving after the start1 or it stops irregularly4 ,if the power be too great1 the load is drawn with an accelerated velocity4 ,the correct amount is easily recognised by the uniformity of the movement1 and even when the slide is heavily laden1 a few tenths of a pound on the power hook cause an appreciable difference of velocity4 #abf4 ,the accuracy with which the friction can be measured may be appreciated by inspecting ,table ,,ii4 ,,table ,,ii4--,,friction4 ,smooth horizontal surface of pine #gb #aa2 slide also of pine #i #i2 grain crosswise2 slide started2 force applied is sufficient to maintain uniform motion of the slide4 ---------------------------------------- ------------------- ,load on slide ,force ,force ,number of in lbs41 necessary necessary ,mean ,experiment4 including to maintain to maintain values4 weight of motion4 motion4 slide4 #ast ,series4 #bnd ,series4 ---------------------------------------- ------------------- #a #ad #di #di #di #b #bh #he #hf #he #c #db #abf #abd #abe #d #ef #afc #afb #afb #e #gj #aig #bjj #aih #f #hd #bcg #bcj #bcd #g #ih #bfe #bfa #bfc #h #aab #big #bii #bih ---------------------------------------- ------------------- #abg4 ,two series of experiments to determine the power necessary to maintain the motion have been recorded4 ,thus1 in experiment #g1 the load on the slide being #ih lbs41 it was found that #bfe lbs4 was sufficient to sustain the motion1 and a second trial being made independently1 the power found was #bfa lbs43 a mean of the two values1 #bfc lbs41 is adopted as being near the truth4 ,the greatest difference between the two series1 amounting to #jg lb41 is found in experiment #f2 a third value was therefore obtained for the friction of #hd lbs43 this amounted to #bce lbs41 which is intermediate between the two former results1 and #bcd lbs41 a mean of the three1 is adopted as the final result4 #abh4 ,the close accordance of the experiments in this table shows that the means of the fifth column are probably very near the true values of the friction for the corresponding loads upon the slide4 #abi4 ,the mean frictions must1 however1 be slightly diminished before we can assert that they represent only the friction of the wood upon the wood4 ,the pulley over which the rope passes turns round its axle with a small amount of friction1 which must also be overcome by the power4 ,the mode of estimating this amount1 which in these experiments never exceeds #je lb41 need not now be discussed4 ,the corrected values of the friction are shown in the third column of ,table ,,iii4 ,thus1 for example1 the #di lbs4 of friction in experiment #a consists of #dg1 the true friction of the wood1 and #jb1 which is the friction of the pulley2 and #bfc of experiment #g is similarly composed of #beh and #je4 ,it is the corrected frictions which will be employed in our subsequent calculations4 ,,friction ,,is ,,proportional ,,to ,,the ,,pressure4 #acj4 ,having ascertained the values of the force of friction for eight different weights1 we proceed to inquire into the laws which may be founded on our results4 ,it is evident that the friction increases with the load1 of which it is always greater than a fourth1 and less than a third4 ,it is natural to surmise that the friction 7,f7 is really a constant fraction of the load 7,r7--in other words1 that ,f k,r1 where k is a constant number4 #aca4 ,to test this supposition we must try to determine k2 it may be ascertained by dividing any value of ,f by the corresponding value of ,r4 ,if this be done1 we shall find that each of the experiments yields a different quotient2 the first gives #jccf1 and the last #jbfb1 while the other experiments give results between these extreme values4 ,these numbers are tolerably close together1 but there is still sufficient discrepancy to show that it is not strictly true to assert that the friction is proportional to the load4 #acb4 ,but the law that the friction varies proportionally to the pressure is so approximately true as to be sufficient for most practical purposes1 and the question then arises1 which of the different values of k shall we adopt8 ,by a method which is described in the ,appendix we can determine a value for k which1 while it does not represent any one of the experiments precisely1 yet represents them collectively better than it is possible for any other value to do4 ,the number thus found is #jbg4 ,it is intermediate between the two values already stated to be extreme4 ,the character of this result is determined by an inspection of ,table ,,iii4 ,the fourth column of this table has been calculated from the formula ,f #jbg ,r4 ,thus1 for example1 in experiment #e1 the friction of a load of #gj lbs4 is #aid lbs41 and the product of #gj and #jbg is #ahi1 which is #je lb4 less than the true amount4 ,in the last column of this table the discrepancies between the observed and the calculated values are recorded1 for facility of comparison4 ,it will be observed that the greatest difference is under #a lb4 ,,table ,,iii4--,,friction4 ,friction of pine upon pine2 the mean values of the friction given in ,table ,,ii4 7corrected for the friction of the pulley7 compared with the formula ,f #jbg ,r4 ---------------------------------------- ---------------- ,number of ,r4 ,corrected ,f4 ,discrepancies ,experiment4,total loadmean value,calculated between the on slide of value of observed and in lbs4 friction4 friction4 calculated frictions4 ---------------------------------------- ---------------- #a #ad #dg #ch -#ji #b #bh #hb #gf -#jf #c #db #abb #aac -#ji #d #ef #aeh #aea -#jg #e #gj #aid #ahi -#je #f #hd #bcj #bbg -#jc #g #ih #beh #bfe #jg #h #aab #bic #cjb #ji ---------------------------------------- ---------------- #acc4 ,hence the law ,f #jbg ,r represents the experiments with tolerable accuracy2 and the numerical ratio ,o#bg is called the coefficient of friction4 ,we may apply this law to ascertain the friction in any case where the load lies between #ad lbs4 and #aab lbs42 for example1 if the load be #fc lbs41 the friction is #fc #jbg #agj4 #acd4 ,the coefficient of friction would have been slightly different had the grain of the slide been parallel to that of the plank2 and it of course varies with the nature of the surfaces4 ,experimenters have given tables of the coefficients of friction of various substances1 wood1 stone1 metals1 c4 ,the use of these coefficients depends upon the assumption of the ordinary law of friction1 namely1 that the friction is proportional to the pressure3 this law is accurate enough for most purposes1 especially when used for loads that lie between the extreme weights employed in calculating the value of the coefficient which is employed4 ,a ,,more ,,accurate ,,law ,,of ,,friction4 #ace4 ,in making one of our measurements with care1 it is unusual to have an error of more than a few tenths of #a lb4 and it is hardly possible that any of the mean frictions we have found should be in error to so great an extent as #je lb4 ,but with the value of the coefficient of friction which is used in ,table ,,iii41 the discrepancies amount sometimes to #ji lbs4 ,with any other numerical coefficient than #jbg1 the discrepancies would have been even still more serious4 ,as these are too great to be attributed to errors of experiment1 we have to infer that the law of friction which has been assumed cannot be strictly true4 ,the signs of the discrepancies indicate that the law gives frictions which for small loads are too small1 and for large loads are too great4 #acf4 ,we are therefore led to inquire whether some other relation between ,f and ,r may not represent the experiments with greater fidelity than the common law of friction4 ,if we diminished the coefficient by a small amount1 and then added a constant quantity to the product of the coefficient and the load1 the effect of this change would be that for small loads the calculated values would be increased1 while for large loads they would be diminished4 ,this is the kind of change which we have indicated to be necessary for reconciliation between the observed and calculated values4 #acg4 ,we therefore infer that a relation of the form ,f x y ,r will probably express a more correct law1 provided we can find x and y4 ,one equation between x and y is obtained by introducing any value of ,r with the corresponding value of ,f1 and a second equation can be found by taking any other similar pair4 ,from these two equations the values of x and of y may be deduced by elementary algebra1 but the best formula will be obtained by combining together all the pairs of corresponding values4 ,for this reason the method described in the ,appendix must be used1 which1 as it is founded on all the experiments1 must give a thoroughly representative result4 ,the formula thus determined1 is ,f #add #jbeb ,r4 ,this formula is compared with the experiments in ,table ,,iv4 ,,table ,,iv4--,,friction4 ,friction of pine upon pine2 the mean values of the friction given in ,table ,,ii4 7corrected for the friction of the pulley7 compared with the formula ,f #add #jbeb ,r4 ---------------------------------------- ----------------- ,discrepancies ,number of ,r4 ,corrected ,f4 between the ,experiment4,total load mean ,calculatedobserved and on slide value of value of calculated in lbs4 friction4 friction4 frictions4 ---------------------------------------- ----------------- #a #ad #dg #ej #jc #b #bh #hb #he #jc #c #db #abb #abj -#jb #d #ef #aeh #aef -#jb #e #gj #aid #aia -#jc #f #hd #bcj #bbf -#jd #g #ih #beh #bfa #jc #h #aab #bic #big #jd ---------------------------------------- ----------------- ,the fourth column contains the calculated values3 thus1 for example1 in experiment #d1 where the load is #ef lbs41 the calculated value is #add #jbeb #ef #aef2 the difference #jb between this and the observed value #aeh is shown in the last column4 #ach4 ,it will be noticed that the greatest discrepancy in this column is #jd lbs41 and that therefore the formula represents the experiments with considerable accuracy4 ,it is undoubtedly nearer the truth than the former law 7,art4 #acb72 in fact1 the differences are now such as might really belong to errors unavoidable in making the experiments4 #aci4 ,this formula may be used for calculating the friction for any load between #ad lbs4 and #aab lbs4 ,thus1 if the load be #fc lbs41 the friction is #add #jbeb #fc #agc lbs41 which does not differ much from #agj lbs41 the value found by the more ordinary law4 ,we must1 however1 be cautious not to apply this formula to weights which do not lie between the limits of the greatest and least weight used in those experiments by which the numerical values in the formula have been determined2 for example1 to take an extreme case1 if ,r #j1 the formula would indicate that the friction was #add1 which is evidently absurd2 here the formula errs in excess1 while if the load were very large it is certain the formula would err in defect4 ,,the ,,coefficient ,,varies ,,with ,,the ,,weights ,,used4 #adj4 ,in a subsequent lecture we shall employ as an inclined plane the plank we have been examining1 and we shall require to use the knowledge of its friction which we are now acquiring4 ,the weights which we shall then employ range from #g lbs4 to #ef lbs4 ,assuming the ordinary law of friction1 we have found that #jbg is the best value of its coefficient when the loads range between #ad lbs4 and #aab lbs4 ,suppose we only consider loads up to #ef lbs41 we find that the coefficient #jbhh will best represent the experiments within this range1 though for #aab lbs4 it would give an error of nearly #c lbs4 ,the results calculated by the formula ,f #jbhh ,r are shown in ,table ,v41 where the greatest difference is #jg lb4 ,,table ,v4--,,friction4 ,friction of pine upon pine2 the mean values of the friction given in ,table ,,ii4 7corrected for the friction of the pulley7 compared with the formula ,f #jbhh ,r ---------------------------------------- ------------- ,discrepancies ,number of ,r4 ,corrected ,f4 between the ,experiment4,total load mean ,calculated observed and on slide value of value of calculated in lbs4 friction4 friction4 frictions4 ---------------------------------------- ------------- #a #ad #dg #dj -#jg #b #bh #hb #ha -#ja #c #db #abb #aba -#ja #d #ef #aeh #afa #jc ---------------------------------------- ------------- #ada4 ,but we can replace the common law of friction by the more accurate law of ,art4 #acg1 and the formula computed so as to best harmonise the experiments up to #ef lbs41 disregarding the higher loads1 is ,f #ji #jbff ,r4 ,this formula is obtained by the method referred to in ,art4 #acg4 ,we find that it represents the experiments better than that used in ,table ,v4 ,between the limits named1 this formula is also more accurate than that of ,table ,,iv4 ,it is compared with the experiments in ,table ,,vi41 and it will be noticed that it represents them with great precision1 as no discrepancy exceeds #ja4 ,,table ,,vi4--,,friction4 ,friction of pine upon pine2 the mean values of the friction given in ,table ,,ii4 7corrected for the friction of the pulley7 compared with the formula ,f #ji #jbff ,r4 ---------------------------------------- ------------- ,discrepancies ,number of ,r4 ,corrected ,f4 between the ,experiment4,total load mean ,calculated observed and on slide value of value of calculated in lbs4 friction4 friction4 frictions4 ---------------------------------------- ------------- #a #ad #dg #df -#ja #b #bh #hb #hc #ja #c #db #abb #aba -#ja #d #ef #aeh #aeh #jj ---------------------------------------- ------------- ,illustration3 ,,fig4 #cc4 ,,the ,,angle ,,of ,,friction4 #adb4 ,there is another mode of examining the action of friction besides that we have been considering4 ,the apparatus for this purpose is shown in ,fig4 #cc1 in which ,b ,c represents the plank of pine we have already used4 ,it is now mounted so as to be capable of turning about one end ,b2 the end ,c is suspended from one hook of the chain from the 8epicycloidal0 pulley-block ,e4 ,this block is very convenient for the purpose4 ,by its means the inclination of the plank can be adjusted with the greatest nicety1 as the raising chain ,g is held in one hand and the lowering chain ,f in the other4 ,another great convenience of this block is that the load does not run down when the lifting chain is left free4 ,the plank is clamped to the hinge about which it turns4 ,the frames by which both the hinge and the block are supported are weighted in order to secure steadiness4 ,the inclination of the plane is easily ascertained by measuring the difference in height of its two ends above the floor1 and then making a drawing on the proper scale4 ,the starting-screw ,d1 whose use has been already mentioned1 is also fastened to the framework in the position shown in the figure4 #adc4 ,suppose the slide ,a be weighted and placed upon the inclined plane ,b ,c2 if the end ,c be only slightly elevated1 the slide remains at rest2 the reason being that the friction between the slide and the plane neutralizes the force of gravity4 ,but suppose1 by means of the pulley-block1 ,c be gradually raised2 an elevation is at last reached at which the slide starts off1 and runs with an accelerating velocity to the bottom of the plane4 ,the angle of elevation of the plane when this occurs is called the angle of statical friction4 #add4 ,the weights with which the slide was laden in these experiments were #ad lbs41 #ef lbs41 and #aab lbs41 and the results are given in ,table ,,vii4 ,we see that a load of #ef lbs4 started when the plane reached an inclination of #bja in the first series1 and of #agb in the second1 the mean value #ahf being given in the fifth column4 ,these means for the three different weights agree so closely that we assert the remarkable law that the angle of friction does not depend upon the magnitude of the load4 ,,table ,,vii4--,,angle ,,of ,,statical ,,friction4 ,a smooth plane of pine #gb #aa carries a loaded slide of pine #i #i2 one end of the plane is gradually elevated until the slide starts off4 ---------------------------------------- ----------------- ,number of ,total load on ,angle of ,angle of ,mean values ,experiment4 the slide elevation4 elevation4 of the in lbs4 #ast ,series4#bnd ,series4 angles4 ---------------------------------------- ----------------- #a #ad #aie ---- #aie #b #ef #bja #agb #ahf #c #aab #bjc #ahi #aif ---------------------------------------- ----------------- #ade4 ,we might1 however1 proceed differently in determining the angle of friction1 by giving the load a start1 and ascertaining if the motion will continue4 ,to do so requires the aid of an assistant1 who will start the load with the help of the screw1 while the elevation of the plane is being slowly increased4 ,the result of these experiments is given in ,table ,,viii4 ,,table ,,viii4--,,angle ,,of ,,friction4 ,a smooth plane of pine #gb #aa carries a loaded slide of pine #i #i2 one end of the plane is gradually elevated until the slide1 having received a start1 moves off uniformly4 ---------------------------------------- ,number of ,total load on ,angle of ,experiment4the slide in lbs4inclination4 ---------------------------------------- #a #ad #adc #b #ef #acj #c #aab #acj ---------------------------------------- ,we see from this table also that the angle of friction is independent of the load1 but the angle is in this case less by #e or #f than was found necessary to impart motion when a start was not given4 #adf4 ,it is commonly stated that the coefficient of friction equals the tangent of the angle of friction1 and this can be proved to be true when the ordinary law of friction is assumed4 ,but as we have seen that the law of friction is only approximately correct1 we need not expect to find this other law completely verified4 #adg4 ,when the slide is started1 the mean value of the angle of friction is #acd4 ,the tangent of this angle is #jbd3 this is about #aa per cent4 less than the coefficient of friction #jbg1 which we have already determined4 ,the mean value of the angle of friction when the slide is not started is #aib1 and its tangent is #jce4 ,the experiments of ,table ,i4 are1 as already pointed out1 rather unsatisfactory1 but we refer to them here to show that1 so far as they go1 the coefficient of friction is in no sense equal to the tangent of the angle of friction4 ,if we adopt the mean values given in the last column of ,table ,i41 the best coefficient of friction which can be deduced is #jda4 ,whether1 therefore1 the slide be started or not started1 the tangent of the angle of friction is smaller than the corresponding coefficient of friction4 ,when the slide is started1 the tangent is about #aa per cent4 less than the coefficient2 and when the slide is not started1 it is about #ad per cent4 less4 ,there are doubtless many cases in which these differences are sufficiently small to be neglected1 and in which1 therefore1 the law may be received as true4 ,,another ,,law ,,of ,,friction4 #adh4 ,the area of the wooden slide is #i #i1 but we would have found that the friction under a given load was practically the same whatever were the area of the slide1 so long as its material remained unaltered4 ,this follows as a consequence of the approximate law that the friction is proportional to the pressure4 ,suppose that the weight were #ajj lbs41 and the area of the slide #ajj inches1 there would then be a pressure of #a lb4 per square inch over the surface of the slide1 and therefore the friction to be overcome on each square inch would be #jbg lb41 or for the whole slide #bg lbs4 ,if1 however1 the slide had only an area of #ej square inches1 the load would produce a pressure of #b lbs41 per square inch2 the friction would therefore be #b #jbg #jed lb4 for each square inch1 and the total friction would be #ej #jed #bg lbs41 the same as before3 hence the total friction is independent of the extent of surface4 ,this would remain equally true even though the weight were not1 as we have supposed1 uniformly distributed over the surface of the slide4 ,,concluding ,,remarks4 #adi4 ,the importance of friction in mechanics arises from its universal presence4 ,we often recognize it as a destroyer or impeder of motion1 as a waster of our energy1 and as a source of loss or inconvenience4 ,but1 on the other hand1 friction is often indirectly the means of producing motion1 and of this we have a splendid example in the locomotive engine4 ,the engine being very heavy1 the wheels are pressed closely to the rails2 there is friction enough to prevent the wheels slipping1 consequently when the engines force the wheels to turn round they must roll onwards4 ,the coefficient of friction of wrought iron upon wrought iron is about #jb4 ,suppose a locomotive weigh #cj tons1 and the share of this weight borne by the driving wheels be #aj tons1 the friction between the driving wheels and the rails is #b tons4 ,this is the greatest force the engine can exert on a level line4 ,a force of #aj lbs4 for every ton weight of the train is known to be sufficient to sustain the motion1 consequently the engine we have supposed should draw along the level a load of #ddh tons4 #aej4 ,but we need not invoke the steam-engine to show the use of friction4 ,we could not exist without it4 ,in the first place we could not move about1 for walking is only possible on account of the friction between the soles of our boots and the ground2 nor if we were once in motion could we stop without coming into collision with some other object1 or grasping something to hold on by4 ,objects could only be handled with difficulty1 nails would not remain in wood1 and screws would be equally useless4 ,buildings could hardly be erected1 nay1 even hills and mountains would gradually disappear1 and finally dry land would be immersed beneath the level of the sea4 ,friction is1 so far as we are concerned1 quite as essential a law of nature as the law of gravitation4 ,we must not seek to evade it in our mechanical discussions because it makes them a little more difficult4 ,friction obeys laws2 its action is not vague or uncertain4 ,when inconvenient it can be diminished1 when useful it can be increased2 and in our lectures on the mechanical powers1 to which we now proceed1 we shall have opportunities of describing machines which have been devised in obedience to its laws4 ,,lecture ,,vi4 ,,the ,,pulley4 ,introduction4--,friction between a ,rope and an ,iron ,bar4--,the use of the ,pulley4--,large and ,small ,pulleys4--,the ,law of ,friction in the ,pulley4--,wheels4--,energy4 ,,introduction4 #aea4 ,the pulley forms a good introduction to the important subject of the mechanical powers4 ,but before entering on the discussions of the next few chapters1 it will be necessary for us to explain what is meant in mechanics by 8work10 and by 8energy10 which is the capacity for performing work1 and we shall therefore include a short outline of this subject in the present lecture4 #aeb4 ,the pulley is a machine which is employed for the purpose of changing the direction of a force4 ,we frequently wish to apply a force in a different direction from that in which it is convenient to exert it1 and the pulley enables us to do so4 ,we are not now speaking of these arrangements for increasing power in which pulleys play an important part2 these will be considered in the next lecture3 we at present refer only to change of direction4 ,in fact1 as we shall shortly see1 some force is even wasted when the single fixed pulley is used1 so that this machine certainly cannot be called a mechanical power4 #aec4 ,the occasions upon which a single fixed pulley is used are numerous and familiar4 ,let us suppose a sack of corn has to be elevated from the lower to one of the upper stories of a building4 ,it may of course be raised by a man who carries it1 but he has to lift his own weight in addition to that of the sack1 and therefore the quantity of exertion used is greater than absolutely necessary4 ,but supposing there be a pulley at the top of the building over which a rope passes2 then1 if a man attach one end of the rope to the sack and pull the other1 he raises the sack without raising his own weight4 ,the pulley has thus provided the means by which the downward force has been changed in direction to an upward force4 #aed4 ,the weights1 ropes1 and pulleys which are used in our windows for counterpoising the weight of the sash afford a very familiar instance of how a pulley changes the direction of a force4 ,here the downward force of the weight is changed by means of the pulley into an upward force1 which nearly counterbalances the weight of the sash4 ,,friction ,,between ,a ,,rope ,,and ,,an ,,iron ,,bar4 #aee4 ,every one is familiar with the ordinary form of the pulley2 it consists of a wheel capable of turning freely on its axle1 and it has a groove in its circumference in which the rope lies4 ,but why is it necessary to give the pulley this form8 ,why could not the direction of the rope be changed by simply passing it over a bar1 as well as by the more complicated pulley8 ,we shall best answer this question by actually trying the experiment1 which we can do by means of the apparatus of ,fig4 #cd 7see page #ij74 ,in this are shown two iron studs1 ,g1 ,h1 #jf diameter1 and about #h apart2 over these passes a rope1 which has a hook at each end4 ,if ,i suspend a weight of #ad lbs4 from one hook ,a1 and pull the hook ,b1 ,i can by exerting sufficient force raise the weight on ,a1 but with this arrangement ,i am conscious of having to exert a very much larger force than would have been necessary to raise #ad lbs4 by merely lifting it4 #aef4 ,in order to study the question exactly1 we shall ascertain what weight suspended from the hook ,b will suffice to raise ,a4 ,i find that in order to raise #ad lbs4 on ,a no less than #dg lbs4 is necessary on ,b1 consequently there is an enormous loss of force3 more than two-thirds of the force which is exerted is expended uselessly4 ,if instead of the #ad lbs4 weight ,i substitute any other weight1 ,i find the same result1 viz4 that more than three times its amount is necessary to raise it by means of the rope passing over the studs4 ,if a labourer1 in raising a sack1 were to pass a rope over two bars such as these1 then for every stone the sack weighed he would have to exert a force of more than three stones1 and there would be a very extravagant loss of power4 #aeg4 ,whence arises this loss8 ,the rope in moving slides over the surface of the iron studs4 ,although these are quite smooth and polished1 yet when there is a strain on the rope it presses closely upon them1 and there is a certain amount of force necessary to make the rope slide along the iron4 ,in other words1 when ,i am trying to raise up #ad lbs4 with this contrivance1 ,i not only have its weight opposed to me1 but also another force due to the sliding of the rope on the iron3 this force is due to friction4 ,were it not for friction1 a force of #ad lbs4 on one hook would exactly balance #ad lbs4 on the other1 and the slightest addition to either weight would make it descend and raise the other4 ,if1 then1 we are obliged to change the direction of a force1 we must devise some means of doing so which does not require so great a sacrifice as the arrangement we have just used4 ,,the ,,use ,,of ,,the ,,pulley4 #aeh4 ,we shall next inquire how it is that we are enabled to obviate friction by means of a pulley4 ,it is evident we must provide an arrangement in which the rope shall not be required to slide upon an iron surface4 ,this end is attained by the pulley1 of which we may take ,i1 ,fig4 #cd1 as an example4 ,this represents a cast iron wheel #ad in diameter1 with a ,v-shaped groove in its circumference to receive the rope3 this wheel turns on a inch wrought iron axle1 which is well oiled4 ,the rope used is about #jbe in diameter4 #aei4 ,from the hooks ,e1 ,f at each end of the rope a #ad lb4 weight is suspended4 ,these equal weights balance each other4 ,according to our former experiment with the studs1 it would be necessary for me to treble the weight on one of these hooks in order to raise the other1 but now ,i find that an additional #je lb4 placed on either hook causes it to descend and make the other ascend4 ,this is a great improvement2 #je lb4 now accomplishes what #cc lbs4 was before required for4 ,we have avoided a great deal of friction1 but we have not got rid of it altogether1 for #jbe lb4 is incompetent1 when added to either weight1 to make that weight descend4 #afj4 ,to what is the improvement due8 ,when the weight descends the rope does not slide upon the wheel1 but it causes the wheel to revolve with it1 consequently there is little or no friction at the circumference of the pulley2 the friction is transferred to the axle4 ,we still have some resistance to overcome1 but for smooth oiled iron axles the friction is very small1 hence the advantage of the pulley4 ,there is in every pulley a small loss of power from the force expended in bending the rope2 this need not concern us at present1 for with the pliable plaited rope that we have employed the effect is inappreciable1 but with large strong ropes the loss becomes of importance4 ,the amount of loss by using different kinds of ropes has been determined by careful experiments4 ,,large ,,and ,,small ,,pulleys4 #afa4 ,there is often a considerable advantage obtained by using large rather than small pulleys4 ,the amount of force necessary to overcome friction varies inversely as the size of the pulley4 ,we shall demonstrate this by actual experiment with the apparatus of ,fig4 #cd4 ,a small pulley ,k is attached to the large pulley ,i2 they are in fact one piece1 and turn together on the same axle4 ,hence if we first determine the friction with the rope over the large pulley1 and then with the rope over the small pulley1 any difference can only be due to the difference in size1 as all the other circumstances are the same4 #afb4 ,in making the experiments we must attend to the following point4 ,the pulleys and the socket on which they are mounted weigh several pounds1 and consequently there is friction on the axle arising from the weight of the pulleys1 quite independently of any weights that may be placed on the hooks4 ,we must then1 if possible1 evade the friction of the pulley itself1 so that the amount of friction which is observed will be entirely due to the weights raised4 ,this can be easily done4 ,the rope and hooks being on the large pulley ,i1 ,i find that #jaf lb4 attached to one of the hooks1 ,e1 is sufficient to overcome the friction of the pulley1 and to make that hook descend and raise ,f4 ,if therefore we leave #jaf lb4 on ,e1 we may consider the friction due to the weight of the pulley1 rope1 and hooks as neutralized4 ,illustration3 ,,fig4 #cd4 #afc4 ,i now place a stone weight on each of the hooks ,e and ,f4 ,the amount necessary to make the hook ,e and its load descend is #jbh lb4 ,this does not of course include the weight of #jaf lb4 already referred to4 ,we see therefore that with the large pulley the amount of friction to be overcome in raising one stone is #jbh lb4 #afd4 ,let us now perform precisely the same experiment with the small pulley4 ,i transfer the same rope and hooks to ,k1 and ,i find that #jaf lb4 is not now sufficient to overcome the friction of the pulley1 but ,i add on weights until ,c will just descend1 which occurs when the load reaches #jie lb4 ,this weight is to be left on ,c as a counterpoise1 for the reasons already pointed out4 ,i place a stone weight on ,c and another on ,d1 and you see that ,c will descend when it receives an additional load of #ace lbs42 this is therefore the amount of friction to be overcome when a stone weight is raised over the pulley ,k4 #afe4 ,let us compare these results with the dimensions of the pulleys4 ,the proper way to measure the effective circumference of a pulley when carrying a certain rope is to measure the length of that rope which will just embrace it4 ,the length measured in this way will of course depend to a certain extent upon the size of the rope4 ,i find that the circumferences of the two pulleys are #dcj and #ie4 ,the ratio of these is #de2 the corresponding resistances from friction we have seen to be #jbh lb4 and #ace lbs4 ,the larger of these quantities is #dh times the smaller4 ,this number is very close to #de2 we must not1 as already explained1 expect perfect accuracy in experiments in friction4 ,in the present case the agreement is within the _/th of the whole1 and we may regard it as a proof of the law that the friction of a pulley is inversely proportional to its circumference4 #aff4 ,it is easy to see the reason why friction should diminish when the size of the pulley is increased4 ,the friction acts at the circumference of the axle about which the wheel turns2 it is there present as a force tending to retard motion4 ,now the larger the wheel the greater will be the distance from the axis at which the force acts which overcomes the friction1 and therefore the less need be the magnitude of the force4 ,you will perhaps understand this better after the principle of the lever has been discussed4 #afg4 ,we may deduce from these considerations the practical maxim that large pulleys are economical of power4 ,this rule is well known to engineers2 large pulleys should be used1 not only for diminishing friction1 but also to avoid loss of power by excessive bending of the rope4 ,a rope is bent gradually around the circumference of a large pulley with far less force than is necessary to accommodate it to a smaller pulley3 the rope also is apt to become injured by excessive bending4 ,in coal pits the trucks laden with coal are hoisted to the surface by means of wire ropes which pass from the pit over a pulley into the engine-house3 this pulley is of very large dimensions1 for the reasons we have pointed out4 ,,the ,,law ,,of ,,friction ,,in ,,the ,,pulley4 #afh4 ,i have here a wooden pulley #ce in diameter2 the hole is lined with brass1 and the pulley turns very freely on an iron spindle4 ,i place the rope and hooks upon the groove4 ,brass rubbing on iron has but little friction1 and when #g lbs4 is placed on each hook1 #je lb4 added to either will make it descend and raise up the other4 ,let #ad lbs4 be placed on each hook1 #je lb4 is no longer sufficient2 #a lb4 is required3 hence when the weight is doubled the friction is also doubled4 ,repeating the experiment with #ba lbs4 and #bh lbs4 on each side1 the corresponding weights necessary to overcome friction are #ae lbs4 and #b lbs4 ,in the four experiments the weights used are in the proportion #a1 #b1 #c1 #d2 and the forces necessary to overcome friction1 #je lb41 #a lb41 #ae lbs41 and #b lbs41 are in the same proportion4 ,hence the friction is proportional to the load4 ,,wheels4 #afi4 ,the wheel is one of the most simple and effective devices for overcoming friction4 ,a sleigh is an admirable vehicle on a smooth surface such as ice1 but it is totally unadapted for use on common roads2 the reason being that the amount of friction between the sleigh and the road is so great that to move the sleigh the horse would have to exert a force which would be very great compared with the load he was drawing4 ,but a vehicle properly mounted on wheels moves with the greatest ease along the road1 for the circumference of the wheel does not slide1 and consequently there is no friction between the wheel and the road2 the wheel however turns on its axle1 therefore there is sliding1 and consequently friction1 at the axle1 but the axle and the wheel are properly fitted to each other1 and the surfaces are lubricated with oil1 so that the friction is extremely small4 #agj4 ,with large wheels the amount of friction on the axle is less than with small wheels2 other advantages of large wheels are that they do not sink much into depressions in the roads1 and that they have also an increased facility in surmounting the innumerable small obstacles from which even the best road is not free4 #aga4 ,when it is desired to make a pulley turn with extremely small friction1 its axle1 instead of revolving in fixed bearings1 is mounted upon what are called friction wheels4 ,a set of friction wheels is shown in the apparatus of ,fig4 #ff3 as the axle revolves1 the friction between the axles and the wheels causes the latter to turn round with a comparatively slow motion2 thus all the friction is transferred to the axles of the four friction wheels2 these revolve in their bearings with extreme slowness1 and consequently the pulley is but little affected by friction4 ,the amount of friction in a pulley so mounted may be understood from the following experiment4 ,a silk cord is placed on the pulley1 and #a lb4 weight is attached to each of its ends3 these of course balance4 ,a number of fine wire hooks1 each weighing #jjja lb41 are prepared1 and it is found that when a weight of #jjjd lb4 is attached to either side it is sufficient to overcome friction and set the weights in motion4 ,,energy4 #agb4 ,in connection with the subject of friction1 and also as introductory to the mechanical powers1 the notion of 8work10 or as it is more properly called 8energy10 is of great importance4 ,the meaning of this word as employed in mechanics will require a little consideration4 #agc4 ,in ordinary language1 whatever a man does that can cause fatigue1 whether of body or mind1 is called work4 ,in mechanics1 we mean by energy that particular kind of work which is directly or indirectly equivalent to raising weights4 #agd4 ,suppose a weight is lying on the floor and a stool is standing beside it3 if a man raise the weight and place it upon the stool1 the exertion that he expends is energy in the sense in which the word is used in mechanics4 ,the amount of exertion necessary to place the weight upon the stool depends upon two things1 the magnitude of the weight and the height of the stool4 ,it is clear that both these things must be taken into account1 for although we know the weight which is raised1 we cannot tell the amount of exertion that will be required until we know the height through which it is to be raised2 and if we know the height1 we cannot appreciate the quantity of exertion until we know the weight4 #age4 ,the following plan has been adopted for expressing quantities of energy4 ,the small amount of exertion necessary to raise #a lb4 avoirdupois through one ,british foot is taken as a standard1 compared with which all other quantities of energy are estimated4 ,this quantity of exertion is called in mechanics the unit of energy1 and sometimes also the 8foot-pound40 #agf4 ,if a weight of #a lb4 has to be raised through a height of #b feet1 or a weight of #b lbs4 through a height of #a foot1 it will be necessary to expend twice as much energy as would have raised a weight of #a lb4 through #a foot1 that is1 #b foot-pounds4 ,if a weight of #e lbs4 had to be raised from the floor up to a stool #c feet high1 how many units of energy would be required8 ,to raise #e lbs4 through #a foot requires #e foot-pounds1 and the process must be again repeated twice before the weight arrive at the top of the stool4 ,for the whole operation #ae foot-pounds will therefore be necessary4 ,if #ajj lbs4 be raised through #bj feet1 #ajj foot-pounds of energy is required for the first foot1 the same for the second1 third1 c41 up to the twentieth1 making a total of #b1jjj foot-pounds4 ,here is a practical question for the sake of illustration4 ,which would it be preferable to hoist1 by a rope passing over a single fixed pulley1 a trunk weighing #dj lbs4 to a height of #bj feet1 or a trunk weighing #ej lbs4 to a height of #ae feet8 ,we shall find how much energy would be necessary in each case3 #dj times #bj is #hjj2 therefore in the first case the energy would be #hjj foot-pounds4 ,but #ej times #ae is #gej2 therefore the amount of work1 in the second case1 is only #gej lbs4 ,hence it is less exertion to carry #ej lbs4 up #ae feet than #dj lbs4 up #bj feet4 #agg4 ,the rate of working of every source of energy1 whether it lie in the muscles of men or other animals1 in water-wheels1 steam-engines1 or other prime movers1 is to be measured by the number of foot-pounds produced in the unit of time4 ,the power of a steam-engine is defined by its equivalent in horse-power4 ,for example1 it is meant that a steam-engine of #c horse-power1 could1 when working for an hour1 do as much work as #c horses could do when working for the same time4 ,the power of a horse is1 however1 an uncertain quantity1 differing in different animals and not quite uniform in the same individual2 accordingly the selection of this measure for the efficiency of the steam-engine is inconvenient4 ,we replace it by a convenient standard horse-power1 which is1 however1 a good deal larger than that continuously obtainable from any ordinary horse4 ,a one horse-power steam-engine is capable of accomplishing #cc1jjj foot-pounds per minute4 #agh4 ,we shall illustrate the numerical calculation of horse-power by an example3 if a mine be #a1jjj feet deep1 how much water per minute would a #ej horse-power engine be capable of raising to the surface8 ,the engine would yield #ej #cc1jjj units of work per minute1 but the weight has to be raised #a1jjj feet1 consequently the number of pounds of water raised per minute is #ej #cc1jjj ------------------------ #a1fej4 #a1jjj #agi4 ,we shall apply the principle of work to the consideration of the pulley already described 7p4 #ij74 ,in order to raise the weight of #ad lbs41 it is necessary that the rope to which the power is applied should be pulled downwards by a force of #ae lbs41 the extra pound being on account of the friction4 ,to fix our ideas1 we shall suppose the #ad lbs4 to be raised #a foot2 to lift this load directly1 without the intervention of the pulley1 #ad foot-pounds would be necessary1 but when it is raised by means of the pulley1 #ae1 foot-pounds are necessary4 ,hence there is an absolute loss of _/th of the energy when the pulley is used4 ,if a steam-engine of #a horse-power were employed in raising weights by a rope passing over a pulley similar to that on which we have experimented1 only _/ths of the work would be usefully employed2 but we find #cc1jjj #ad #cj1hjj4 -------- #ae ,the engine would therefore perform #cj1hjj foot-pounds of useful work per minute4 #ahj4 ,the effect of friction on a pulley1 or on any other machine1 is always to waste energy4 ,to perform a piece of work directly requires a certain number of foot-pounds1 while to do it by a machine requires more1 on account of the loss by friction4 ,this may at first sight appear somewhat paradoxical1 as it is well known that1 by levers1 pulleys1 c41 an enormous mechanical advantage may be gained4 ,this subject will be fully explained in the next and following lectures1 which relate to the mechanical powers4 #aha4 ,we shall conclude with a few observations on a point of the greatest importance4 ,we have seen a case where #ae foot-pounds of energy only accomplished #ad foot-pounds of work1 and thus #a foot-pound appeared to be lost4 ,we say that this was expended upon the friction2 but what is the friction8 ,the axle is gradually worn away by rubbing in its bearings1 and1 if it be not properly oiled1 it becomes heated4 ,the amount of energy that seems to disappear is partly expended in grinding down the axle1 and is partly transformed into heat2 it is thus not really lost1 but unfortunately assumes a form which we do not require and in which it is rather injurious than otherwise4 ,indeed we know that energy cannot be destroyed1 however it may be transformed2 if it disappear in one shape1 it is only to reappear in another4 ,a so-called loss of energy by friction only means a diversion of a part of the work to some purpose other than that which we wish to accomplish4 ,it has long been known that matter is indestructible3 it is now equally certain that the same may be asserted of energy4 ,,lecture ,,vii4 ,,the ,,pulley-,,block4 ,introduction4--,the ,single ,moveable ,pulley4--,the ,three-sheave ,pulley-block4--,the ,differential ,pulley-block4--,the ,epicycloidal ,pulley-block4 ,,introduction4 #ahb4 ,in the first lecture ,i showed how a large weight could be raised by a smaller weight1 and ,i stated that this subject would again occupy our attention4 ,i now fulfil this promise4 ,the questions to be discussed involve the most advantageous methods of employing a small force to overcome a greater4 ,here is a subject of practical importance4 ,a man of average strength cannot raise more than a hundredweight without great exertion1 yet the weights which it is necessary to lift and move about often weigh many hundredweights1 or even many tons4 ,it is not always practicable to employ numerous hands for the purpose1 nor is a steam-engine or other great source of power at all times available4 ,but what are called the mechanical powers enable the forces at our disposal to be greatly increased4 ,one man1 by their aid1 can exert as much force as several could without such assistance2 and when they are employed to augment the power of several men or of a steam-engine1 gigantic weights1 amounting sometimes to hundreds of tons1 can be managed with facility4 #ahc4 ,in the various arts we find innumerable cases where great resistances have to be overcome2 we also find a corresponding number and variety of devices contrived by human skill to conquer them4 ,the girders of an iron bridge have to be lifted up to their piers2 the boilers and engines of an ocean steamer have to be placed in position2 a great casting has to be raised from its mould2 a railway locomotive has to be placed on the deck of a vessel for transit2 a weighty anchor has to be lifted from the bottom of the sea2 an iron plate has to be rolled or cut or punched3 for all of these cases suitable arrangements must be devised in order that the requisite power may be obtained4 #ahd4 ,we know but little of the means which the ancients employed in raising the vast stones of those buildings which travellers in the ,east have described to us4 ,it is sometimes thought that a large number of men could have transported these stones without the aid of appliances which we would now use for a similar purpose4 ,but it is more likely that some of the mechanical powers were used1 as1 with a multitude of men1 it is difficult to ensure the proper application of their united strength4 ,in ,easter ,island1 hundreds of miles distant from civilised land1 and now inhabited by savages1 vast idols of stone have been found in the hills which must have been raised by human labour4 ,it is useless to speculate on the extinct race by whom this work was achieved1 or on the means they employed4 #ahe4 ,the mechanical powers are usually enumerated as follows3--,the pulley1 the lever1 the wheel and axle1 the wedge1 the inclined plane1 the screw4 ,these different powers are so frequently used in combination that the distinctions cannot be always maintained4 ,the classification will1 however1 suffice to give a general notion of the subject4 #ahf4 ,many of the most valuable mechanical powers are machines in which ropes or chains play an important part4 ,pulleys are usually employed wherever it is necessary to change the direction of a rope or chain which is transmitting power4 ,in the present lecture we shall examine the most important mechanical powers that are produced by the combination of pulleys4 ,,the ,,single ,,moveable ,,pulley4 #ahg4 ,we commence with the most simple case1 that of the single moveable pulley 7,fig4 #ce74 ,the rope is firmly secured at one end ,a2 it then passes down under the moveable pulley ,b1 and upwards over a fixed pulley4 ,to the free end1 which depends from the fixed pulley1 the power is applied while the load to be raised is suspended from the moveable pulley4 ,we shall first study the relation between the power and the load in a simple way1 and then we shall describe a few exact experiments4 #ahh4 ,when the load is raised the moveable pulley itself must of course be also raised1 and a part of the power is expended for this purpose4 ,but we can eliminate the weight of the moveable pulley1 so far as our calculations are concerned1 by first attaching to the power end of the rope a sufficient weight to lift up the moveable pulley when not carrying a load4 ,the weight necessary for doing this is found by trial to be a little over #ae lbs4 ,so that when a load is being raised we must reduce the apparent power by #ae lbs4 to obtain the power really effective4 #ahi4 ,let us suspend #ad lbs4 from the load hook at ,b1 and ascertain what power will raise the load4 ,we leave the weight of the moveable pulley and #ae lbs4 of the power at ,c out of consideration4 ,i then find by experiment that #g lbs4 of effective power is not sufficient to raise the load1 but if one pound more be added1 the power descends1 and the load is raised4 ,here1 then1 is a remarkable result2 a weight of #h lbs4 has overcome #ad lbs4 ,in this we have the first application of the mechanical powers to increase our available forces4 ,illustration3 ,,fig4 #ce4 #aij4 ,let us examine the reason of this mechanical advantage4 ,if the load be raised one foot1 it is plain that the power must descend two feet3 for in order to raise the load the two parts of the rope descending to the moveable pulley must each be shortened one foot1 and this can only be done by the power descending two feet4 ,hence when the load of #ad lbs4 is lifted by the machine1 for every foot it is raised the power must descend two feet3 this simple point leads to a conception of the greatest importance1 on which depends the efficiency of the pulley4 ,in the study of the mechanical powers it is essential to examine the number of feet through which the power must act in order to raise the load one foot3 this number we shall always call the velocity ratio4 #aia4 ,to raise #ad lbs4 one foot requires #ad foot-pounds of energy4 ,hence1 were there no such thing as friction1 #g lbs4 on the power hook would be sufficient to raise the load2 because #g lbs4 descending through two feet yields #ad foot-pounds4 ,but there is a loss of energy on account of friction1 and a power of #g lbs4 is not sufficient3 #h lbs4 are necessary4 ,eight lbs4 in descending two feet performs #af foot-pounds2 of these only #ad are utilised on the load1 the remainder being the quantity of energy that has been diverted by friction4 ,we learn1 then1 that in the moveable pulley the quantity of energy employed is really greater than that which would lift the weight directly1 but that the actual force which has to be exerted is less4 #aib4 ,suppose that #bh lbs4 be placed on the load hook1 a few trials assure us that a power of #af lbs4 7but not less7 will be sufficient for motion2 that is to say1 when the load is doubled1 we find1 as we might have expected1 that the power must be doubled also4 ,it is easily seen that the loss of energy by friction then amounts to #d foot-pounds4 ,we thus verify1 in the case of the moveable pulley1 the approximate law that the friction is proportional to the load4 #aic4 ,by means of a moveable pulley a man is able to raise a weight nearly double as great as he could lift directly4 ,from a series of careful experiments it has been found that when a man is employed in the particular exertion necessary for raising weights over a pulley1 he is able to work most efficiently when the pull he is required to make is about #dj lbs4 ,a man could1 of course1 exert greater force than this1 but in an ordinary day's work he is able to perform more foot-pounds when the pull is #dj lbs4 than when it is larger or smaller4 ,if therefore the weights to be lifted amount to about #hj lbs41 energy may really be economized by the use of the single moveable pulley1 although by so doing a greater quantity of energy would be actually expended than would have been necessary to raise the weights directly4 #aid4 ,some experiments on larger loads have been tried with the moveable pulley we have just described2 the results are recorded in ,table ,,ix4 ,,table ,,ix4--,,single ,,moveable ,,pulley4 ,moveable pulley of cast iron #cbe diameter1 groove #jf wide1 wrought iron axle #jf diameter2 fixed pulley of cast iron #e diameter1 groove #jd wide1 wrought iron axle #jf diameter1 axles oiled2 flexible plaited rope #jbe diameter2 velocity ratio #b1 mechanical efficiency #ah1 useful effect #ij per cent42 formula ,p #bba #jedec ,r4 ---------------------------------------- ------------------ ,p4 ,discrepancies ,number of ,r4 ,observed ,calculated between observed ,experiment4 ,load power power and calculated in lbs4 in lbs4 in lbs4 powers4 ---------------------------------------- ------------------ #a #bh #age #age #jj #b #eg #cce #ccc - #jb #c #he #dhe #dhf #ja #d #aac #fdj #fch - #jb #e #adb #hjj #gif - #jd #f #agj #ide #idi #jd #g #aih #aaje #aajb - #jc #h #bbf #abee #abee #jj ---------------------------------------- ------------------ ,the dimensions of the pulleys are precisely stated because1 for pulleys of different construction1 the numerical coefficients would not necessarily be the same4 ,an attentive study of this table will1 however1 show the general character of the relation between the power and the load in all arrangements of this class4 ,the table consists of five columns4 ,the first contains merely the numbers of the experiments for convenience of reference4 ,in the second column1 headed ,r1 the loads1 expressed in pounds1 which are raised in each experiment1 are given2 that is1 the weight attached to the hook1 not including the weight of the lower pulley4 ,the weight of this pulley is not included in the stated loads4 ,in the third column the powers are recorded1 which were found to be sufficient to raise the corresponding loads in the second column4 ,thus1 in experiment #g1 it is found that a power of #aaje lbs4 will be sufficient to raise a load of #aih lbs4 ,the third column has thus been determined by gradually increasing the power until motion begins4 #aie4 ,from an examination of the columns showing the power and the load1 we see that the power always amounts to more than half the load4 ,the excess is partly due to a small portion of the power 7about #ae lbs47 being employed in raising the lower block1 and partly to friction4 ,for example1 in experiment #g1 if there had been no friction and if the lower block were without weight1 a power of #ii lbs4 would have been sufficient2 but1 owing to the presence of these disturbing causes1 #aaje lbs4 are necessary3 of this amount #ae lbs4 is due to the weight of the pulley1 #aj lbs4 is the force of friction1 and the remaining #ii lbs4 raises the load4 #aif4 ,by a calculation based on this table we have ascertained a certain relation between the power and the load2 they are connected by the formula which may be enunciated as follows3 ,the power is found by multiplying the weight of the load into #jedec1 and adding #bba to the product4 ,calling ,p the power and ,r the load1 we may express the relation thus3 ,p #bba #jedec ,r4 ,for example1 in experiment #e1 the product of #adb and #jedec is #ggdc1 to which1 when #bba is added1 we find for ,p #gifd1 very nearly the same as #hj lbs41 the observed value of the power4 ,in the fourth column the values of ,p calculated by means of this formula are given1 and in the last we exhibit the discrepancies between the observed and the calculated values for the sake of comparison4 ,it will be seen that the discrepancy in no case amounts to #je lb41 consequently the formula expresses the experiments very well4 ,the mode of deducing it is given in the ,appendix4 #aig4 ,the quantity #bba is partly that portion of the power expended in overcoming the weight of the moveable pulley1 and partly arises from friction4 #aih4 ,we can readily calculate from the formula how much power will be required to raise a given weight2 for example1 suppose #bjj lbs4 be attached to the moveable pulley1 we find that #aaa lbs4 must be applied as the power4 ,but in order to raise #bjj lbs4 one foot1 the power exerted must act over two feet2 hence the number of foot-pounds required is #b #aaa #bbb4 ,the quantity of energy that is lost is #bb foot-pounds4 ,out of every #bbb foot-pounds applied1 #bjj are usefully employed2 that is to say1 about #ij per cent4 of the applied energy is utilized1 while the remaining #aj per cent4 is lost by friction4 ,,the ,,three-,,sheave ,,pulley-,,block4 #aii4 ,the next arrangement we shall employ is a pair of pulley-blocks ,s ,t1 ,fig4 #ce1 each containing three sheaves1 as the small pulleys are termed4 ,a rope is fastened to the upper block1 ,s2 it then passes down to the lower block ,t under one sheave1 up again to the upper block and over a sheave1 and so on1 as shown in the figure4 ,to the end of the rope from the last of the upper sheaves the power ,h is applied1 and the load ,g is suspended from the hook attached to the lower block4 ,when the rope is pulled1 it gradually raises the lower block2 and to raise the load one foot1 each of the six parts of the rope from the upper block to the lower block must be shortened one foot1 and therefore the power must have pulled out six feet of rope4 ,hence1 for every foot that the load is raised the power must have acted through six feet2 that is to say1 the velocity ratio is #f4 #bjj4 ,if there were no friction1 the power would only be one-sixth of the load4 ,this follows at once from the principles already explained4 ,suppose the load be #fj lbs41 then to raise it one foot would require #fj foot-pounds2 and the power must therefore exert #fj foot-pounds2 but the power moves over six feet1 therefore a power of #aj lbs4 would be sufficient4 ,owing1 however1 to friction1 some energy is lost1 and we must have recourse to experiment in order to test the real efficiency of the machine4 ,the single moveable pulley nearly doubled our power2 we shall prove that the three-sheave pulley-block will quadruple it4 ,in this case we deal with larger weights1 with reference to which we may leave the weight of the lower block out of consideration4 #bja4 ,let us first attach #a cwt4 to the load hook2 we find that #bi lbs4 on the power hook is the smallest weight that can produce motion3 this is only #a lb4 more than one-quarter of the load raised4 ,if #b cwt4 be the load1 we find that #ef lbs4 will just raise it3 this time the power is exactly one-quarter of the load4 ,the experiment has been tried of placing #d cwt4 on the hook2 it is then found that #aji lbs4 will raise it1 which is only #c lbs4 short of #a cwt4 ,these experiments demonstrate that for a three-sheave pulley-block of this construction we may safely apply the rule1 that the power is one-quarter of the load4 #bjb4 ,we are thus enabled to see how much of our exertion in raising weights must be expended in merely overcoming friction1 and how much may be utilized4 ,suppose for example that we have to raise a weight of #ajj lbs4 one foot by means of the pulley-block2 the power we must apply is #be lbs41 and six feet of rope must be drawn out from between the pulleys3 therefore the power exerts #aej foot-pounds of energy4 ,of these only #ajj foot-pounds are usefully employed1 and thus #ej foot-pounds1 one-third of the whole1 have been expended on friction4 ,here we see that notwithstanding a small force overcomes a large one1 there is an actual loss of energy in the machine4 ,the real advantage of course is that by the pulley-block ,i can raise a greater weight than ,i could move without assistance1 but ,i do not create energy2 ,i merely modify it1 and lose by the process4 #bjc4 ,the result of another series of experiments made with this pair of pulley-blocks is given in ,table ,x4 ,,table ,x4--,,three-,,sheave ,,pulley-,,blocks4 ,sheaves cast iron #be diameter2 plaited rope #jbe diameter2 velocity ratio #f2 mechanical advantage #d2 useful effect #fg per cent42 formula ,p #bcf #jbch ,r4 ---------------------------------------- ------------------ ,p4 ,discrepancies ,number of ,r4 ,observed ,calculated between observed ,experiment4 ,load power power and calculated in lbs4 in lbs4 in lbs4 powers4 ---------------------------------------- ------------------ #a #eg #aee #aei #jd #b #aad #bie #bie #jj #c #aga #dce #dca - #jd #d #bbh #efj #eff #jf #e #bha #gjj #fib - #jh #f #cch #hcj #hbh - #jb #g #cie #igj #ifd - #jf #h #deb #ajij #ajii #ji ---------------------------------------- ------------------ #bjd4 ,this table contains five columns2 the weights raised 7shown in the second column7 range up to somewhat over #d cwt4 ,the observed values of the power are given in the third column2 each of these is generally about one-quarter of the corresponding value of the load4 ,there is1 however1 a more accurate rule for finding the power2 it is as follows4 #bje4 ,to find the power necessary to raise a given load1 multiply the loads in lbs4 by #jbch1 and add #bcf lbs4 to the product4 ,we may express the rule by the formula ,p #bcf #jbch ,r4 #bjf4 ,to find the power which would raise #bbh lbs42 the product of #bbh and #jbch is #edbf2 adding #bcf1 we find #eff lbs4 for the power required2 the actual observed power is #ef lbs41 so that the rule is accurate to within about half a pound4 ,in the fourth column will be found the values of ,p calculated by means of this rule4 ,in the fifth column1 the discrepancies between the observed and the calculated values of the powers are given1 and it will be seen that the difference in no case reaches #a lb4 ,of course it will be understood that this formula is only reliable for loads which lie between those employed in the first and last of the experiments4 ,we can calculate the power for any load between #eg lbs4 and #deb lbs41 but for loads much larger than #deb or less than #eg it would probably be better to use the simple fourth of the load rather than the power computed by the formula4 #bjg4 ,i will next perform an experiment with the three-sheave pulley-block1 which will give an insight into the exact amount of friction without calculation by the help of the velocity ratio4 ,we first counterpoise the weight of the lower block by attaching weights to the power4 ,it is found that about #af lbs4 is sufficient for this purpose4 ,i attach a #ef lb4 weight as a load1 and find that #aca lbs4 is sufficient power for motion4 ,this amount is partly composed of the force necessary to raise the load if there were no friction1 and the rest is due to the friction4 ,i next gradually remove the power weights3 when ,i have taken off a pound1 you see the power and the load balance each other2 but when ,i have reduced the power so low as #ee lbs4 7not including the counterpoise for the lower block71 the load is just able to overhaul the power1 and run down4 ,we have therefore proved that a power of #aca lbs4 or greater raises #ef lbs41 that any power between #aca lbs4 and #ee lbs4 balances #ef lbs41 and that any power less than #ee lbs4 is raised by #ef lbs4 ,when the power is raised1 the force of friction1 together with the power1 must be overcome by the load4 ,let us call ,x the real power that would be necessary to balance #ef lbs4 in a perfectly frictionless machine1 and ,y the force of friction4 ,we shall be able to determine ,x and ,y by the experiments just performed4 ,when the load is raised a power equal to ,x ,y must be applied1 and therefore ,x ,y #aca4 ,on the other hand1 when the power is raised1 the force ,x is just sufficient to overcome both the friction ,y and the weight #ee2 therefore ,x ,y #ee4 ,solving this pair of equations1 we find that ,x #ic and ,y #ch4 ,hence we infer that the power in the frictionless machine would be #ic2 but this is exactly what would have been deduced from the velocity ratio1 for #ef #f #ic lbs4 ,in this result we find a perfect accordance between theory and experiment4 ,,the ,,differential ,,pulley-,,block4 #bjh4 ,by increasing the number of sheaves in a pair of pulley-blocks the power may be increased2 but the length of rope 7or chain7 requisite for several sheaves becomes a practical inconvenience4 ,there are also other reasons which make the differential pulley-block1 which we shall now consider1 more convenient for many purposes than the common pulley blocks when a considerable augmentation of power is required4 #bji4 ,the principle of the differential pulley is very ancient1 and in modern times it has been embodied in a machine of practical utility4 ,the object is to secure1 that while the power moves over a considerable distance1 the load shall only be raised a short distance4 ,when this has been attained1 we then know by the principle of energy that we have gained a mechanical advantage4 #baj4 ,let us consider the means by which this is effected in that ingenious contrivance1 ,weston's differential pulley-block4 ,the principle of this machine will be understood from ,fig4 #cf and ,fig4 #cg4 ,illustration3 ,,fig4 #cf4 ,it consists of three parts1--an upper pulley-block1 a moveable pulley1 and an endless chain4 ,we shall briefly describe them4 ,the upper block ,p is furnished with a hook for attachment to a support4 ,the sheave it contains resembles two sheaves1 one a little smaller than the other1 fastened together3 they are in fact one piece4 ,the grooves are provided with ridges1 adapted to prevent the chain from slipping4 ,the lower pulley ,q consists of one sheave1 which is also furnished with a groove2 it carries a hook1 to which the load is attached4 ,the endless chain performs a part that will be understood from the sketch of the principle in ,fig4 #cf4 ,the chain passes from the hand at ,a up to ,l over the larger groove in the upper pulley1 then downwards at ,b1 under the lower pulley1 up again at ,c1 over the smaller groove in the upper pulley at ,a1 and then back again by ,d to the hand at ,a4 ,when the hand pulls the chain downwards1 the two grooves of the upper pulley begin to turn together in the direction shown by the arrows on the chain4 ,the large groove is therefore winding up the chain1 while the smaller groove is lowering4 #baa4 ,in the pulley which has been employed in the experiments to be described1 the effective circumference of the large groove is found to be #aahd1 while that of the small groove is #ajcf4 ,when the upper pulley has made one revolution1 the large groove must have drawn up #aahd of chain1 since the chain cannot slip on account of the ridges2 but in the same time the small groove has lowered #ajcf of chain3 hence when the upper pulley has revolved once1 the chain between the two must have been shortened by the difference between #aahd and #ajcf1 that is by #adh2 but this can only have taken place by raising the moveable pulley through half #adh1 that is1 through a space #jgd4 ,the power has then acted through #aahd1 and has raised the resistance #jgd4 ,the power has therefore moved through a space #af times greater than that through which the load moves4 ,in fact1 it is easy to verify by actual trial that the power must be moved through #af feet in order that the load may be raised #a foot4 ,we express this by saying that the velocity ratio is #af4 #bab4 ,by applying power to the chain at ,d proceeding from the smaller groove1 the chain is lowered by the large groove faster than it is raised by the small one1 and the lower pulley descends4 ,the load is thus raised or lowered by simply pulling one chain ,a or the other ,d4 ,illustration3 ,,fig4 #cg4 #bac4 ,we shall next consider the mechanical efficiency of the differential pulley-block4 ,the block 7,fig4 #cg7 which we shall use is intended to be worked by one man1 and will raise any weight not exceeding a quarter of a ton4 ,we have already learned that with this block the power must act through sixteen feet for the load to be raised one foot4 ,hence1 were it not for friction1 the power need only be the sixteenth part of the load4 ,a few trials will show us that the real efficiency is not so large1 and that in fact more than half the work exerted is merely expended upon overcoming friction4 ,this will lead afterwards to a result of considerable practical importance4 #bad4 ,placing upon the load hook a weight of #bjj lbs41 ,i find that #ch lbs4 attached to a hook fastened on the power chain is sufficient to raise the load2 that is to say1 the power is about one-sixth of the load4 ,if ,i make the load #djj lbs4 ,i find the requisite power to be #fd lbs41 which is only about #c lbs4 less than one-sixth of #djj lbs4 ,we may safely adopt the practical rule1 that with this differential pulley-block a man would be able to raise a weight six times as great as he could raise without such assistance4 #bae4 ,a series of experiments carefully tried with different loads have given the results shown in ,table ,,xi4 ,,table ,,xi4--,,the ,,differential ,,pulley-,,block4 ,circumference of large groove #aahd1 of small groove #ajcf2 velocity ratio #af2 mechanical efficiency #fjg2 useful effect #ch per cent42 formula ,p #chg #jaejh ,r4 ---------------------------------------- --------------- ,p4 ,difference ,number of ,r4 ,observed ,calculatedof the observed ,experiment4 ,load power power and calculated in lbs4 in lbs4 in lbs4 values4 ---------------------------------------- --------------- #a #ef #aj #abc #bc #b #aab #bj #bjh #jh #c #afh #ca #bib - #ah #d #bbd #ch #cgg - #jc #e #bhj #dh #dfa - #ai #f #ccf #ed #edf #jf #g #cib #fd #fca - #ji #h #ddh #gb #gae - #je #i #ejd #hj #hjj #jj #aj #efj #hf #hhd #bd ---------------------------------------- --------------- ,the first column contains the numbers of the experiments1 the second the weights raised1 the third the observed values of the corresponding powers4 ,from these the following rule for finding the power has been obtained3-- #baf4 ,to find the power1 multiply the load by #jaejh1 and add #chg lbs4 to the product2 this rule may be expressed by the formula ,p #chg #jaejh ,r4 7,see ,appendix47 #bag4 ,the calculated values of the powers are given in the fourth column1 and the differences between the observed and calculated values in the last column4 ,the differences do not in any case amount to #be lbs41 and considering that the loads raised are up to a quarter of a ton1 the formula represents the experiments with satisfactory precision4 #bah4 ,suppose for example #bhj lbs4 is to be raised2 the product of #bhj and #jaejh is #dbbb1 to which1 when #chg is added1 we find #dfji to be the requisite power4 ,the mechanical efficiency found by dividing #dfji into #bhj is #fjg4 #bai4 ,to raise #bhj lbs4 one foot #bhj foot-pounds of energy would be necessary1 but in the differential pulley-block #dfji lbs4 must be exerted for a distance of #af feet in order to accomplish this object4 ,the product of #dfji and #af is #gcgd4 ,hence the differential pulley-block requires #gcgd foot-pounds of energy to be applied in order to yield #bhj useful foot-pounds2 but #bhj is only #ch per cent4 of #gcgd1 and therefore with a load of #bhj lbs4 only #ch per cent4 of the energy applied to a differential pulley-block is utilized4 ,in general1 we may state that not more than about #dj per cent4 is profitably used1 and that the remainder is expended in overcoming friction4 #bbj4 ,it is a remarkable and useful property of the differential pulley1 that a weight which has been hoisted will remain suspended when the hand is removed1 even though the chain be not secured in any manner4 ,the pulleys we have previously considered do not possess this convenient property4 ,the weight raised by the three-sheave pulley-block1 for example1 will run down unless the free end of the rope be properly secured4 ,the difference in this respect between these two mechanical powers is not a consequence of any special mechanism2 it is simply caused by the excessive friction in the differential pulley-block4 #bba4 ,the reason why the load does not run down in the differential pulley may be thus explained4 ,let us suppose that a weight of #djj lbs4 is to be raised one foot by the differential pulley-block2 #djj units of work are necessary1 and therefore #a1jjj units of work must be applied to the power chain to produce the #djj units 7since only #dj per cent4 is utilized74 ,the friction will thus have consumed #fjj units of work when the load has been raised one foot4 ,if the power-weight be removed1 the pressure supported by the upper pulley-block is diminished4 ,in fact1 since the power-weight is about _/th of the load1 the pressure on the axle when the power-weight has been removed is only _/ths of its previous value4 ,the friction is nearly proportional to that pressure3 hence when the power has been removed the friction on the upper axle is _/ths of its previous value1 while the friction on the lower pulley remains unaltered4 ,we may therefore assume that the total friction is at least _/ths of what it was before the power-weight was removed4 ,will friction allow the load to descend8 #fjj foot-pounds of work were required to overcome the friction in the ascent3 at least _/ #fjj #ead foot-pounds would be necessary to overcome friction in the descent4 ,but where is this energy to come from8 ,the load in its descent could only yield #djj units1 and thus descent by the mere weight of the load is impossible4 ,to enable the load to descend we have actually to aid the movement by pulling the chain ,d 7,figs4 #cf and #cg71 which proceeds from the small groove in the upper pulley4 #bbb4 ,the principle which we have here established extends to other mechanical powers1 and may be stated generally4 ,whenever more than half the applied energy is consumed by friction1 the load will remain without running down when the machine is left free4 ,,the ,,epicycloidal ,,pulley-,,block4 #bbc4 ,we shall conclude this lecture with some experiments upon a useful mechanical power introduced by ,mr4 ,eade under the name of the epicycloidal pulley-block4 ,it is shown in ,fig4 #cc1 and also in ,fig4 #di4 ,in this machine there are two chains3 one a slight endless chain to which the power is applied2 the other a stout chain which has a hook at each end1 from either of which the load may be suspended4 ,each of these chains passes over a sheave in the block3 these sheaves are connected by an ingenious piece of mechanism which we need not here describe4 ,this mechanism is so contrived that1 when the power causes the sheave to revolve over which the slight chain passes1 the sheave which carries the large chain is also made to revolve1 but very slowly4 #bbd4 ,by actual trial it is ascertained that the power must be exerted through twelve feet and a half in order to raise the load one foot2 the velocity ratio of the machine is therefore #abe4 #bbe4 ,if the machine were frictionless1 its mechanical efficiency would be of course equal to its velocity ratio2 owing to the presence of friction the mechanical efficiency is less than the velocity ratio1 and it will be necessary to make experiments to determine the exact value4 ,i attach to the load hook a weight of #bhj lbs41 and insert a few small hooks into the links of the power chain in order to receive weights3 #ef lbs4 is sufficient to produce motion1 hence the mechanical efficiency is #e4 ,had there been no friction a power of #ef lbs4 would have been capable of overcoming a load of #abe #ef #gjj lbs4 ,thus #gjj units of energy must be applied to the machine in order to perform #bhj units of work4 ,in other words1 only #dj per cent4 of the applied energy is utilized4 #bbf4 ,an extended series of experiments upon the epicycloidal pulley-block is recorded in ,table ,,xii4 ,,table ,,xii4--,,the ,,epicycloidal ,,pulley-,,block4 ,size adapted for lifting weights up to #e cwt42 velocity ratio #abe2 mechanical efficiency #e2 useful effect #dj per cent42 calculated formula ,p #eh #jahe ,r4 ---------------------------------------- --------------- ,p4 ,difference ,number of ,r4 ,observed ,calculatedof the observed ,experiment4 ,load power power and calculated in lbs4 in lbs4 in lbs4 values4 ---------------------------------------- --------------- #a #ef #ae #afb #ab #b #aab #bg #bfe - #je #c #afh #dj #cfi - #ca #d #bbd #dg #dgb #jb #e #bhj #ef #egf #af #f #ccf #ff #fhj #bj #g #cib #gh #ghc #jc #h #ddh #hh #hhf #jf #i #ejd #ajj #iij - #aj #aj #efj #aaj #ajid - #jf ---------------------------------------- --------------- ,the fourth column shows the calculated values of the powers derived from the formula4 ,it will be seen by the last column that the formula represents the experiments with but little error4 #bbg4 ,since #fj per cent4 of energy is consumed by friction1 this machine1 like the differential pulley-block1 sustains its load when the chains are free4 ,the differential pulley-block gives a mechanical efficiency of #f1 while the epicycloidal pulley-block has only a mechanical efficiency of #e1 and so far the former machine has the advantage2 on the other hand1 that the epicycloidal pulley contains but one block1 and that its lifting chain has two hooks1 are practical conveniences strongly in its favour4 ,,lecture ,,viii4 ,,the ,,lever4 ,the ,lever of the ,first ,order4--,the ,lever of the ,second ,order4--,the ,shears4--,the ,lever of the ,third ,order4 ,,the ,,lever ,,of ,,the ,,first ,,order4 #bbh4 ,there are many cases in which a machine for overcoming great resistance is necessary where pulleys would be quite inapplicable4 ,to meet these various demands a correspondingly various number of contrivances has been devised4 ,amongst these the lever in several different forms holds an important place4 #bbi4 ,the lever of the first order will be understood by reference to ,fig4 #ch4 ,it consists of a straight rod1 to one end of which the power is applied by means of the weight ,c4 ,at another point ,b the load is raised1 while at ,a the rod is supported by what is called the fulcrum4 ,in the case represented in the figure the rod is of iron1 #a #a in section and #f' long2 it weighs #ai lbs4 ,the power is produced by a #ef lb4 weight3 the fulcrum consists of a moderately sharp steel edge firmly secured to the framework4 ,the load in this case is replaced by a spring balance ,h1 and the hook of the balance is attached to the frame4 ,the spring is strained by the action of the lever1 and the index records the magnitude of the force produced at the short end4 ,this is the lever with which we shall commence our experiments4 ,illustration3 ,,fig4 #ch4 #bcj4 ,in examining the relation between the power and the load1 the question is a little complicated by the weight of the lever itself 7#ai lbs471 but we shall be able to evade the difficulty by means similar to those employed on a former occasion 7,art4 #fj72 we can counterpoise the weight of the iron bar4 ,this is easily done by applying a hook to the middle of the bar at ,d1 thence carrying a rope over a pulley ,f1 and suspending a weight ,g of #ai lbs4 from its free extremity4 ,thus the bar is balanced1 and we may leave its weight out of consideration4 #bca4 ,we might also adopt another plan analogous to that of ,art4 #ea1 which is however not so convenient4 ,the weight of the bar produces a certain strain upon the spring balance4 ,i may first read off the strain produced by the bar alone1 and then apply the weight ,c and read again4 ,the observed strain is due both to the weight ,c and to the weight of the bar4 ,if ,i subtract the known effect of the bar1 the remainder is the effect of ,c4 ,it is1 however1 less complicated to counterpoise the bar1 and then the strains indicated by the balance are entirely due to the power4 #bcb4 ,the lever is #f' long2 the point ,b is #f from the end1 and ,b ,c is #e' long4 ,b ,c is divided into #e equal portions of #a'2 ,a is at one of these divisions1 #a' distant from ,b1 and ,c is #e' distant from ,b in the figure2 but ,c is capable of being placed at any position1 by simply sliding its ring along the bar4 #bcc4 ,the mode of experimenting is as follows3--,the weight is placed on the bar at the position ,c3 a strain is immediately produced upon ,h2 the spring stretches a little1 and the bar becomes inclined4 ,it may be noticed that the hook of the spring balance passes through the eye of a wire-strainer1 so that by a few turns of the nut upon the strainer the lever can be restored to the horizontal position4 #bcd4 ,the power of #ef lbs4 being #d' from the fulcrum1 while the load is #a' from the fulcrum1 it is found that the strain indicated by the balance is #bbd lbs42 that is1 four times the amount of the power4 ,if the weight be moved1 so as to be #c' from the fulcrum1 the strain is observed to be #afh lbs42 and whatever be the distance of the power from the fulcrum1 we find that the strain produced is obtained by multiplying the magnitude of the power in pounds by the distance expressed in feet1 and fractional parts of a foot4 ,this law may be expressed more generally by stating that the power is to the load as the distance of the load from the fulcrum is to the distance of the power from the fulcrum4 #bce4 ,we can verify this law under varied circumstances4 ,i move the steel edge which forms the fulcrum of the lever until the edge is #b' from ,b1 and secure it in that position4 ,i place the weight ,c at a distance of #c' from the fulcrum4 ,i now find that the strain on the balance is #hd lbs42 but #hd is to #ef as #c is to #b1 and therefore the law is also verified in this instance4 #bcf4 ,there is another aspect in which we may express the relation between the power and the load4 ,the law in this form is thus stated3 ,the power multiplied by its distance from the fulcrum is equal to the load multiplied by its distance from the fulcrum4 ,thus1 in the case we have just considered1 the product of #ef and #c is #afh1 and this is equal to the product of #hd and #b4 ,the distances from the fulcrum are commonly called the arms of the lever1 and the rule is expressed by stating that ,the power multiplied into its arm is equal to the load multiplied into its arm3 hence the load may be found by dividing the product of the power and the power arm by the load arm4 ,this simple law gives a very convenient method of calculating the load1 when we know the power and the distances of the power and the load from the fulcrum4 #bcg4 ,when the power arm is longer than the load arm1 the load is greater than the power2 but when the power arm is shorter than the load arm1 the power is greater than the load4 ,we may regard the strain on the balance as a power which supports the weight1 just as we regard the weight to be a power producing the strain on the balance4 ,we see1 then1 that for the lever of the first order to be efficient as a mechanical power it is necessary that the power arm be longer than the load arm4 #bch4 ,the lever is an extremely simple mechanical power2 it has only one moving part4 ,friction produces but little effect upon it1 so that the laws which we have given may be actually applied in practice1 without making any allowance for friction4 ,in this we notice a marked difference between the lever and the pulley-blocks already described4 #bci4 ,in the lever of the first order we find an excellent machine for augmenting power4 ,a power of #ad lbs4 can by its means overcome a resistance of a hundredweight1 if the power be eight times as far from the fulcrum as the load is from the fulcrum4 ,this principle it is which gives utility to the crowbar4 ,the end of the bar is placed under a heavy stone1 which it is required to raise2 a support near that end serves as a fulcrum1 and then a comparatively small force exerted at the power end will suffice to elevate the stone4 #bdj4 ,the applications of the lever are innumerable4 ,it is used not only for increasing power1 but for modifying and transforming it in various ways4 ,the lever is also used in weighing machines1 the principles of which will be readily understood1 for they are consequences of the law we have explained4 ,into these various appliances it is not our intention to enter at present2 the great majority of them may1 when met with1 be easily understood by the principle we have laid down4 ,,the ,,lever ,,of ,,the ,,second ,,order4 #bda4 ,in the lever of the second order the power is at one end1 the fulcrum at the other end1 and the load lies between the two3 this lever therefore differs from the lever of the first order1 in which the fulcrum lies between the two forces4 ,the relation between the power and the load in the lever of the second order may be studied by the arrangement in ,fig4 #ci4 #bdb4 ,the bar ,a ,c is the same rod of iron #gb #a #a which was used in the former experiment4 ,the fulcrum ,a is a steel edge on which the bar rests2 the power consists of a spring balance ,h1 in the hook of which the end ,c of the bar rests2 the spring balance is sustained by a wire-strainer1 by turning the nut of which the bar may be adjusted horizontally4 ,the part of the bar between the fulcrum ,a and the power ,c is divided into five portions1 each #a' long1 and the points ,a and ,c are each #f distant from the extremities of the bar4 ,the load employed is #ef lbs42 through the ring of this weight the bar passes1 and thus the bar supports the load4 ,the bar is counterpoised by the weight of #ai lbs4 at ,g1 in the manner already explained 7,art4 #bca74 #bdc4 ,the mode of experimenting is as follows3--,let the weight ,b be placed #a' from the fulcrum2 the strain shown by the spring balance is about #aa lbs4 ,if we calculate the value of the power by the rule already given1 we should have found the same result4 ,the product of the load by its distance from the fulcrum is #ef1 the distance of the power from the fulcrum is #e2 hence the value of the power should be #ef #e #aab4 ,illustration3 ,,fig4 #ci4 #bdd4 ,if the weight be placed #b' from the fulcrum the strain is about #bbe lbs4 and it is easy to ascertain that this is the same amount as would have been found by the application of the rule4 ,a similar result would have been obtained if the #ef lb4 weight had been placed upon any other part of the bar2 and hence we may regard the rule proved for the lever of the second order as well as for the lever of the first order3 that1 the power multiplied by its distance from the fulcrum is equal to the load multiplied by its distance from the fulcrum4 ,in the present case the load is uniformly #ef lbs41 while the power by which it is sustained is always less than #ef lbs4 ,illustration3 ,,fig4 #dj4 #bde4 ,the lever of the second order is frequently applied to practical purposes2 one of the most instructive of these applications is illustrated in the shears shown in ,fig4 #dj4 ,the shears consist of two levers of the second order1 which by their united action enable a man to exert a greatly increased force1 sufficient1 for example1 to cut with ease a rod of iron #jbe square4 ,the mode of action is simple4 ,the first lever ,a ,f has a handle at one end ,f1 which is #bb distant from the other end ,a1 where the fulcrum is placed4 ,at a point ,b on this lever1 #ah distant from the fulcrum ,a1 a short link ,b ,c is attached2 the end of the link ,c is jointed to a second lever ,c ,d2 this second lever is #h long2 it forms one edge of the cutting shears1 the other edge being fixed to the framework4 #bdf4 ,i place a rod of iron #jbe square between the jaws of the shears in the position ,e1 the distance ,d ,e being #ce1 and proceed to cut the iron by applying pressure to the handle4 ,let us calculate the amount by which the levers increase the power exerted upon ,f4 ,suppose for example that ,i press downwards on the handle with a force of #aj lbs41 what is the magnitude of the pressure upon the piece of iron8 ,the effect of each lever is to be calculated separately4 ,we may ascertain the power exerted at ,b by the rule of moments already explained2 the product of the power and its arm is #bb #aj #bbj2 this divided by the number of inches1 #ah in the line ,a ,b1 gives a quotient #abb1 and this quotient is the number of pounds pressure which is exerted by means of the link upon the second lever4 ,we proceed in the same manner to find the magnitude of the pressure upon the iron at ,e4 ,the product of #abb and #h is #igf4 ,this is divided by #ce1 and the quotient found is #bgi4 ,hence the exertion of a pressure of #aj lbs4 at ,f produces a pressure of #bgi lbs4 at ,e4 ,in round numbers1 we may say that the pressure is magnified #bh-;fold by means of this combination of levers of the second order4 #bdg4 ,a pressure of #aj lbs4 is not sufficient to shear across the bar of iron1 even though it be magnified to #bgi lbs4 ,i therefore suspend weights from ,f1 and gradually increase the load until the bar is cut4 ,i find at the first trial that #aab lbs4 is sufficient1 and a second trial with the same bar gives #aad lbs42 #aac lbs41 the mean between these results1 may be considered an adequate force4 ,this is the load on ,f2 the real pressure on the bar is #aac #bgi #caec lbs43 thus the actual pressure which was necessary to cut the bar amounted to more than a ton4 #bdh4 ,we can calculate from this experiment the amount of force necessary to shear across a bar one square inch in section4 ,we may reasonably suppose that the necessary power is proportional to the section1 and therefore the power will bear to #caec lbs4 the proportion which a square of one inch bears to the square of a quarter inch2 but this ratio is #af3 hence the force is #af #caec lbs41 equal to about #bbe tons4 #bdi4 ,it is noticeable that #bbe tons is nearly the force which would suffice to tear the bar in sunder by actual tension4 ,we shall subsequently return to the subject of shearing iron in the lecture upon ,inertia 7,lecture ,,xvi474 ,,the ,,lever ,,of ,,the ,,third ,,order4 #bej4 ,the lever of the third order may be easily understood from ,fig4 #ci1 of which we have already made use4 ,in the lever of the third order the fulcrum is at one end1 the load is at the other end1 while the power lies between the two4 ,in this case1 then1 the power is represented by the #ef lb4 weight1 while the load is indicated by the spring balance4 ,the power always exceeds the load1 and consequently this lever is to be used where speed is to be gained instead of power4 ,thus1 for example2 when the power1 #ef lbs41 is #b' distant from the fulcrum1 the load indicated by the spring balance is about #bc lbs4 #bea4 ,the treadle of a grindstone is often a lever of the third order4 ,the fulcrum is at one end1 the load is at the other end1 and the foot has only to move through a small distance4 #beb4 ,the principles which have been discussed in ,lecture ,,iii4 with respect to parallel forces explain the laws now laid down for levers of different orders1 and will also enable us to express these laws more concisely4 #bec4 ,a comparison between ,figs4 #bj and #ci shows that the only difference between the contrivances is that in ,fig4 #bj we have a spring balance ,c in the same place as the steel edge ,a in ,fig #ci4 ,we may in ,fig4 #bj regard one spring balance as the power1 the other as the fulcrum1 and the weight as the load4 ,nor is there any essential difference between the apparatus of ,fig4 #ch and that of ,fig4 #bj4 ,in ,fig4 #ch the bar is pulled down by a force at each end1 one a weight1 the other a spring balance1 while it is supported by the upward pressure of the steel edge4 ,in ,fig4 #bj the bar is being pulled upwards by a force at each end1 and downwards by the weight4 ,the two cases are substantially the same4 ,in each of them we find a bar acted upon by a pair of parallel forces applied at its extremities1 and retained in equilibrium by a third force4 #bed4 ,we may therefore apply to the lever the principles of parallel forces already explained4 ,we showed that two parallel forces acting upon a bar could be compounded into a resultant1 applied at a certain point of the bar4 ,we have defined the moment of a force 7,art4 #fd71 and proved that the moments of two parallel forces about the point of application of their resultant are equal4 #bee4 ,in the lever of the first order there are two parallel forces1 one at each end2 these are compounded into a resultant1 and it is necessary that this resultant be applied exactly over the steel edge or fulcrum in order that the bar may be maintained at rest4 ,in the levers of the second and third orders1 the power and the load are two parallel forces acting in opposite directions2 their resultant1 therefore1 does not lie between the forces1 but is applied on the side of the greater1 and at the point where the steel edge supports the bar4 ,in all cases the moment of one of the forces about the fulcrum must be equal to that of the other4 ,from the equality of moments it follows that the product of the power and the distance of the power from the fulcrum equals the product of the load1 and the distance of the load from the fulcrum3 this principle suffices to demonstrate the rules already given4 #bef4 ,the laws governing the lever may be deduced from the principle of work2 the load1 if nearer than the power to the fulcrum1 is moved through a smaller distance than the power4 ,thus1 for example1 in the lever of the first order3 if the load be #ab times as far as the power from the fulcrum1 then for every inch the load moves it can be demonstrated that the power must move #ab inches4 ,the number of units of work applied at one end of a machine is equal to the number yielded at the other1 always excepting the loss due to friction1 which is1 however1 so small in the lever that we may neglect it4 ,if then a power of #a lb4 be applied to move the power end through #ab inches1 one unit of work will have been put into the machine4 ,hence one unit of work must be done on the load1 but the load only moves through _/ of a foot1 and therefore a load of #ab lbs4 could be overcome3 this is the same result as would be given by the rule 7,art4 #bcf74 #beg4 ,to conclude3 we have first determined by actual experiment the relation between the power and the load in the lever2 we have seen that the law thus obtained harmonizes with the principle of the composition of parallel forces2 and1 finally1 we have shown how the same result can be deduced from the fertile and important principle of work4 ,,lecture ,,ix4 ,,the ,,inclined ,,plane ,,and ,,the ,,screw4 ,the ,inclined ,plane without ,friction4--,the ,inclined ,plane with ,friction4--,the ,screw4--,the ,screw-jack4--,the ,bolt and ,nut4 ,,the ,,inclined ,,plane ,,without ,,friction4 #beh4 ,the mechanical powers now to be considered are often used for other purposes beside those of raising great weights4 ,for example3 the parts of a structure have to be forcibly drawn together1 a powerful compression has to be exerted1 a mass of timber or other material has to be riven asunder by splitting4 ,for purposes of this kind the inclined plane in its various forms1 and the screw1 are of the greatest use4 ,the screw also1 in the form of the screw-jack1 is sometimes used in raising weights4 ,it is principally convenient when the weight is enormously great1 and the distance through which it has to be raised comparatively small4 ,illustration3 ,,fig4 #da4 #bei4 ,we shall commence with the study of the inclined plane4 ,the apparatus used is shown in ,fig4 #da4 ,a ,b is a plate of glass #d' long1 mounted on a frame and turning round a hinge at ,a2 ,b ,d is a circular arc1 with its centre at ,a1 by which the glass may be supported2 ,d ,c is a vertical rod1 to which the pulley ,c is clamped4 ,this pulley can be moved up and down1 to be accommodated to the position of ,a ,b2 the pulley is made of brass1 and turns very freely4 ,a little truck ,r is adapted to run on the plane of glass4 ,the truck is laden to weigh #a lb41 and this weight is unaltered throughout the experiments2 the wheels are very free1 so that the truck runs with but little friction4 #bfj4 ,but the friction1 though small1 is appreciable1 and it will be necessary to measure the amount and then endeavour to counteract its effect upon the motion4 ,the silk cord attached to the truck is very fine1 and its weight is neglected4 ,a series of weights is provided2 they are made from pieces of brass wire1 and weigh #ja lb4 and #jja lb43 these can easily be hooked into the loop on the cord at ,p4 ,we first make the plane ,a ,b horizontal1 and bring down the pulley ,c so that the cord shall be parallel to the plane2 we find that a force must be applied by the cord in order to draw the truck along the plane3 this force is of course the friction1 and by a suitable weight at ,p the friction may be said to be counterbalanced4 ,but we cannot expect that the friction will be the same when the plane is horizontal as when the plane is inclined4 ,we must therefore examine this question by a method analogous to that used in ,art4 #bjg4 #bfa4 ,let the plane be elevated until ,b ,e1 the elevation of ,b above ,a ,d1 is #bj2 let ,c be properly adjusted3 it is found that when ,p is ,o#de lb4 ,r is just pulled up2 and on the other hand1 when ,p is only #jdj lb4 the truck descends and raises ,p2 and when ,p has any value intermediate between these two1 the truck remains in equilibrium4 ,let us denote the force of gravity acting down the plane by ,r1 and it follows that ,r must be #jdbe lb41 and the friction #jjbe lb4 ,for when ,p raises ,r1 it must overcome friction as well as ,r2 therefore the power must be #jjbe #jdbe #jde4 ,on the other hand1 when ,r raises ,p1 it must also overcome the friction #jjbe1 therefore ,p can only be #jdbe - #jjbe #jdj2 and ,r is thus found to be a mean between the greatest and least values of ,p consistent with equilibrium4 ,if the plane be raised so that the height ,b ,e is #cc1 the greatest and least values of ,p are #jff and #jga2 therefore ,r is #jfhe friction #jjbe1 the same as before4 ,finally1 making the height ,b ,e only #b1 the friction is found to be #jjbj1 which is not much less than the previous determinations4 ,these experiments show that we may consider this very small friction to be practically constant at these inclinations4 7,were the friction large1 other methods are necessary1 see ,art4 #bfe47 ,as in the experiments ,r is always raised we shall give ,p the permanent load of #jjbe lb41 thus sufficiently counteracting friction1 which we may therefore dismiss from consideration4 ,it is hardly necessary to remark that1 in afterwards recording the weights placed at ,p1 this counterpoise is not to be included4 #bfb4 ,we have now the means of studying the relation between the power and the load in the frictionless inclined plane4 ,the incline being set at different elevations1 we shall observe the force necessary to draw up the constant load of #a lb4 ,our course will be guided by first making use of the principle of energy4 ,suppose ,b ,e to be #b'2 when the truck has been moved from the bottom of the plane to the top1 it will have been raised vertically through a height of #b'1 and two units of energy must have been consumed4 ,but the plane being #d' long1 the force which draws up the truck need only be #je lb41 for #je lb4 acting over #d' produces two units of work4 ,in general1 if l be the length of the plane and h its height1 ,r the load1 and ,p the power1 the number of units of energy necessary to raise the load is ,r h1 and the number of units expended in pulling it up the plane is ,p l3 hence ,r h ,p l1 and consequently ,p 3 h 33 ,r 3 l2 that is1 the power is to the height of the plane as the load is to its length4 ,in the present case ,r #a lb41 l #dh2 therefore ,p #jjbjh h1 where h is the height of the plane in inches1 and ,p the power in pounds4 #bfc4 ,we compare the powers calculated by this formula with the actual observed values3 the result is given in ,table ,,xiii4 ,,table ,,xiii4--,,inclined ,,plane4 ,glass ,plane #dh long1 truck #a lb4 in weight1 friction counterpoised2 formula ,p #jjbjh h4 ---------------------------------------- --------------------- ,number of ,height,observed ,p4 ,difference of the ,experiment4 of power ,calculated observed and calculated plane4 in lbs4power in lbs4 powers ---------------------------------------- --------------------- #a #b #jjd #jjd #jjj #b #d #jjh #jjh #jjj #c #f #jac #jab -#jja #d #h #jaf #jag #jja #e #aj #jba #jba #jjj #f #ae #jca #jca #jjj #g #bj #jdb #jdb #jjj #h #cc #jga #jfi -#jjb ---------------------------------------- --------------------- ,thus for example1 in experiment #f1 where the height ,b ,e is #ae1 it is observed that the power necessary to draw the truck is #jca lb4 ,the truck is placed in the middle of the plane1 and the power is adjusted so as to be sufficient to draw the truck to the top with certainty2 the necessary power calculated by the formula is also #jca lbs41 so that the theory is verified4 #bfd4 ,the fifth column of the table shows the difference between the observed and the calculated powers4 ,the very slight differences1 in no case exceeding the fiftieth part of a pound1 may be referred to the inevitable errors of experiment4 ,,the ,,inclined ,,plane ,,with ,,friction4 #bfe4 ,the friction of the truck upon the glass plate is always very small1 and is shown to have but little variation at those inclinations of the plane which we used4 ,but when the friction is large1 we shall not be justified in neglecting its changes at different elevations1 and we must adopt more rigorous methods4 ,for this inquiry we shall use the pine plank and slide already described in ,art4 #aag4 ,we do not in this case attempt to diminish friction by the aid of wheels1 and consequently it will be of considerable amount4 #bff4 ,in another respect the experiments of ,table ,,xiii4 are also in contrast with those now to be described4 ,in the former the load was constant1 while the elevation was changed4 ,in the latter the elevation remains constant while a succession of different loads are tried4 ,we shall find in this inquiry also that when the proper allowance has been made for friction1 the theoretical law connecting the power and the load is fully verified4 #bfg4 ,the apparatus used is shown in ,fig4 #cc2 the plane1 is1 however1 secured at one inclination1 and the pulley ,c shown in ,fig4 #cb is adjusted to the apparatus1 so that the rope from the pulley to the slide is parallel to the incline4 ,the elevation of the plane in the position adopted is #agb1 so that its length1 base1 and height are in the proportions of the numbers #a1 #jiee1 and #jbif4 ,weights ranging from #g lbs4 to #ef lbs4 are placed upon the slide1 and the power is found which1 when the slide is started by the screw1 will draw it steadily up the plane4 ,the requisite power consists of two parts1 that which is necessary to overcome gravity acting down the plane1 and that which is necessary to overcome friction4 ,illustration3 ,,fig4 #db4 #bfh4 ,the forces are shown in ,fig4 #db4 ,r ,g1 the force of gravity1 is resolved into ,r ,l and ,r ,m2 ,r ,l is evidently the component acting down the plane1 and ,r ,m the pressure against the plane2 the triangle ,g ,l ,r is similar to ,a ,b ,c1 hence if ,r be the load1 the force ,r ,l acting down the plane must be #jbif ,r1 and the pressure upon the plane #jiee ,r4 #bfi4 ,we shall first make a calculation with the ordinary law that the friction is proportional to the pressure4 ,the pressure upon the plane ,a ,b1 to which the friction is proportional1 is not the weight of the load4 ,the pressure is that component 7,r ,m7 of the load which is perpendicular to the plane ,a ,b4 ,when the weights do not extend beyond #ef lbs41 the best value for the coefficient of friction is #jbhh 7,art4 #ada73 hence the amount of friction upon the plane is #jbhh #jiee ,r #jbge ,r4 ,this force must be overcome in addition to #jbif ,r 7the component of gravity acting down the plane73 hence the expression for the power is #jbge ,r #jbif ,r #jega ,r4 #bgj4 ,the values of the observed powers compared with the powers calculated from the expression #jega ,r are shown in ,table ,,xiv4 ,,table ,,xiv4--,,inclined ,,plane4 ,smooth plane of pine #gb #aa2 angle of inclination #agb2 slide of pine1 grain crosswise2 slide started2 formula ,p #jega ,r4 ---------------------------------------- --------------- ,r4 ,power in ,p4 ,difference of ,number of ,total loadlbs4 which,calculated the observed ,experiment4 on slide just draws value of and calculated in lbs4 up slide4the power4 powers4 ---------------------------------------- --------------- #a #g #df #dj -#jf #b #ad #hc #hj -#jc #c #ba #abc #abj -#jc #d #bh #afe #afj -#je #e #ce #bjj #bjj #jj #f #db #bdb #bdj -#jb #g #di #bhj #bhj #jj #h #ef #cah #cbj #jb ---------------------------------------- --------------- #bga4 ,thus for example1 in experiment #f1 a load of #db lbs4 was raised by a force of #bdb lbs41 while the calculated value is #bdj lbs42 the difference1 #jb lbs41 is shown in the last column4 #bgb4 ,the calculated values are found to agree tolerably well with the observed values1 but the presence of the large differences in ,no4 #a and ,no4 #d leads us to inquire whether by employing the more accurate law of friction 7,art4 #ada7 a better result may not be obtained4 ,in ,table ,,vi4 we have shown that the friction for weights not exceeding #ef lbs4 is expressed by the formula ,f #ji #jbff pressure1 but the pressure is in this case #jiee ,r1 and hence the friction is #ji #jbed ,r4 ,to this must be added #jbif ,r1 the component of the force of gravity which must be overcome1 and hence the total force necessary is #ji #jee ,r4 ,the powers calculated from this expression are compared with those actually observed in ,table ,,xv4 ,,table ,,xv4--,,inclined ,,plane4 ,smooth plane of pine #gb #aa2 angle of inclination #agb2 slide of pine1 grain crosswise2 slide started2 formula ,p #ji #jee ,r4 ---------------------------------------- --------------- ,r4 ,power in ,p4 ,difference of ,number of ,total loadlbs4 which,calculated the observed ,experiment4 on slide just draws value of and calculated in lbs4 up slide4the power4 powers4 ---------------------------------------- --------------- #a #g #df #dg #ja #b #ad #hc #hf #jc #c #ba #abc #abe #jb #d #bh #afe #afc -#jb #e #ce #bjj #bja #ja #f #db #bdb #bdj -#jb #g #di #bhj #bgh -#jb #h #ef #cah #cag -#ja ---------------------------------------- --------------- ,for example3 in experiment #e1 a load of #ce lbs4 is found to be raised by a power of #bjj lbs41 while the calculated power is #ji #jee #ce #bja lbs4 #bgc4 ,the calculated values of the powers are shown by this table to agree extremely well with the observed values1 the greatest difference being only ,o#c lb4 ,hence there can be no doubt that the principles on which the formula has been calculated are correct4 ,this table may therefore be regarded as verifying both the law of friction1 and the rule laid down for the relation between the power and the load in the inclined plane4 #bgd4 ,the inclined plane is properly styled a mechanical power4 ,for let the weight be #cj lbs41 we calculate by the formula that #agd lbs4 would be sufficient to raise it1 so that1 notwithstanding the loss by friction1 we have here a smaller force overcoming a larger one1 which is the essential feature of a mechanical power4 ,the mechanical efficiency is #cj #agd #agb4 #bge4 ,the velocity ratio in the inclined plane is the ratio of the distance through which the power moves to the height through which the weight is raised1 that is #a #jbif #cch4 ,to raise #cj lbs4 one foot1 a force of #agd lbs4 must therefore be exerted through #cch feet4 ,the number of units of work expended is thus #agd #cch #ehh4 ,of this #cj units1 equivalent to #ea per cent41 are utilized4 ,the remaining #bhh units1 or #di per cent41 are absorbed by friction4 #bgf4 ,we have pointed out in ,art4 #bbb that a machine in which less than half the energy is lost by friction will permit the load to run down when free3 this is the case in the present instance2 hence the weight will run down the plane unless specially restrained4 ,that it should do so agrees with ,art4 #adg1 for it was there shown that at about #acd1 and still more at any greater inclination1 the slide would descend when started4 ,,the ,,screw4 #bgg4 ,the inclined plane as a mechanical power is often used in the form of a wedge or in the still more disguised form of a screw4 ,a wedge is an inclined plane which is forced under the load2 it is usually moved by means of a hammer1 so that the efficiency of the wedge is augmented by the dynamical effect of a blow4 #bgh4 ,the screw is one of the most useful mechanical powers which we possess4 ,its form may be traced by wrapping a wedge-shaped piece of paper around a cylinder1 and then cutting a groove in the cylinder along the spiral line indicated by the margin of the paper4 ,such a groove is a screw4 ,in order that the screw may be used it must revolve in a nut which is made from a hollow cylinder1 the internal diameter of which is equal to that of the cylinder from which the screw is cut4 ,the nut contains a spiral ridge1 which fits into the corresponding thread in the screw2 when the nut is turned round1 it moves backwards or forwards according to the direction of the rotation4 ,large screws of the better class1 such as those upon which we shall first make experiments1 are always turned in a lathe1 and are thus formed with extreme accuracy4 ,small screws are made in a simpler manner by means of dies and other contrivances4 #bgi4 ,a characteristic feature of a screw is the inclination of the thread to the axis4 ,this is most conveniently described by the number of complete turns which the thread makes in a specified length of the screw1 usually an inch4 ,for example3 a screw is said to have ten threads to the inch when it requires #aj revolutions of the nut in order to move it one inch4 ,the shape of the thread itself varies with the purposes for which the screw is intended2 the section may be square or triangular1 or1 as is generally the case in small screws1 of a rounded form4 #bhj4 ,there is so much friction in the screw that experiments are necessary for the determination of the law connecting the power and the load4 #bha4 ,we shall commence with an examination of the screw by the apparatus shown in ,fig4 #dc4 ,the nut ,a is secured upon a stout frame2 to the end of the screw hooks are attached1 in order to receive the load1 which in this apparatus does not exceed #bbd lbs42 at the top of the screw is an arm ,e by which the screw is turned2 to the end of the arm a rope is attached1 which passing over a pulley ,d1 carries a hook for receiving the power ,c4 ,illustration3 ,,fig4 #dc4 #bhb4 ,we first apply the principle of work to this screw1 and calculate the relation between the power and the load as it would be found if friction were absent4 ,the diameter of the circle described by the end of the arm is #bje2 its circumference is therefore #fdd4 ,the screw contains three threads in the inch1 hence in order to raise the load #a the power moves #c #fdd #aic very nearly2 therefore the velocity ratio is #aic1 and were the screw capable of working without friction1 #aic would represent the mechanical efficiency4 ,in actually performing the experiments the arm ,e is placed at right angles to the rope leading to the pulley1 and the power hook is weighted until1 with a slight start1 the arm is steadily drawn2 but the power will only move the arm a few inches1 for when the cord ceases to be perpendicular to the arm the power acts with diminished efficiency2 consequently the load is only raised in each experiment through a small fraction of an inch1 but quite sufficient for our purpose4 ,,table ,,xvi4--,,the ,,screw4 ,wrought iron screw1 square thread1 diameter #abe1 with #c threads to the inch1 length of arm #ajbe2 nut of cast iron1 bearing surfaces oiled1 velocity ratio #aic1 useful effect #cf per cent41 mechanical efficiency #gj2 formula ,p #jjadc ,r4 ---------------------------------------- -------------------- ,p4 ,difference of ,number of ,r4 ,observed ,calculated the observed and ,experiment4 ,load power power calculated in lbs4 in lbs4 in lbs4 powers4 ---------------------------------------- -------------------- #a #bh #jd #jd #jj #b #ef #jh #jh #jj #c #hd #ab #ab #jj #d #aab #af #af #jj #e #adj #bj #bj #jj #f #afh #bd #bd #jj #g #aif #bg #bh #ja #h #bbd #cc #cb -#ja ---------------------------------------- -------------------- #bhc4 ,the results of the experiments are shown in ,table ,,xvi4 ,if the motion had not been aided by a start the powers would have been greater4 ,thus in experiment #f1 #bd lbs4 is the power with a start1 when without a start #cb lbs4 was found to be necessary4 ,the experiments have all been aided by a start1 and the results recorded have been corrected for the friction of the pulley over which the rope passes3 this correction is very small1 in no case exceeding #jb lb4 ,the fourth column contains the values of the powers computed by the formula ,p #jjadc ,r4 ,this formula has been deduced from the observations in the manner described in the ,appendix4 ,the fifth column proves that the experiments are truly represented by the formula3 in each of the experiments #g and #h1 the difference between the calculated and observed values amounts to #ja lb41 but this is quite inconsiderable in comparison with the weights we are employing4 #bhd4 ,in order to lift #ajj lbs4 the expression #jjadc ,r shows that #adc lbs4 would be necessary3 hence the mechanical efficiency of the screw is #ajj #adc #gj4 ,thus this screw is vastly more powerful than any of the pulley systems which we have discussed4 ,a machine so capable1 so compact1 and so strong as the screw1 is invaluable for innumerable purposes in the ,arts1 as well as in multitudes of appliances in daily use4 #bhe4 ,it is evident1 however1 that the distance through which the screw can raise a weight must be limited by the length of the screw itself1 and that in the length of lift the screw cannot compete with many of the other contrivances used in raising weights4 #bhf4 ,we have seen that the velocity ratio is #aic2 therefore1 to raise #ajj lbs4 #a foot1 we find that #adc #aic #bgf units of energy must be expended3 of this only #ajj units1 or #cf per cent41 is usefully employed2 the rest being consumed in overcoming the friction of the screw4 ,thus nearly two-thirds of the energy applied to such a screw is wasted4 ,hence we find that friction does not permit the load to run down1 since less than fifty per cent4 of the applied energy is usefully employed 7,art4 #bbb74 ,this is one of the valuable properties which the screw possesses4 #bhg4 ,we may contrast the screw with the pulley-block 7,art4 #aii74 ,they are both powerful machines3 the latter is bulky and economical of power1 the former is compact and wasteful of power2 the latter is adapted for raising weights through considerable distances1 and the former for exerting pressures through short distances4 ,illustration3 ,,fig4 #dd4 ,,the ,,screw-,,jack4 #bhh4 ,the importance of the screw as a mechanical power justifies us in examining another of its useful forms1 the screw-jack4 ,this machine is used for exerting great pressures1 such for example as starting a ship which is reluctant to be launched1 or replacing a locomotive upon the line from which its wheels have slipped4 ,these machines vary in form1 as well as in the weights for which they are adapted2 one of them is shown at ,d in ,fig4 #dd1 and a description of its details is given in ,table ,,xvii4 ,we shall determine the powers to be applied to this machine for overcoming resistances not exceeding half a ton4 #bhi4 ,to employ weights so large as half a ton would be inconvenient if not actually impossible in the lecture room1 but the required pressures can be produced by means of a lever4 ,in ,fig4 #dd is shown a stout wooden bar #af' long4 ,it is prevented from bending by means of a chain2 at ,e the lever is attached to a hinge1 about which it turns freely2 at ,a a tray is placed for the purpose of receiving weights4 ,the screw-jack is #b' distant from ,e1 consequently the bar is a lever of the second order1 and any weight placed in the tray exerts a pressure eightfold greater upon the top of the screw-jack4 ,thus each stone in the tray produces a pressure of #a cwt4 at the point ,d4 ,the weight of the lever and the tray is counterpoised by the weight ,c1 so that until the tray receives a load there is no pressure upon the top of the screw-jack1 and thus we may omit the lever itself from consideration4 ,the screw-jack is furnished with an arm ,d ,g2 at the extremity ,g of this arm a rope is attached1 which passes over a pulley and supports the power ,b4 #bij4 ,the velocity ratio for this screw-jack with an arm of #cc1 is found to be #dad1 by the method already described 7,art4 #bhc74 #bia4 ,to determine its mechanical efficiency we must resort to experiment4 ,the result is given in ,table ,,xvii4 ,,table ,,xvii4--,,the ,,screw-,,jack4 ,wrought iron screw1 square thread1 diameter #b1 pitch #b threads to the inch1 arm #cc2 nut brass1 bearing surfaces oiled2 velocity ratio #dad2 useful effect1 #bh per cent42 mechanical efficiency #aaf2 formula ,p #jff #jjjge ,r4 ---------------------------------------- --------------- ,p4 ,difference of the ,number of ,r4 ,observed ,calculated observed and ,experiment4 ,load power power calculated in lbs4 in lbs4 in lbs4 powers4 ---------------------------------------- --------------- #a #aab #ad #ae #ja #b #bbd #bb #bc #ja #c #ccf #cc #cb -#ja #d #ddh #da #dj -#ja #e #efj #ej #di -#ja #f #fgb #eg #eg #jj #g #ghd #fe #fe #jj #h #hif #gd #gd #jj #i #ajjh #ha #hb #ja #aj #aabj #ij #ia #ja ---------------------------------------- --------------- #bib4 ,it may be seen from the column of differences how closely the experiments are represented by the formula4 ,the power which is required to raise a given weight1 say #fjj lbs41 may be calculated by this formula2 it is #jff #jjjge #fjj #eaf4 ,hence the mechanical efficiency of the screw-jack is #fjj #eaf #aaf4 ,thus the screw is very powerful1 increasing the force applied to it more than a hundredfold4 ,in order to raise #fjj lbs4 one foot1 a quantity of work represented by #eaf #dad #bacf units must be expended2 of this only #fjj1 or #bh per cent41 is utilized1 so that nearly three-quarters of the energy applied is expended upon friction4 #bic4 ,this screw does not let the load run down1 since less than #ej per cent4 of energy is utilised2 to lower the weight the lever has actually to be pressed backwards4 #bid4 ,the details of an experiment on this subject will be instructive1 and afford a confirmation of the principles laid down4 ,in experiment #aj we find that #ij lbs4 suffice to raise #a1abj lbs42 now by moving the pulley to the other side of the lever1 and placing the rope perpendicularly to the lever1 ,i find that to produce motion the other way--that is1 of course to lower the screw--a force of #cd lbs4 must be applied4 ,hence1 even with the assistance of the load1 a force of #cd lbs4 is necessary to overcome friction4 ,this will enable us to determine the amount of friction in the same manner as we determined the friction in the pulley-block 7,art4 #bjg74 ,let ,x be the force usefully employed in raising1 and ,y the force of friction1 which acts equally in either direction against the production of motion2 then to raise the load the power applied must be sufficient to overcome both ,x and ,y1 and therefore we have ,x ,y #ij4 ,when the weight is to be lowered the force ,x of course aids in the lowering1 but ,x alone is not sufficient to overcome the friction2 it requires the addition of #cd lbs41 and we have therefore ,x #cd ,y1 and hence ,x #bh1 ,y #fb4 ,that is1 #bh is the amount of force which with a frictionless screw would have been sufficient to raise half a ton4 ,but in the frictionless screw the power is found by dividing the load by the velocity ratio4 ,in this case #aabj #dad #bg1 which is within #ja lb4 of the value of ,x4 ,the agreement of these results is satisfactory4 ,,the ,,screw ,,bolt ,,and ,,nut4 ,illustration3 ,,fig4 #de4 #bie4 ,one of the most useful applications of the screw is met with in the common bolt and nut1 shown in ,fig4 #de4 ,it consists of a wrought iron rod with a head at one end and a screw on the other1 upon which the nut works4 ,bolts in many different sizes and forms represent the stitches by which machines and frames are most readily united4 ,there are several reasons why the bolt is so convenient4 ,it draws the parts into close contact with tremendous force2 it is itself so strong that the parts united practically form one piece4 ,it can be adjusted quickly1 and removed as readily4 ,the same bolt by the use of washers can be applied to pieces of very different sizes4 ,no skilled hand is required to use the simple tool that turns the nut4 ,adding to this that bolts are cheap and durable1 we shall easily understand why they are so extensively used4 #bif4 ,we must remark in conclusion that the bolt owes its utility to friction2 screws of this kind do not overhaul1 hence when the nut is screwed home it does not recoil4 ,if it were not that more than half the power applied to a screw is consumed in friction1 the bolt and the nut would either be rendered useless1 or at least would require to be furnished with some complicated apparatus for preventing the motion of the nut4 ,,lecture ,x4 ,,the ,,wheel ,,and ,,axle4 ,introduction4--,experiments upon the ,wheel and ,axle4--,friction upon the ,axle4--,the ,wheel and ,barrel4--,the ,wheel and ,pinion4--,the ,crane4--,conclusion4 ,,introduction4 #big4 ,the mechanical powers discussed in these lectures may be grouped into two classes1--the first where ropes or chains are used1 and the second where ropes or chains are absent4 ,belonging to that class in which ropes are not employed1 we have the screw discussed in the last lecture2 and the lever discussed in ,lecture ,,viii42 while among those machines in which ropes or chains form an essential part of the apparatus1 the pulley and the wheel and axle hold a prominent place4 ,we have already examined several forms of the pulley1 and we now proceed to the not less important subject of the wheel and axle4 #bih4 ,where great resistances have to be overcome1 but where the distance through which the resistance must be urged is short1 the lever or the screw is generally found to be the most appropriate means of increasing power4 ,when1 however1 the resistance has to be moved a considerable distance1 the aid of the pulley1 or the wheel and axle1 or sometimes of both combined1 is called in4 ,the wheel and axle is the form of mechanical power which is generally used when the distance is considerable through which a weight must be raised1 or through which some resistance must be overcome4 ,illustration3 ,,fig4 #df4 #bii4 ,the wheel and axle assumes very many forms corresponding to the various purposes to which it is applied4 ,the general form of the arrangement will be understood from ,fig4 #df4 ,it consists of an iron axle ,b1 mounted in bearings1 so as to be capable of turning freely2 to this axle a rope is fastened1 and at the extremity of the rope is a weight ,d1 which is gradually raised as the axle revolves4 ,attached to the axle1 and turning with it1 is a wheel ,a with hooks in its circumference1 upon which lies a rope2 one end of this rope is attached to the circumference of the wheel1 and the other supports a weight ,e4 ,this latter weight may be called the power1 while the weight ,d suspended from the axle is the load4 ,when the power is sufficiently large1 ,e descends1 making the wheel to revolve2 the wheel causes the axle to revolve1 and thus the rope is wound up and the load ,d is raised4 #cjj4 ,when compared with the differential pulley as a means of raising a weight1 this arrangement appears rather bulky and otherwise inconvenient1 but1 as we shall presently learn1 it is a far more economical means of applying energy4 ,in its practical application1 moreover1 the arrangement is simplified in various ways1 two of which may be mentioned4 #cja4 ,the capstan is essentially a wheel and axle2 the power is not in this case applied by means of a rope1 but by direct pressure on the part of the men working it2 nor is there actually a wheel employed1 for the pressure is applied to what would be the extremities of the spokes of the wheel if a wheel existed4 #cjb4 ,in the ordinary winch1 the power of the labourer is directly applied to the handle which moves round in the circumference of a circle4 #cjc4 ,there are innumerable other applications of the principle which are constantly met with1 and which can be easily understood with a little attention4 ,these we shall not stop to describe1 but we pass on at once to the important question of the relation between the power and the load4 ,,experiments ,,upon ,,the ,,wheel ,,and ,,axle4 #cjd4 ,we shall commence a series of experiments upon the wheel ,a and axle ,b of ,fig4 #df4 ,we shall first determine the velocity ratio1 and then ascertain the mechanical efficiency by actual experiment4 ,the wheel is of wood2 it is about #cj in diameter4 ,the string to which the power is attached is coiled round a series of hooks1 placed near the margin of the wheel2 the effective circumference is thus a little less than the real circumference4 ,i measure a single coil of the string and find the length to be #hhe4 ,this length1 therefore1 we shall adopt for the effective circumference of the wheel4 ,the axle is #jge in diameter1 but its effective circumference is larger than the circle of which this length is the diameter4 #cje4 ,the proper mode of finding the effective circumference of the axle in a case where the rope bears a considerable proportion to the axle is as follows4 ,attach a weight to the extremity of the rope sufficient to stretch it thoroughly4 ,make the wheel and axle revolve suppose #bj times1 and measure the height through which the weight is lifted2 then the one-twentieth part of that height is the effective circumference of the axle4 ,by this means ,i find the circumference of the axle we are using to be #bhg4 #cjf4 ,we can now ascertain the velocity ratio in this machine4 ,when the wheel and axle have made one complete revolution the power has been lowered through a distance of #hhe1 and the load has been raised through #bhg4 ,this is evident because the wheel and axle are attached together1 and therefore each completes one revolution in the same time2 hence the ratio of the distance which the power moves over to that through which the load is raised is #hhe #bhg #ca very nearly4 ,we shall therefore suppose the velocity ratio to be #ca4 ,thus this wheel and axle has a far higher velocity ratio than any of the systems of pulleys which we have been considering4 #cjg4 ,were friction absent the velocity ratio of #ca would necessarily express the mechanical efficiency of this wheel and axle2 owing to the presence of friction the real efficiency is less than this--how much less1 we must ascertain by experiment4 ,i attach a load of #ef lbs4 to the hook which is borne by the rope descending from the axle3 this load is shown at ,d in ,fig4 #df4 ,i find that a power of #bf lbs4 applied at ,e is just sufficient to raise ,d4 ,we infer from this result that the mechanical efficiency of this machine is #ef #bf #bae4 ,i add a second #ef lb4 weight to the load1 and ,i find that a power of #ej lbs4 raises the load of #aab lbs4 ,the mechanical efficiency in this case is #aab #ee #bbe4 ,we adopt the mean value #bb4 ,hence the mechanical efficiency is reduced by friction from #ca to #bb4 #cjh4 ,we may compute from this result the number of units of energy which are utilized out of every #ajj units applied4 ,let us suppose a load of #ajj lbs4 is to be raised one foot2 a force of #ajj #bb #df lbs4 will suffice to raise this load4 ,this force must be exerted through a space of #ca'1 and consequently #ca #df #adc units of energy must be expended2 of this amount #ajj units are usefully employed1 and therefore the percentage of energy utilized is #ajj #adc #ajj #gj4 ,it follows that #cj per cent4 of the applied energy is consumed in overcoming friction4 #cji4 ,we can see the reason why the wheel and axle overhauls--that is1 runs down of its own accord--when allowed to do so2 it is because less than half the applied energy is expended upon friction4 #caj4 ,a series of experiments which have been carefully made with this wheel and axle are recorded in ,table ,,xviii4 ,,table ,,xviii4--,,wheel ,,and ,,axle4 ,wheel of wood2 axle of iron1 in oiled brass bearings2 weight of wheel and axle together1 #afe lbs42 effective circumference of wheel1 #hhe2 effective circumference of axle1 #bhg2 velocity ratio1 #ca2 mechanical efficiency1 #bb2 useful effect1 #gj per cent42 formula1 ,p #jbjd #jjdbf ,r4 ---------------------------------------- -------------- ,p4 ,difference of ,number of ,r4 ,observed ,calculated the observed ,experiment4 ,load power power and calculated in lbs4 in lbs4 in lbs4 values4 ---------------------------------------- -------------- #a #bh #ad #ad #jj #b #db #bj #bj #jj #c #ef #bf #bf #jj #d #gj #cb #cb #jj #e #hd #cg #ch #ja #f #ih #dd #dd #jj #g #aab #ej #ej #jj ---------------------------------------- -------------- ,by the method of the ,appendix a relation connecting the power and the load has been determined2 it is expressed in the form-- ,p #jbjd #jjdbf ,r4 #caa4 ,thus for example in experiment #e a load of #hd lbs4 was found to be raised by a power of #cg lbs4 ,the value calculated by the formula is #jbjd #jjdbf #hd #ch4 ,the calculated value only differs from the observed value by #ja lb41 which is shown in the fifth column4 ,it will be seen from this column that the values calculated from the formula represents the experiments with fidelity4 #cab4 ,we have deduced the relation between the power and the load from the principle of energy1 but we might have obtained it from the principle of the lever4 ,the wheel and axle both revolve about the centre of the axle2 we may therefore regard the centre as the fulcrum of a lever1 and the points where the cords meet the wheel and axle as the points of application of the power and the load respectively4 #cac4 ,by the principle of the lever of the first order 7,art4 #bcg71 the power is to the load in the inverse proportion of the arms2 in this case1 therefore1 the power is to the load in the inverse proportion of the radii of the wheel and the axle4 ,but the circumferences of circles are in proportion to their radii1 and therefore the power must be to the load as the circumference of the axle is to the circumference of the wheel4 #cad4 ,this mode of arriving at the result is a little artificial2 it is more natural to deduce the law directly from the principle of energy4 ,in a mechanical power of any complexity it would be difficult to trace exactly the transmission of power from one part to the next1 but the principle of energy evades this difficulty2 no matter what be the mechanical arrangement1 simple or complex1 of few parts or of many1 we have only to ascertain by trial how many feet the power must traverse in order to raise the load one foot2 the number thus obtained is the theoretical efficiency of the machine4 ,,friction ,,upon ,,the ,,axle4 #cae4 ,in the wheel and axle upon which we have been experimenting1 we have found that about #cj per cent4 of the power is consumed by friction4 ,we shall be able to ascertain to what this loss is due1 and then in some degree to remove its cause4 ,from the experiments of ,art4 #afe we learned that the friction of a small pulley was very much greater than that of a large pulley--in fact1 the friction is inversely proportional to the diameter of the pulley4 ,we infer from this that by winding the rope upon a barrel instead of upon the axle1 the friction may be diminished4 ,illustration3 ,,fig4 #dg4 #caf4 ,we can examine experimentally the effect of friction on the axle by the apparatus of ,fig4 #dg4 ,b is a shaft #jge diameter1 about which a rope is coiled several times2 the ends of this rope hang down freely1 and to each of them hooks ,e1 ,f are attached4 ,this shaft revolves in brass bearings1 which are oiled4 ,in order to investigate the amount of power lost by winding the rope upon an axle of this size1 ,i shall place a certain weight--suppose #ef lbs4--upon one hook ,f1 and then ,i shall ascertain what amount of power hung upon the other hook ,e will be sufficient to raise ,f4 ,there is here no mechanical advantage1 so that the excess of load which ,e must receive in order to raise ,f is the true measure of the friction4 #cag4 ,i add on weights at ,e until the power reaches #he lbs41 when ,e descends4 ,we thus see that to raise #ef lbs4 an excess of #bi lbs4 was necessary to overcome the friction4 ,we may roughly enunciate the result by stating that to raise a load in this way1 half as much again is required for the power4 ,this law is verified by suspending #bh lbs4 at ,f1 when it is found that a power of #dc lbs4 at ,e is required to lift it4 ,had the power been #db lbs41 it would have been exactly half as much again as the load4 #cah4 ,hence in raising ,f upon this axle1 about one-third of the power which must be applied at the circumference of the axle is wasted4 ,this experiment teaches us where the loss lies in the wheel and axle of ,art4 #cjd1 and explains how it is that about a third of its efficiency is lost4 #he lbs4 was only able to raise two-thirds of its own weight1 owing to the friction2 and hence we should expect to find1 as we actually have found1 that the power applied at the circumference of the wheel has an effect which is only two-thirds of its theoretical efficiency4 #cai4 ,from this experiment we should infer that the proper mode of avoiding the loss by friction is to wind the rope upon a barrel of considerable diameter rather than upon the axle itself4 ,i place upon a similar axle to that on which we have been already experimenting a barrel of about #ae circumference4 ,i coil the rope two or three times about the barrel1 and let the ends hang down as before4 ,i then attach to each end #ef lbs4 weight1 and ,i find that #aj lbs4 added to either of the weights is sufficient to overcome friction1 to make it descend1 and raise the other weight4 ,the apparatus is shown in ,fig4 #dg4 ,a is the barrel1 ,c and ,d are the weights4 ,in this arrangement #aj lbs4 is sufficient to overcome the friction which required #bi lbs4 when the rope was simply coiled around the axle4 ,in other words1 by the barrel the loss by friction is reduced to one-third of its amount4 ,,the ,,wheel ,,and ,,barrel4 #cbj4 ,we next place the barrel upon the axis already experimented upon and shown in ,fig4 #df at ,b4 ,the circumference of the wheel is #hhe2 the circumference of the barrel is #adi4 ,the proper mode of finding the circumference of the barrel is to suspend a weight from the rope1 then raise this weight by making one revolution of the wheel1 and the distance through which the weight is raised is the effective circumference of the barrel4 ,the velocity ratio of the wheel and barrel is then found1 by dividing #adi into #hhe1 to be #eid4 #cba4 ,the mechanical efficiency of this machine is determined by experiment4 ,i suspend a weight of #ef lbs4 from the hook1 and apply power to the wheel4 ,i find that #aja lbs4 is just sufficient to raise the load4 #cbb4 ,the mechanical efficiency is to be found by dividing #aja into #ef2 the quotient thus obtained is #eed4 ,the mechanical efficiency does not differ much from #eid1 the velocity ratio2 and consequently in this machine but little power is expended upon friction4 #cbc4 ,we can ascertain the loss by computing the percentage of applied energy which is utilized4 ,let us suppose a weight of #ajj lbs4 has to be raised one foot3 for this purpose a force of #ajj #eed #aha lbs4 must be applied4 ,this is evident from the definition of the mechanical efficiency2 but since the load has to be raised one foot1 it is clear from the meaning of the velocity ratio that the power must move over #e'id3 hence the number of units of work to be applied is to be measured by the product of #eid and #aha1 that is1 by #ajge2 in order therefore to accomplish #ajj units of work #ajge units of work must be applied4 ,the percentage of energy usefully employed is #ajj #ajge #ajj #ic4 ,this is far more than #gj1 which is the percentage utilized when the axle was used without the barrel 7,art4 #cji74 #cbd4 ,a series of experiments made with care upon the wheel and barrel are recorded in ,table ,,xix4 ,,table ,,xix4--,,the ,,wheel ,,and ,,barrel4 ,wheel of wood1 #hhe in circumference1 on the same axle as a cast iron barrel of #adi circumference2 axle is of wrought iron1 #jge in diameter1 mounted in oiled brass bearings2 power is applied to the circumference of the wheel1 load raised by rope round barrel2 velocity ratio1 #eid2 mechanical efficiency1 #eed2 useful effect1 #ic per cent42 formula1 ,p #je #jafi ,r4 ---------------------------------------- ----------------- ,p4 ,difference of ,number of ,r4 ,observed ,calculated the observed ,experiment4 ,load power power and calculated in lbs4 in lbs4 in lbs4 values4 ---------------------------------------- ----------------- #a #ad #bg #bi #jb #b #bh #ec #eb - #ja #c #db #gg #gf - #ja #d #ef #aja #ajj - #ja #e #gj #abd #abd #jj #f #hd #adg #adg #jj #g #ih #aga #aga #jj #h #aab #aid #aie #ja ---------------------------------------- ----------------- ,the formula which represents the experiments with the greatest amount of accuracy is ,p #je #jafi ,r4 ,this formula is compared with the experiments1 and the column of differences shows that the calculated and the observed values agree very closely4 ,the constant part #je is partly due to the constant friction of the heavy barrel and wheel1 and partly1 it may be1 to small irregularities which have prevented the centre of gravity of the whole mass from being strictly in the axle4 #cbe4 ,though this machine is more economical of power than the wheel and axle of ,art #cje1 yet it is less powerful2 in fact1 the mechanical efficiency1 #eed1 is only about one-fourth of that of the wheel and axle4 ,it is therefore necessary to inquire whether we cannot devise some method by which to secure the advantages of but little friction1 and at the same time have a large mechanical efficiency3 this we shall proceed to investigate4 ,,the ,,wheel ,,and ,,pinion4 #cbf4 ,by means of what are called cog-wheels or toothed wheels1 we are enabled to combine two or more wheels and axles together1 and thus greatly to increase the power which can be produced by a single wheel and axle4 ,toothed wheels are used for a great variety of purposes in mechanics2 we have already had some illustration of their use during these lectures 7,fig4 #cj74 ,the wheels which we shall employ are those often used in lathes and other small machines2 they are what are called #aj-pitch wheels1--that is to say1 a wheel of this class contains ten times as many teeth in its circumference as there are inches in its diameter4 ,i have here a wheel #bj diameter1 and consequently it has #bjj teeth2 here is another which is #be diameter1 and which consequently contains #be teeth4 ,we shall mount these wheels upon two parallel shafts1 so that they gear one into the other in the manner shown in ,fig4 #df3 ,f is the large wheel containing #bjj teeth1 and ,g the pinion of #be teeth4 ,the axles are #jge diameter2 around each of them a rope is wound1 by which a hook is suspended4 #cbg4 ,a small weight at ,k is sufficient to raise a much larger weight on the other shaft2 but before experimenting on the mechanical efficiency of this arrangement1 we shall as usual calculate the velocity ratio4 ,the wheel contains eight times as many teeth as the pinion2 it is therefore evident that when the wheel has made one revolution1 the pinion will have made eight revolutions1 and conversely the pinion must turn round eight times to turn the wheel round once3 hence the power which is turning the pinion round must be lowered through eight times the circumference of the axle1 while the load is raised through a length equal to one circumference of the axle4 ,we thus find the velocity ratio of the machine to be #h4 #cbh4 ,we determine the mechanical efficiency by trial4 ,attaching a load of #ef lbs4 to the axle of the large wheel1 it is observed that a power of #acg lbs4 at ,k will raise it2 the mechanical efficiency of the machine is therefore about #da1 which is almost exactly half the velocity ratio4 ,we note that the load will only just run down when the power is removed2 from this we might have inferred1 by ,art4 #bbb1 that nearly half the power is expended on friction1 and that therefore the mechanical efficiency is about half the velocity ratio4 ,the actual percentage of energy that is utilised with this particular load is #ea4 ,if we suspend #aab lbs4 from the load hook1 #bf lbs4 is just enough to raise it2 the mechanical efficiency that would be deduced from this result is #aab #bf #dc1 which is slightly in excess of the amount obtained by the former experiment4 ,it is often found to be a property of the mechanical powers1 that as the load increases the mechanical efficiency slightly improves4 #cbi4 ,in ,table ,,xx4 will be found a record of experiments upon the relation between the power and the load with the wheel and pinion2 the table will sufficiently explain itself1 after the description of similar tables already given 7,arts4 #caj1 #cbd74 ,,table ,,xx4--,,the ,,wheel ,,and ,,pinion4 ,wheel 7#aj-pitch71 #bjj teeth2 pinion1 #be teeth2 axles equal1 effective circumference of each being #bhg2 oiled brass bearings2 velocity ratio1 #h2 mechanical efficiency1 #da2 useful effect1 #ea per cent42 formula1 ,p #bdf #jba ,r4 ---------------------------------------- --------------- ,p4 ,difference of ,number of ,r4 ,observed ,calculated the observed ,experiment4 ,load power power and calculated in lbs4 in lbs4 in lbs4 powers4 ---------------------------------------- --------------- #a #ad #ed #ed #jj #b #bh #hg #hc - #jd #c #db #aaj #aac #jc #d #ef #acg #adb #je #e #gj #age #agb - #jc #f #hd #bjj #bja #ja #g #ih #bcj #bcj #jj #h #aab #bfj #bfj #jj ---------------------------------------- --------------- #ccj4 ,the large amount of friction present in this contrivance is the consequence of winding the rope directly upon the axle instead of upon a barrel1 as already pointed out in ,art4 #cai4 ,we might place barrels upon these axles and demonstrate the truth of this statement2 but we need not delay to do so1 as we use the barrel in the machines which we shall next describe4 ,,the ,,crane4 #cca4 ,we have already explained 7,art4 #ch7 the construction of the lifting crane1 so far as its framework is concerned4 ,we now examine the mechanism by which the load is raised4 ,we shall employ for this purpose the model which is represented in ,fig4 #dh4 ,the jib is supported by a wooden bar as a tie1 and the crane is steadied by means of the weights placed at ,h3 some such counterpoise is necessary1 for otherwise the machine would tumble over when a load is suspended from the hook4 #ccb4 ,the load is supported by a rope or chain which passes over the pulley ,e and thence to the barrel ,d1 upon which it is to be wound4 ,this barrel receives its motion from a large wheel ,a1 which contains #bjj teeth4 ,the wheel ,a is turned by the pinion ,b which contains #be teeth4 ,in the actual use of the crane1 the axle which carries this pinion would be turned round by means of a handle2 but for the purpose of experiments upon the relation of the power to the load1 the handle would be inconvenient1 and therefore we have placed upon the axle of the pinion a wheel ,c containing a groove in its circumference4 ,around this groove a string is wrapped1 so that when a weight ,g is suspended from the string it will cause the wheel to revolve4 ,this weight ,g will constitute the power by which the load may be raised4 #ccc4 ,let us compute the velocity ratio of this machine before commencing experiments upon its mechanical efficiency4 ,the effective circumference of the barrel ,d is found by trial to be #adi4 ,since there are #bjj teeth on ,a and #be on ,b1 it follows that the pinion ,b must revolve eight times to produce one revolution of the barrel4 ,hence the wheel ,c at the circumference of which the power is applied must also revolve eight times for one revolution of the barrel4 ,the effective circumference of ,c is #dc2 the power must therefore have been applied through #h #dc #cdd1 in order to raise the load #aei4 ,the velocity ratio is #cdd #adi #bc very nearly4 ,we can easily verify this value of the velocity ratio by actually raising the load #a'1 when it appears that the number of revolutions of the wheel ,b is such that the power must have moved #bc'4 ,illustration3 ,,fig4 #dh4 #ccd4 ,the mechanical efficiency is to be found as usual by trial4 #ef lbs4 placed at ,f is raised by #ca lbs4 at ,g2 hence the mechanical efficiency deduced from this experiment is #ef #ca #ah4 ,the percentage of useful effect is easily shown to be #gh by the method of ,art4 #cbc4 ,here1 then1 we have a machine possessing very considerable efficiency1 and being at the same time economical of energy4 ,,table ,,xxi4--,,the ,,crane4 ,circumference of wheel to which the power is applied1 #dc2 train of wheels1 #be #bjj2 circumference of drum on which rope is wound1 #adi2 velocity ratio1 #bc2 mechanical efficiency1 #ah2 useful effect1 #gh per cent42 formula1 ,p #jjeef ,r4 ---------------------------------------- ----------- ,p4 ,difference of ,number of ,r4 ,observed,calculated the observed ,experiment4 ,load power power and calculated in lbs4 in lbs4 in lbs4 values4 ---------------------------------------- ----------- #a #ad #ji #jh -#ja #b #bh #af #af #jj #c #db #bd #bc -#ja #d #ef #ca #ca #jj #e #gj #ch #ci #ja #f #hd #de #dg #jb #g #ih #ec #ee #jb #h #aab #fb #fb #jj ---------------------------------------- ----------- #cce4 ,a series of experiments made with this crane is recorded in ,table ,,xxi41 and a comparison of the calculated and observed values will show that the formula ,p #jjeef ,r represents the experiments with considerable accuracy4 #ccf4 ,it may be noticed that in this formula the term independent of ,r1 which we frequently meet with in the expression of the relation between the power and the load1 is absent4 ,the probable explanation is to be found in the fact that some minute irregularity in the form of the barrel or of the wheel has been constantly acting like a small weight in favour of the power4 ,in each experiment the motion is always started from the same position of the wheels1 and hence any irregularity will be constantly acting in favour of the power or against it2 here the former appears to have happened4 ,in other cases doubtless the latter has occurred2 the difference is1 however1 of extremely small amount4 ,the friction of the machine itself when without a load is another source for the production of the constant term2 it has happened in the present case that this friction has been almost exactly balanced by the accidental influence referred to4 #ccg4 ,in cranes it is usual to provide means of adding a second train of wheels1 when the load is very heavy4 ,in another model we applied the power to an axle with a pinion of #be teeth1 gearing into a wheel of #bjj teeth2 on the axle of the wheel with #bjj teeth is a pinion of #cj teeth1 which gears into a wheel of #ahj teeth2 the barrel is on the axle of the last wheel4 ,a series of experiments with this crane is shown in ,table ,,xxii4 ,,table ,,xxii4--,,the ,,crane ,,for ,,heavy ,,loads4 ,circumference of wheel to which power is applied1 #dc2 train of wheels1 #be #bjj #cj #ahj2 circumference of drum on which rope is wound1 #adi2 velocity ratio1 #acg2 mechanical efficiency1 #hg2 useful effect1 #fc per cent42 formula1 ,p #jahe #jjjghb ,r4 ---------------------------------------- ----------- ,p4 ,difference of ,number of ,r4 ,observed,calculated the observed ,experiment4 ,load power power and calculated in lbs4 in lbs4 in lbs4 values4 ---------------------------------------- ----------- #a #ad #jcj #jbi -#jja #b #bh #jdj #jdj #jjj #c #db #jej #jea #jja #d #ef #jfj #jfb #jjb #e #gj #jge #jgc -#jjb #f #hd #jhe #jhd -#jja #g #ih #jie #jie #jjj #h #aab #aje #ajf #jja ---------------------------------------- ----------- ,the velocity ratio is now #acg1 and the mechanical efficiency is #hg2 one man could therefore raise a ton with ease by applying a power of #bf lbs4 to a crane of this kind4 ,,conclusion4 #cch4 ,it will be useful to contrast the wheel and axle on which we have experimented 7,art4 #cjd7 with the differential pulley 7,art4 #bji74 ,the velocity ratio of the former machine is nearly double that of the latter1 and its mechanical efficiency is nearly four times as great4 ,less than half the applied power is wasted in the wheel and axle1 while more than half is wasted in the differential pulley4 ,this makes the wheel and axle both a more powerful machine1 and a more economical machine than the differential pulley4 ,on the other hand1 the greater compactness of the latter1 its facility of application1 and the practical conveniences arising from the property of not allowing the load to run down1 do often more than compensate for the superior mechanical advantage of the wheel and axle4 #cci4 ,we may also contrast the wheel and axle with the screw 7,art4 #bgg74 ,the screw is remarkable among the mechanical powers for its very high velocity ratio1 and its excessive friction4 ,thus we have seen in ,art4 #bia how the velocity ratio of a screw-jack with an arm attached amounted to #dad1 while its mechanical efficiency was little more than one-fourth as great4 ,no single wheel and axle could conveniently be made to give a mechanical efficiency of #aaf2 but from ,art4 #ccg we could easily design a combination of wheels and axles to yield an efficiency of quite this amount4 ,the friction in the wheel and axle is very much less than in the screw1 and consequently energy is saved by the use of the former machine4 #cdj4 ,in practice1 however1 it generally happens that economy of energy does not weigh much in the selection of a mechanical power for any purpose1 as there are always other considerations of greater consequence4 #cda4 ,for example1 let us take the case of a lifting crane employed in loading or unloading a vessel1 and inquire why it is that a train of wheels is generally used for the purpose of producing the requisite power4 ,the answer is simple1 the train of wheels is convenient1 for by their aid any length of chain can be wound upon the barrel2 whereas if a screw were used1 we should require a screw as long as the greatest height of lift4 ,this screw would be inconvenient1 and indeed impracticable1 and the additional circumstance that a train of wheels is more economical of energy than a screw has no influence in the matter4 #cdb4 ,on the other hand1 suppose that a very heavy load has to be overcome for a short distance1 as for example in starting a ship launch1 a screw-jack is evidently the proper machine to employ2 it is easily applied1 and has a high mechanical efficiency4 ,the want of economy of energy is of no consequence in such an operation4 ,,lecture ,,xi4 ,,the ,,mechanical ,,properties ,,of ,,timber4 ,introduction4--,the ,general ,properties of ,timber4--,resistance to ,extension4--,resistance to ,compression4--,condition of a ,beam strained by a ,transverse ,force4 ,,introduction #cdc4 ,in the lectures on the mechanical powers which have been just completed1 we have seen how great weights may be raised or other large resistances overcome4 ,we are now to consider the important subject of the application of mechanical principles to structures4 ,these are fixtures1 while machines are adapted for motion2 a roof or a bridge is a structure1 but a crane or a screw-jack is a machine4 ,structures are employed for supporting weights1 and the mechanical powers give the means of raising them4 #cdd4 ,a structure has to support both its own weight and also any load that is to be placed upon it4 ,thus a railway bridge must at all times sustain what is called the permanent load1 and frequently1 of course1 the weight of one or more trains4 ,the problem which the engineer solves is to design a bridge which shall be sufficiently strong1 and1 at the same time1 economical2 his skill is shown by the manner in which he can attain these two ends in the same structure4 #cde4 ,in the four lectures of the course which will be devoted to this subject it will only be possible to give a slight sketch1 and therefore but few details can be introduced4 ,an extended account of the properties of different materials used in structures would be beyond our scope1 but there are some general principles relating to the strength of materials which may be discussed4 ,timber1 as a building material1 has1 in modern times1 been replaced to a great extent by iron in large structures1 but timber is more capable than iron of being experimented upon in the lecture room4 ,the elementary laws which we shall demonstrate with reference to the strength of timber1 are also1 substantially the same as the corresponding laws for the strength of iron or any other material4 ,hence we shall commence the study of structures by two lectures on timber4 ,the laws which we shall prove experimentally will afterwards be applied to a few simple cases of bridges and other actual structures4 ,,the ,,general ,,properties ,,of ,,timber4 #cdf4 ,the uses of timber in the arts are as various as its qualities4 ,some woods are useful for their beauty1 and others for their strength or durability under different circumstances4 ,we shall only employ 8pine0 in our experimental inquiries4 ,this wood is selected because it is so well known and so much used4 ,a knowledge of the properties of pine would probably be more useful than a knowledge of the properties of any other wood1 and at the same time it must be remembered that the laws which we shall establish by means of slips of pine may be generally applied4 #cdg4 ,a transverse section of a tree shows a number of rings1 each of which represents the growth of wood in one year4 ,the age of the tree may sometimes be approximately found by counting the number of distinguishable rings4 ,the outer rings are the newer portions of the wood4 #cdh4 ,when a tree is felled it contains a large quantity of sap1 which must be allowed to evaporate before the wood is fit for use4 ,with this object the timber is stored in suitable yards for two or more years according to the purposes for which it is intended2 sometimes the process of seasoning1 as it is called1 is hastened by other means4 ,wood1 when seasoning1 contracts2 hence blocks of timber are often found split from the circumference to the centre1 for the outer rings1 being newer and containing more sap1 contract more than the inner rings4 ,for the same reason a plank is found to warp when the wood is not thoroughly seasoned4 ,the side of the plank which was farthest from the centre of the tree contracts more than the other side1 and becomes concave4 ,this can be easily verified by looking at the edge of the plank1 for we there see the rings of which it is composed4 #cdi4 ,timber may be softened by steaming4 ,i have here a rod of pine1 #bd #je #je1 and here a second rod cut from the same piece and of the same size1 which has been exposed to steam of boiling water for more than an hour3 securing these at one end to a firm stand1 ,i bend them down together1 and you see that after the dry rod has broken the steamed rod can be bent much farther before it gives way4 ,this property of wood is utilized in shaping the timbers of wooden ships4 ,we shall be able to understand the action of steam if we reflect that wood is composed of a number of fibres ranged side by side and united together4 ,a rope is composed of a number of fibres laid together and twisted1 but the fibres are not coherent as they are in wood4 ,hence we find that a rod of wood is stiff1 while a rope is flexible4 ,the steam finds its way into the interstices between the fibres of the wood2 it softens their connections1 and increases the pliability of the fibres themselves1 and thus1 the operation of steaming tends to soften a piece of timber and render it tractable4 #cej4 ,the structure of wood is exhibited by the following simple experiment3--,here are two pieces of pine1 each #i #a #a4 ,one of them ,i easily snap across with a blow1 while my blows are unable to break the other4 ,the difference is merely that one of these pieces is cut against the grain1 while the other is with it4 ,in the first case ,i have only to separate the connection between the fibres1 which is quite easy4 ,in the other case ,i would have to tear asunder the fibres themselves1 which is vastly more difficult4 ,to a certain extent the grained structure is also found in wrought iron1 but the contrast between the strength of iron with the grain and against the grain is not so marked as it is in wood4 ,,resistance ,,to ,,extension4 #cea4 ,it will be necessary to explain a little more definitely what is meant by the strength of timber4 ,we may conceive a rod to be broken in three different ways4 ,in the first place the rod may be taken by a force at each end and torn asunder by pulling1 as a thread may be broken4 ,to do this requires very great power1 and the strength of the material with reference to such a mode of destroying it is called its resistance to extension4 ,in the second place1 it may be broken by longitudinal pressure at each end1 as a pillar may be crushed by the superincumbent weight being too large2 the strength that relates to this form of force is called resistance to compression3 finally1 the rod may be broken by a force applied transversely4 ,the strength of pine with reference to these different applications of force will be considered successively4 ,the rods that are to be used have been cut from the same piece of timber1 which has been selected on account of its straightness of grain and freedom from knots4 ,they are of different rectangular sections1 #a #je and #je #je being generally used1 but sometimes #a #a is employed4 ,illustration3 ,,fig4 #di4 #ceb4 ,with reference to the strength of timber in its capacity to resist extension1 we can do but little in the lecture room4 ,i have here a pine rod ,a ,b1 of dimensions #dh #j4e #je1 ,fig4 #di4 ,each end of this rod is firmly secured between two cheeks of iron1 which are bolted together3 the rod is suspended by its upper extremity from the hook of the epicycloidal pulley-block 7,art4 #bac71 which is itself supported by a tripod2 hooks are attached to the lower end of the rod for carrying the weights4 ,by placing #c cwt4 on these hooks and pulling the hand chain of the pulley-block1 ,i find that ,i can raise the weight safely1 and therefore the rod will resist at all events a tension of #c cwt4 ,from experiments which have been made on the subject1 it is ascertained that about a ton would be necessary to tear such a rod asunder2 hence we see that pine is enormously strong in resisting a force of extension4 ,the tensile strength of the rod does not depend upon its length1 but upon the area of the cross section4 ,that of the rod we have used is one-fourth of a square inch1 and the breaking weight of a rod one square inch in section is about four tons4 #cec4 ,a rod of any material generally elongates to some extent under the action of a suspended weight2 and we shall ascertain whether this occurs perceptibly in wood4 ,before the rod was strained ,i had marked two points upon it exactly #b feet apart4 ,when the rod supports #c cwt4 ,i find that the distance between the two points has not appreciably altered1 though by more delicate measurement ,i have no doubt we should find that the distance had elongated to an insignificant extent4 #ced4 ,let us contrast the resistance of a rod of timber to extension with the effect upon a rope under the same circumstances4 ,i have here a rope about #j4be diameter2 it is suspended from a point1 and bears a #ad lb4 weight in order to be completely stretched4 ,i mark points upon the rope #b' apart4 ,i now change the stone weight for a weight of #a cwt41 and on measurement ,i find that the two points which before were #b' apart1 are now #b' #b2 thus the rope has stretched at the rate of an inch per foot for a strain of #a cwt41 while the timber did not stretch perceptibly for a strain of #c cwt4 #cee4 ,we have already explained in ,art4 #cg the meaning of the word 8tie40 ,the material suitable for a tie should be capable of offering great resistance1 not only to actual rupture by tension1 but even to appreciable elongation4 ,these qualities we have found to be possessed by wood4 ,they are1 however1 possessed in a much higher degree by wrought iron1 which possesses other advantages in durability and facility of attachment4 ,,resistance ,,to ,,compression4 #cef4 ,we proceed to examine into the capability of timber to resist forces of longitudinal compression1 either as a pillar or in any other form of 8strut10 such for instance1 as the jib of the crane represented in ,fig4 #ag4 ,the use of timber as a strut depends in a great degree upon the coherence of the fibres to each other1 as well as upon their actual rigidity4 ,the action of timber in resisting forces of compression is thus very different from its action when resisting forces of extension2 we can examine1 by actual experiment1 the strength of timber under the former conditions1 as the weights which it will be necessary to employ are within the capabilities of our lecture room apparatus4 #ceg4 ,the apparatus is shown in ,fig4 #ej4 ,it consists of a lever of the second order1 #aj' long1 the mechanical advantage of which is threefold2 the resistance of the pillar ,d ,e to crushing is the load to be overcome1 and the power consists of weights1 to receive which the tray ,b is used2 every pound placed in the tray produces a compressive force of #c lbs4 on the pillar at ,d4 ,the fulcrum is at ,a and guides at ,g4 ,the lever and the tray would somewhat complicate our calculations unless their weights were counterpoised4 ,a cord attached to the extremity of the lever passes over a pulley ,f2 at the other end of this cord1 sufficient weights ,c are attached to neutralize the weight of the apparatus4 ,in fact1 the lever and tray now swing as if they had no weight1 and we may therefore leave them out of consideration4 ,the pillar to be experimented upon is fitted at its lower end ,e into a hole in a cast iron bracket3 this bracket can be adjusted so as to take in pieces of different lengths2 the upper end of the pillar passes through a hole in a second piece of cast iron1 which is bolted to the lever3 thus our little experimental column is secured at each end1 and the risk of slipping is avoided4 ,the stands are heavily weighted to secure the stability of the arrangement ,illustration3 ,,fig4 #ej4 #ceh4 ,the first experiment we shall make with this apparatus is upon a pine rod #dj long and #je square2 the lower bracket is so placed that the lever is horizontal when just resting upon the top of the rod4 ,weights placed in the tray produce a pressure three times as great down the rod1 the effect of which will first be to bend the rod1 and1 when the deflection has reached a certain amount1 to break it across4 ,i place #bh lbs4 in the tray3 this produces a pressure of #hd lbs4 upon the rod1 but the rod still remains perfectly straight1 so that it bears this pressure easily4 ,when the pressure is increased to #if lbs4 a very slight amount of deflection may be seen4 ,when the strain reaches #aad lbs4 the rod begins to bend into a curved form1 though the deflection of the middle of the rod from its original position is still less than #jbe4 ,gradually augmenting the pressure1 ,i find that when it reaches #acb lbs4 the deviation has reached #je2 and finally1 when #dh lbs4 is placed in the tray1 that is1 when the rod is subjected to #add lbs41 it breaks across the middle4 ,hence we see that this rod sustained a load of #if lbs4 without sensibly bending1 but that fracture ensued when the load was increased about half as much again4 ,another experiment with a similar rod gave a slightly less value 7#acb lbs47 for the breaking load4 ,if ,i add these results together1 and divide the sum by #b1 ,i find #ach lbs4 as the mean value of the breaking load1 and this is a sufficiently exact determination4 #cei4 ,let us next try the resistance of a shorter rod of the same section4 ,i place a piece of pine #bj long and #je square in the apparatus1 firmly securing each end as in the former case4 ,the lower bracket is adjusted so as to make the lever horizontal2 the counterpoise1 of course1 remains the same1 and weights are placed in the tray as before4 ,no deflection is noticed when the rod supports #abf lbs42 a very slight amount of bending is noticeable with #ahf lbs42 with #bbh lbs41 the amount by which the centre of the rod has deviated laterally from its original position is about #jb2 and finally1 when the load reaches #bid lbs41 the rod breaks4 ,fracture first occurs in the middle1 but is immediately followed by other fractures near where the ends of the rod are secured4 #cfj4 ,hence the breaking load of a rod #bj long is more than double the breaking load of a rod of #dj long the same section2 from this we learn that the sections being equal1 short pillars are stronger than long pillars4 ,it has been ascertained by experiment that the strength of a square pillar to resist compression is proportional to the square of its sectional area4 ,hence a rod of pine1 #dj long and #a square1 having four times the section of the rod of the same length we have experimented on1 would be sixteen times as strong1 and consequently its breaking weight would amount to nearly a ton4 ,the strength of a rod used as a tie depends only on its section1 while the strength of a rod used as a strut depends on its length as well as on its section4 ,,condition ,,of ,a ,,beam ,,strained ,,by ,a ,,transverse ,,force4 #cfa4 ,we next come to the important practical subject of the strength of timber when supporting a transverse strain2 that is1 when used as a beam4 ,the nature of a transverse strain may be understood from ,fig4 #ea1 which represents a small beam1 strained by a load at its centre4 ,fig4 #eb shows two supports #dj apart1 across which a rod of pine #dh #a #a is laid2 at the middle of this rod a hook is placed1 from which a tray for the reception of weights is suspended4 ,a rod thus supported1 and bearing weights1 is said to be strained transversely4 ,a rafter of a roof1 the flooring of a room1 a gangway from the wharf to a ship1 many forms of bridge1 and innumerable other examples1 might be given of beams strained in this manner4 ,to this important subject we shall devote the remainder of this lecture and the whole of the next4 #cfb4 ,the first point to be noticed is the deflection of the beam from which a weight is suspended4 ,the beam is at first horizontal2 but as the weight in the tray is augmented1 the beam gradually curves downwards until1 when the weight reaches a certain amount1 the beam breaks across in the middle and the tray falls4 ,illustration3 ,,fig4 #ea4 ,for convenience in recording the experiments the tray chain and hooks have been adjusted to weigh exactly #ad lbs4 7,fig4 #eb74 ,a ,b is a cord which is kept stretched by the little weights ,d3 this cord gives a rough measure of the deflection of the beam from its horizontal position when strained by a load in the tray4 ,in order to observe the deflection accurately an instrument is used called the cathetometer 7,g74 ,it consists of a small telescope1 always directed horizontally1 though capable of being moved up and down a vertical triangular pillar2 on one of the sides of the pillar a scale is engraved1 so that the height of the telescope in any position can be accurately determined4 ,the cathetometer is levelled by means of the screws ,h ,h1 so that the triangular pillar on which the telescope slides is accurately vertical3 the dotted line shows the direction of the visual ray when the centre ,c of the beam is seen by the observer through the telescope4 ,illustration3 ,,fig4 #eb4 ,inside the telescope and at its focus a line of spider's web is fixed horizontally2 on the bar to be observed1 and near its middle point ,c1 a cross of two fine lines is marked4 ,the tray being removed1 the beam becomes horizontal2 the telescope of the cathetometer is then directed towards the beam1 so that the lines marked upon it can be seen distinctly4 ,by means of a screw the telescope may be raised or lowered until the spider's web inside the telescope is observed to pass through the image of the intersection of the lines4 ,the scale then indicates precisely how high the telescope is on the pillar4 #cfc4 ,while ,i look through the telescope my assistant suspends the tray from the beam4 ,instantly ,i see the cross descend in the field of view4 ,i lower the telescope until the spider's web again passes through the image of the intersection of the lines1 and then by looking at the scale ,i see that the telescope has been moved down #jai1 that is1 about one-fifth of an inch3 this is1 therefore1 the distance by which the cross lines on the beam1 and therefore the centre of the beam itself1 must have descended4 ,indeed1 even a simpler apparatus would be competent to measure the amount of deflection with some degree of precision4 ,by placing successively one stone after another upon the tray1 the beam is seen to deflect more and more1 until even without the telescope you see the beam has deviated from the horizontal4 #cfd4 ,by carefully observing with the telescope1 and measuring in the way already described1 the deflections shown in ,table ,,xxiii4 were determined4 ,the scale along the vertical pillar was read after the spider's web had been adjusted for each increase in the weight4 ,the movement from the original position is recorded as the deflection for each load4 ,,table ,,xxiii4--,,deflection ,,of ,a ,,beam4 ,a rod of pine #dh #a #a2 resting freely on supports #dj apart2 and laden in the middle4 ------------------------------------- ,number of ,magnitude ,deflection4 ,experiment4 of load4 ------------------------------------- #a #ad #jai #b #bh #jcg #c #db #jee #d #ef #jgd #e #gj #jid #f #hd #aac #g #ih #ace #h #aab #afa #i #abf #aie #aj #adj #bcg ------------------------------------- #cfe4 ,the first column records the number of the experiment4 ,the second represents the load1 and the third contains the corresponding deflections4 ,it will be seen that up to #ih lbs4 the deflection is about #jb for every stone weight1 but afterwards the deflection increases more rapidly4 ,when the weight reaches #adj lbs4 the deflection at first indicated is #bcg2 but gradually the cross lines are seen to descend in the field of the telescope1 showing that the beam is yielding and finally it breaks across4 ,this experiment teaches us that a beam is at first deflected by an amount proportional to the weight it supports2 but that when two-thirds of the breaking weight is reached1 the beam is deflected more rapidly4 #cff4 ,it is a question of the utmost importance to ascertain the greatest load a beam can sustain without injury to its strength4 ,this subject is to be studied by examining the effect of different deflections upon the fibres of a beam4 ,a beam is always deflected whatever be the load it supports2 thus by looking through the telescope of the cathetometer ,i can detect an increase of deflection when a single pound is placed in the tray3 hence whenever a beam is loaded we must have some deflection4 ,an experiment will show what amount of deflection may be experienced without producing any permanently injurious effect4 #cfg4 ,a pine rod #dj #a #a is freely supported at each end1 the distances between the supports being #ch1 and the tray is suspended from its middle point4 ,a fine pair of cross lines is marked upon the beam1 and the telescope of the cathetometer is adjusted so that the spider's line exactly passes through the image of the intersection4 #ad lbs4 being placed in the tray1 the cross is seen to descend2 the weight being removed1 the cross returns precisely to its original position with reference to the spider's line3 hence1 after this amount of deflection1 the beam has clearly returned to its initial condition1 and is evidently just as good as it was before4 ,the tray next received #ef lbs42 the beam was1 of course1 considerably deflected1 but when the weight was removed the cross again returned1--at all events1 to within #jja of where the spider's line was left to indicate its former position4 ,we may consider that the beam is in this case also restored to its original condition1 even though it has borne a strain which1 including the tray1 amounted to #gj lbs4 ,but when the beam has been made to carry #hd lbs4 for a few seconds1 the cross does not completely return on the removal of the load from the tray1 but it shows that the beam has now received a permanent deflection of #jjc4 ,this is still more apparent after the beam has carried #ih lbs41 for when this load is removed the centre of the beam is permanently deflected by #jac4 ,here1 then1 we may infer that the fibres of the beam are beginning to be strained beyond their powers of resistance1 and this is verified when we find that with #bh additional pounds in the tray a collapse ensues4 #cfh4 ,reasoning from this experiment1 we might infer that the elasticity of a beam is not affected by a weight which is less than half that which would break it1 and that1 therefore1 it may bear without injury a weight not exceeding this amount4 ,as1 however1 in our experiments the weight was only applied once1 and then but for a short time1 we cannot be sure that a longer-continued or more frequent application of the same load might not prove injurious2 hence1 to be on the safe side1 we assume that one-third of the breaking weight of a beam is the greatest load it should be made to bear in any structure4 ,in many cases it is found desirable to make the beam much stronger than this ratio would indicate4 #cfi4 ,we next consider the condition of the fibres of a beam when strained by a transverse force4 ,it is evident that since the fracture commences at the lower surface of the beam1 the fibres there must be in a state of tension1 while those at the concave upper surface of the beam are compressed together4 ,this condition of the fibres may be proved by the following experiment4 #cgj4 ,i take two pine rods1 each #dh #a #a1 perfectly similar in all respects1 cut from the same piece of timber1 and therefore probably of very nearly identical strength4 ,with a fine tenon saw ,i cut each of the rods half through at its middle point4 ,i now place one of these beams on the supports #dj apart1 with the cut side of the beam upwards4 ,i suspend from it the tray1 which ,i gradually load with weights until the beam breaks1 which it does when the total weight is #ha lbs4 ,if ,i were to place the second beam on the same supports with the cut upwards1 then there can be no doubt that it would require as nearly as possible the same weight to break it4 ,i place it1 however1 with the cut downwards1 ,i suspend the tray1 and find that the beam breaks with a load of #ca lbs4 ,this is less than half the weight that would have been required if the cut had been upwards4 #cga4 ,what is the cause of this difference8 ,the fibres being compressed together on the upper surface1 a cut has no tendency to open there2 and if the cut could be made with an extremely fine saw1 so as to remove but little material1 the beam would be substantially the same as if it had not been tampered with4 ,on the other hand1 the fibres at the lower surface are in a state of tension2 therefore when the cut is below it yawns open1 and the beam is greatly weakened4 ,it is1 in fact1 no stronger than a beam of #dh #je #a1 placed with its shortest dimension vertical4 ,if we remember that an entire beam of the same size required about #adj lbs4 to break it 7,art4 #cff71 we see that the strength of a beam is reduced to one-fourth by being cut half-way through and having the cut underneath4 #cgb4 ,we may learn from this the practical consequence that the sounder side of a beam should always be placed downwards4 ,any flaw on the lower surface will seriously weaken the beam3 so that the most knotty face of the wood should certainly be placed uppermost4 ,if a portion of the actual substance of a beam be removed--for example1 if a notch be cut out of it--this will be almost equally injurious on either side of the beam4 #cgc4 ,we may illustrate the condition of the upper surface of the beam by a further experiment4 ,i make two cuts #je deep in the middle of a pine rod #dh #a #a4 ,these cuts are #je apart1 and slightly inclined2 the piece between them being removed1 a wedge is shaped to fit tightly into the space2 the wedge is long enough to project a little on one side4 ,if the wedge be uppermost when the beam is placed on the supports1 the beam will be in the same condition as if it had two fine cuts on the upper surface4 ,i now load the beam with the tray in the usual manner1 and ,i find it to bear #gj lbs4 securely4 ,on examining the beam1 which has curved down considerably1 ,i find that the wedge is held in very tightly by the pressure of the fibres upon it1 but1 by a sharp tap at the end1 ,i knock out the wedge1 and instantly the load of #gj lbs4 breaks the beam2 the reason is simple--the piece being removed1 there is no longer any resistance to the compressive strain of the upper fibres1 and consequently the beam gives way4 #cgd4 ,the collapse of a beam by a transverse strain commences by fracture of the fibres on the lower surface1 followed by a rupture of all fibres up to a considerable depth4 ,here we see that by a transverse force the fibres in a beam of #dh #a #a have been broken by a strain of #adj lbs4 7,art4 #cff72 but we have already stated 7,art4 #cec7 that to tear such a rod across by a direct pull at each end a force of about four tons is necessary4 ,the breaking strain of the fibres must be a certain definite quantity1 yet we find that to overcome it in one way four tons is necessary1 while by another mode of applying the strain #adj lbs4 is sufficient4 #cge4 ,to explain this discrepancy we may refer to the experiment of ,art4 #bh1 wherein a piece of string was broken by the transverse pull of a piece of thread in illustration of the fact that one force may be resolved into two others1 each of them very much greater than itself4 ,a similar resolution of force occurs in the transverse deflection of the beam1 and the force of #adj lbs4 is changed into two other forces1 each of them enormously greater and sufficiently strong to rupture the fibres4 ,we need not suppose that the force thus developed is so great as four tons1 because that is the amount required to tear across a square inch of fibres simultaneously1 whereas in the transverse fracture the fibres appear to be broken row after row2 the fracture is thus only gradual1 nor does it extend through the entire depth of the beam4 #cgf4 ,we shall conclude this lecture with one more remark1 on the condition of a beam when strained by a transverse force4 ,we have seen that the fibres on the upper surface are compressed1 while those on the lower surface are extended2 but what is the condition of the fibres in the interior8 ,there can be no doubt that the following is the state of the case3--,the fibres immediately beneath the upper surface are in compression2 at a greater depth the amount of compression diminishes until at the middle of the beam the fibres are in their natural condition2 on approaching the lower surface the fibres commence to be strained in extension1 and the amount of the extension gradually increases until it reaches a maximum at the lower surface4 ,,lecture ,,xii4 ,,the ,,strength ,,of ,a ,,beam4 ,a ,beam free at the ,ends and loaded in the ,middle4--,a ,beam uniformly loaded4--,a ,beam loaded in the ,middle1 whose ,ends are secured4--,a ,beam supported at one end and loaded at the other4 ,a ,,beam ,,free ,,at ,,the ,,ends ,,and ,,loaded ,,in ,,the ,,middle4 #cgg4 ,in the preceding lecture we have examined some general circumstances in connection with the condition of a beam acted on by a transverse force2 we proceed in the present to inquire more particularly into the strength under these conditions4 ,we shall1 as before1 use for our experiments rods of pine only1 as we wish rather to illustrate the general laws than to determine the strength of different materials4 ,the strength of a beam depends upon its length1 breadth1 and thickness2 we must endeavour to distinguish the effects of each of these elements on the capacity of the beam to sustain its load4 ,we shall only employ beams of rectangular section2 this being generally the form in which beams of wood are used4 ,beams of iron1 when large1 are usually not rectangular1 as the material can be more effectively disposed in sections of a different form4 ,it is important to distinguish between the stiffness of a beam in its capacity to resist flexure1 and the strength of a beam in its capacity to resist fracture4 ,thus the stiffest beam which can be made from the cylindrical trunk of a tree #a' in diameter is #f broad and #aje deep1 while the strongest beam is #g broad and #ige deep4 ,we are now discussing the strength 7not the stiffness7 of beams4 #cgh4 ,we shall commence the inquiry by making a number of experiments3 these we shall record in a table1 and then we shall endeavour to see what we can learn from an examination of this table4 ,i have here ten pieces of pine1 of lengths varying from #a' to #d'1 and of three different sections1 viz4 #a #a1 #a #je1 and #je #je4 ,i have arranged four different stands1 on which we can break these pieces3 on the first stand the distance between the points of support is #dj1 and on the other stands the distances are #cj1 #bj1 and #aj respectively2 the pieces being #d'1 #c'1 #b'1 and #a' long1 will just be conveniently held on the supports4 #cgi4 ,the mode of breaking is as follows3--,the beam being laid upon the supports1 an ,s hook is placed at its middle point1 and from this ,s hook the tray is suspended4 ,weights are then carefully added to the tray until the beam breaks2 the load in the tray1 together with the weight of the tray1 is recorded in the table as the breaking load4 #chj4 ,in order to guard as much as possible against error1 ,i have here another set of ten pieces of pine1 duplicates of the former4 ,i shall also break these2 and whenever ,i find any difference between the breaking loads of two similar beams1 ,i shall record in the table the mean between the two loads4 ,the results are shown in ,table ,,xxiv4 ,,table ,,xxiv4--,,strength ,,of ,a ,,beam4 ,slips of pine 7cut from the same piece7 supported freely at each end2 the length recorded is the distance between the points of support2 the load is suspended from the centre of the beam1 and gradually increased until the beam breaks2 area of section depth ,formula1 ,p #fjhj ---------------------------------------- -------- span ---------------------------------------- ---------------------- ,mean of the ,p4 ,difference ,dimensions4 observations,calculated of the ,no4 of ------------------- of the breaking observed and ,experi- breaking load calculated ment4 ,span4,breadth4,depth4load in lbs4 in lbs4 values4 ---------------------------------------- -------------------- #a #djj #aj #aj #aeb #aeb #jj #b #djj #je #aj #gg #gf -#aj #c #djj #aj #je #ch #ch #jj #d #djj #je #je #ai #ai #jj #e #cjj #aj #je #ei #ea -#hj #f #cjj #je #je #be #be #jj #g #bjj #aj #je #gd #gf #bj #h #bjj #je #je #cf #ch #bj #i #ajj #aj #je #aed #aeb -#bj #aj #ajj #je #je #fh #gf #hj ---------------------------------------- -------------------- #cha4 ,in the first column is a series of figures for convenience of reference4 ,the next three columns are occupied with the dimensions of the beams4 ,by span is meant the distance between the points of support2 the real length is of course greater2 the depth is that dimension of the beam which is vertical4 ,the fifth column gives the mean of two observations of the breaking load4 ,thus for example1 in experiment ,no4 #e the two beams used were each #cf #a #je1 they were placed on points of support #cj distant1 so the span recorded is #cj3 one of the beams was broken by a load of #eh lbs41 and the second by a load of #fj lbs42 the mean between the two1 #ei lbs41 is recorded as the mean breaking load4 ,in this manner the column of breaking loads has been found4 ,the meaning of the two last columns of the table will be explained presently4 #chb4 ,we shall endeavour to elicit from these observations the laws which connect the breaking load with the span1 breadth1 and depth of the beam4 #chc4 ,let us first examine the effect of the span2 for this purpose we bring together the observations upon beams of the same section1 but of different spans4 ,sections of #je #je will be convenient for this purpose2 ,nos4 #d1 #f1 #h1 and #aj are experiments upon beams of this section4 ,let us first compare #d and #h4 ,here we have two beams of the same section1 and the span of one 7#dj7 is double that of the other 7#bj74 ,when we examine the breaking weights we find that they are #ai lbs4 and #cf lbs42 the former of these numbers is rather more than half of the latter4 ,in fact1 had the breaking load of #dj been lb4 less1 #ahbe lbs41 and had that of #bj been lb4 more1 #cfe lbs41 one of the breaking loads would have been exactly half the other4 #chd4 ,we must not look for perfect numerical accuracy in these experiments2 we must only expect to meet with approximation1 because the laws for which we are in search are in reality only approximate laws4 ,wood itself is variable in quality1 even when cut from the same piece3 parts near the circumference are different in strength from those nearer the centre2 in a young tree they are generally weaker1 and in an old tree generally stronger4 ,minute differences in the grain1 greater or less perfectness in the seasoning1 these are also among the circumstances which prevent one piece of timber from being identical with another4 ,we shall1 however1 generally find that the effect of these differences is small1 but occasionally this is not the case1 and in trying many experiments upon the breaking of timber1 discrepancies occasionally appear for which it is difficult to account4 #che4 ,but you will find1 ,i think1 that1 making reasonable allowances for such difficulties as do occur1 the laws on the whole represent the experiments very closely4 #chf4 ,we shall1 then1 assume that the breaking weight of a bar of #dj is half that of a bar of #bj of the same section1 and we ask1 ,is this generally true8 is it true that the breaking weight is inversely proportional to the span8 ,in order to test this hypothesis1 we can calculate the breaking weight of a bar of #cj 7,no4 #f71 and then compare the result with the observed value2 if the supposition be true1 the breaking weight should be given by the proportion-- #cj 3 #dj 33 #ai 3 ,answer4 ,the answer is #bec lbs42 on reference to the table we find #be lbs4 to be the observed value1 hence our hypothesis is verified for this bar4 #chg4 ,let us test the law also for the #aj bar1 ,no4 #aj-- #aj 3 #dj 33 #ai 3 ,answer4 ,the answer in this case is #gf1 whereas the observed value is #fh1 or #h lbs4 less2 this does not agree very well with the theory1 but still the difference1 though #h lbs41 is only about #aa or #ab per cent4 of the whole1 and we shall still retain the law1 for certainly there is no other that can express the result as well4 #chh4 ,but the table will supply another verification4 ,in experiment ,no4 #c a #dj bar1 #a broad1 and #je deep1 broke with #ch lbs42 and in experiment ,no4 #g a #bj bar of the same section broke with #gd lbs42 but this is so nearly double the breaking weight of the #dj bar1 as to be an additional illustration of the law1 that for a given section the breaking load varies inversely as the span4 #chi4 ,we next inquire as to the effect of the breadth of the beam upon its strength8 ,for this purpose we compare experiments ,nos4 #c and #d3 we there find that a bar #dj #a #je is broken by a load of #ch lbs41 while a bar just half the breadth is broken by #ai lbs4 ,we might have anticipated this result1 for it is evident that the bar of ,no4 #c must have the same strength as two bars similar to that of ,no4 #d placed side by side4 #cij4 ,this view is confirmed by a comparison of ,nos4 #g and #h1 where we find that a #bj bar takes twice the load to break it that is required for a bar of half its breadth4 ,the law is not quite so well verified by ,nos4 #e and #f1 for half the breaking weight of ,no4 #e1 namely #bie lbs41 is more than #be1 the observed breaking weight of ,no4 #f3 a similar remark may be made about ,nos4 #i and #aj4 #cia4 ,supposing we had a beam of #dj span1 #b broad1 and #je deep1 we can easily see that it is equivalent to two bars like that of ,no4 #c placed side by side2 and we infer generally that the strength of a bar is proportional to its breadth2 or to speak-more definitely1 if two beams have the same span and depth1 the ratio of their breaking loads is the same as the ratio of their breadths4 #cib4 ,we next examine the effect of the depth of a beam upon its strength4 ,in experimenting upon a beam placed edgewise1 a precaution must be observed1 which would not be necessary if the same beam were to be broken flatwise4 ,when the load is suspended1 the beam1 if merely laid edgewise on the supports1 would almost certainly turn over2 it is therefore necessary to place its extremities in recesses in the supports1 which will obviate the possibility of this occurrence2 at the same time the ends must not be prevented from bending upwards1 for we are at present discussing a beam free at each end1 and the case where the ends are not free will be subsequently considered4 #cic4 ,let us first compare together experiments ,nos4 #b and #c2 here we have two bars of the same dimensions1 the section in each being #aj #je1 but the first bar is broken edgewise1 and the second flatwise4 ,the first breaks with #gg lbs41 and the second with #ch lbs42 hence the same bar is twice as strong placed edgewise as flatwise when one dimension of the section is twice as great as the other4 ,we may generalize this law1 and assert that the strength of a rectangular beam broken edgewise is to the strength of a beam of like span and section broken flatwise1 as the greater dimension of the section is to the lesser dimension4 #cid4 ,the strength of a beam #dj #je #a is four times as great as the strength of #dj #je #je1 though the quantity of wood is only twice as great in one as in the other4 ,in general we may state that if a beam were bisected by a longitudinal cut1 the strength of the beam would be halved when the cut was horizontal1 and unaltered when the cut was vertical2 thus1 for example1 two beams of experiment ,no4 #d1 placed one on the top of the other1 would break with about #dj lbs41 whereas if the same rods were in one piece1 the breaking load would be nearly #hj lbs4 #cie4 ,this may be illustrated in a different manner4 ,i have here two beams of #dj #a #je superposed2 they form one beam1 equivalent to that of ,no4 #a in bulk1 but ,i find that they break with #hj lbs41 thus showing that the two are only twice as strong as one4 #cif4 ,i take two similar bars1 and1 instead of laying them loosely one on the other1 ,i unite them tightly with iron clamps like those represented in ,fig4 #ef4 ,i now find that the bars thus fastened together require #ajd lbs4 for fracture4 ,we can readily understand this increase of strength4 ,as soon as the bars begin to bend under the action of the weight1 the surfaces which are in contact move slightly one upon the other in order to accommodate themselves to the change of form4 ,by clamping ,i greatly impede this motion hence the beams deflect less1 and require a greater load before they collapse2 the case is therefore to some extent approximated to the state of things when the two rods form one solid piece1 in which case a load of #aeb lbs4 would be required to produce fracture4 #cig4 ,we shall be able by a little consideration to understand the reason why a bar is stronger edgewise than flatwise4 ,suppose ,i try to break a bar across my knee by pulling the ends held one in each hand1 what is it that resists the breaking8 ,it is chiefly the tenacity of the fibres on the convex surface of the bar4 ,if the bar be edgewise1 these fibres are further away from my knee and therefore resist with a greater moment than when the bar is flatwise3 nor is the case different when the bar is supported at each end1 and the load placed in the centre2 for then the reactions of the supports correspond to the forces with which ,i pulled the ends of the bar4 #cih4 ,we can now calculate the strength of any rectangular beam of pine3 ,let us suppose it to be #ab' long1 #e broad1 and #g deep4 ,this is five times as strong as a beam #a broad and #g deep for we may conceive the original beam to consist of #e of these beams placed side by side 7,art #cia72 the beam #a broad and #g deep1 is #g times as strong as a beam #g broad1 #a deep 7,art4 #cic74 ,hence the original beam must be #ce times as strong as a beam #g broad1 #a deep2 but the beam #g broad and #a deep is seven times stronger than a beam the section of which is #a #a1 hence the original beam is #bde times as strong as a beam #ab' long and #a #a in section2 of which we can calculate the strength1 by ,art4 #chh1 from the proportion-- #add 3 #dj 33 #aeb 3 ,answer4 ,the answer is #dbb lbs41 and thus the breaking load of the original beam is about #aj1cjj lbs4 #cii4 ,it will be useful to deduce the general expression for the breaking load of a beam l span1 b broad1 and d deep1 supported freely at the ends and laden in the centre4 ,let us suppose a bar l long1 and #a #a in section4 ,the breaking load is found by the proportion-- l 3 #dj 33 #aeb 3 ,answer2 and the result obtained is #fjhj_/l4 ,a beam which is d broad1 l span1 and #a deep1 would be just as strong as d of the beams l #a #a placed side by side2 of which the collective strength would be-- #fjhj ---------------- d4 l ,if such a beam1 instead of resting flatwise1 were placed edgewise1 its strength would be increased in the ratio of its depth to its breadth--that is1 it would be increased d-fold--and would therefore amount to #fjhj ------------------ d4 l ,we thus learn the strength of a beam #a broad1 d deep1 and l span4 ,the strength of b of these beams placed side by side1 would be the same as the strength of one beam b broad1 d deep1 and l span1 and thus we finally obtain #fjhj ---------------- d b4 l ,since b d is the area of the section1 we can express this result conveniently by saying that the breaking load in lbs4 of a rectangular pine beam is equal to area of section depth2 #fjhj ---------------------------------------- ---------- span the depth and span being expressed in inches linear measure1 and the section in square inches4 #djj4 ,in order to test this formula1 we have calculated from it the breaking loads of all the ten beams given in ,table ,,xxiv4 and the results are given in the sixth column4 ,the difference between the amount calculated and the observed mean breaking weight is shown in the last column4 #dja4 ,thus1 for example1 in experiment ,no4 #g the span is #bj1 breadth1 #a1 depth #je2 the formula gives1 since the area is #je1 #je #je ,p #fjhj -------------------- #gf #bj ,this agrees sufficiently with #gd lbs41 the mean of two observed values4 #djb4 ,except in experiments ,nos4 #e and #aj1 the differences are very small1 and even in these two cases the differences are not sufficient to make us doubt that we have discovered the correct expression for the load generally sufficient to produce fracture4 #djc4 ,we have already pointed out that a beam begins to sustain permanent injury when it is subjected to a load greater than half that which would break it 7,art4 #cfh71 and we may infer that it is not in general prudent to load a beam which is part of a permanent structure with more than about a third or a fourth of the breaking weight4 ,hence if we wanted to calculate a fair working load in lbs4 for a beam of pine1 we might obtain it from the formula4 area of section depth4 #aejj ---------------------------------------- ---------- span ,probably a smaller coefficient than #aejj would often be used by the cautious builder1 especially when the beam was liable to sudden blows or shocks4 ,the coefficient obtained from small selected rods such as we have used would also be greater than that found from large beams in which imperfections are inevitable4 #djd4 ,had we adopted any other kind of wood we should have found a similar formula for the breaking weight1 but with a different numerical coefficient4 ,for example1 had the beams been made of oak the number #fjhj must be replaced by a larger figure4 ,a ,,beam ,,uniformly ,,loaded4 #dje4 ,we have up to the present only considered the case where the load is suspended from the centre of the beam4 ,but in the actual employment of beams the load is not generally applied in this manner4 ,see in the rafters which support a roof how every inch in the entire length has its burden of slates to bear4 ,the beams which support a warehouse floor have to carry their load in whatever manner the goods are disposed3 sometimes1 as for example in a grain-store1 the pressure will be tolerably uniform along the beams1 while if the weights be irregularly scattered on the floor1 there will be corresponding inequalities in the mode in which the loads are distributed over the beams4 ,it will therefore be useful for us to examine the strength of a beam when its load is applied otherwise than at the centre4 #djf4 ,we shall employ1 in the first place1 a beam #dj span1 #je broad1 and #a deep2 and we shall break it by applying a load simultaneously at two points1 as may be most conveniently done by the contrivance shown in the diagram1 ,fig4 #ec4 ,a ,b is the beam resting on two supports2 ,c and ,d are the points of trisection of the span2 from whence loops descend1 which carry an iron bar ,p ,q2 at the centre ,r of which a weight ,w is suspended4 ,the load is thus divided equally between the two points ,c and ,d1 and we may regard ,a ,b as a beam loaded at its two points of trisection4 ,the tray and weights are employed which we have used in the apparatus represented in ,fig4 #eh4 ,illustration3 ,,fig4 #ec4 #djg4 ,we proceed to break this beam4 ,adding weights to the tray1 we see that it yields with #aag lbs41 and cracks across between ,c and ,d4 ,on reference to ,table ,,xxiv4 we find from experiment ,no4 #b that a similar bar was broken by #gg lbs4 at the centre2 now _/ #gg #aaee2 hence we may state with sufficient approximation that the bar is half as strong again when the load is suspended from the two points of trisection as it is when suspended from the centre4 ,it is remarkable that in breaking the beam in this manner the fracture is equally likely to occur at any point between ,c and ,d4 #djh4 ,a beam uniformly loaded requires twice as much load to break it as would be sufficient if the load were merely suspended from the centre4 ,the mode of applying a load uniformly is shown in ,fig4 #ed4 ,illustration3 ,,fig4 #ed4 ,in an experiment actually tried1 a beam #dj #je #a placed edgewise was found to support ten #ad lb4 weights ranged as in the figure2 one or two stone more would1 however1 doubtless produce fracture4 #dji4 ,we infer from these considerations that beams loaded in the manner in which they are usually employed are considerably stronger than would be indicated by the results in ,table ,,xxiv4 ,,effect ,,of ,,securing ,,the ,,ends ,,of ,a ,,beam ,,upon ,,its ,,strength4 #daj4 ,it has been noticed during the experiments that when the weights are suspended from a beam and the beam begins to deflect1 the ends curve upwards from the supports4 ,this bending of the ends is for example shown in ,fig4 #ed4 ,if we restrain the ends of the beam from bending up in this manner1 we shall add very considerably to its strength4 ,this we can do by clamping them down to the supports4 #daa4 ,let us experiment upon a beam #dj #a #a4 ,we clamp each of the ends and then break the beam by a weight suspended from the centre4 ,it requires #bch lbs4 to accomplish fracture4 ,this is a little more than half as much again as #aeb lbs41 which we find from ,table ,,xxiv4 was the weight required to break this bar when its ends were free4 ,calculation shows that the strength of a beam may be even doubled when the ends are kept horizontal by more perfect methods than we have used4 #dab4 ,when the beam gives way under these circumstances1 there is not only a fracture in the centre1 but each of the halves are also found to be broken across near the points of support2 the necessity for three fractures instead of one explains the increase of strength obtained by restraining the ends to the horizontal direction4 #dac4 ,in structures the beams are generally more or less secured at each end1 and are therefore more capable of bearing resistance than would be indicated by ,table ,,xxiv4 ,from the consideration of ,arts4 #djh and #daa1 we can infer that a beam secured at each end and uniformly loaded would require three or four times as much load to break it as would be sufficient if the ends were free and if the load were applied at the centre4 ,,beams ,,secured ,,at ,,one ,,end ,,and ,,loaded ,,at ,,the ,,other4 #dad4 ,a beam1 one end of which is firmly imbedded in masonry or otherwise secured1 is occasionally called upon to support a weight suspended from its extremity4 ,such a beam is shown in ,fig4 #ee4 ,in the case we shall examine1 ,a ,b is a pine beam of dimensions #bj #je #je1 and we find that1 when ,w reaches #aj lbs41 the beam breaks4 ,in experiment ,no4 #h1 ,table ,,xxiv41 a similar beam required #cf lbs42 hence we see that the beam is broken in the manner of ,fig4 #ee1 by about one-fourth of the load which would have been required if the beam had been supported at each end and laden in the centre4 ,illustration3 ,,fig4 #ee4 ,we shall presently have occasion to apply some of the results obtained by the experiments made in the lecture now terminated4 ,,lecture ,,xiii4 ,,the ,,principles ,,of ,,framework4 ,introduction4--,weight sustained by ,tie and ,strut4--,bridge with ,two ,struts4--,bridge with ,four ,struts4--,bridge with ,two ,ties4--,simple ,form of ,trussed ,bridge4 ,,introduction4 #dae4 ,in this lecture and the next we shall experiment upon some of the arts of construction4 ,we shall employ slips of pine #je #je in section for the purpose of making models of simple framework3 these slips can be attached to each other by means of the small clamps about #c long1 shown in ,fig4 #ef1 and the general appearance of the models thus produced may be seen from ,figs4 #eh and #fb4 ,illustration3 ,,fig4 #ef4 #daf4 ,the following experiment shows the tenacity with which these clamps hold4 ,two slips of pine1 each #ab #je #je1 are clamped together1 so that they overlap about #b1 thus forming a length of #bb3 this composite rod is raised by a pulley-block as in ,fig4 #di1 while a load of #b cwt4 is suspended from it4 ,thus the clamped rods bear a direct tension of #b cwt4 ,the efficiency of the clamps depends principally upon friction1 aided doubtless by a slight crushing of the wood1 which brings the surfaces into perfect contact4 #dag4 ,these slips of pine united by the clamps are possessed of strength quite sufficient for the experiments now to be described4 ,models thus constructed have the great advantage of being erected1 varied or pulled down1 with the utmost facility4 ,we have learned that the compressive strength1 and1 still more1 the tensile strength of timber1 is much greater than its transverse strength4 ,this principle is largely used in the arts of construction4 ,we endeavour by means of suitable combinations to turn transverse forces into forces of tension or compression1 and thus strengthen our constructions4 ,we shall illustrate the mode of doing so by simple forms of framework4 ,,weight ,,sustained ,,by ,,tie ,,and ,,strut4 #dah4 ,we begin with the study of a very simple contrivance1 represented in ,fig4 #eg4 ,a ,b is a rod of pine #bj long4 ,in the diagram it is represented1 for simplicity1 imbedded at the end ,a in the support4 ,in reality1 however1 it is clamped to the support1 and the same remark may be made about some other diagrams used in this lecture4 ,were ,a ,b unsupported except at its end ,a1 it would of course break when a weight of #aj lbs4 was suspended at ,b1 as we have already found in ,art4 #dad4 #dai4 ,we must ascertain whether the transverse force on ,a ,b cannot be changed into forces of tension and compression4 ,the tie ,b ,c is attached by means of clamps2 ,a ,b is sustained by this tie2 it cannot bend downwards under the action of the weight ,w1 because we should then require to have on the same base and on the same side of it two triangles having their conterminous sides equal1 but this we know from ,euclid 7,i4 #g7 is impossible4 ,hence ,b is supported1 and we find that #aab lbs4 may be safely suspended1 so that the strength is enormously increased4 ,in fact the transverse force is changed into a compressive force or thrust down ,a ,b1 and a tensile force on ,b ,c4 ,illustration3 ,,fig4 #eg4 #dbj4 ,the actual magnitudes of these can be computed4 ,draw the parallelogram ,c ,d ,e ,b2 if ,b ,d represent the weight ,w1 it may be resolved into two forces1--one1 ,b ,c1 a force of extension on the tie2 the other1 ,b ,e1 a compressive force on ,a ,b1 which is therefore a strut4 ,hence the forces are proportional to the sides of the triangle1 ,a ,b ,c4 ,in the present case ,a ,b #bj1 ,a ,c #ah1 ,b ,c #bg2 therefore1 when ,w is #aab lbs41 we calculate that the force on ,a ,b is #abd lbs41 and on ,c ,b #afh lbs4 ,a ,b would require about #cjj lbs4 to crush it1 and ,c ,b about #b1jjj lbs4 to tear it asunder1 consequently the tie and strut can support #a cwt4 with ease4 ,if1 however1 ,w were increased to about #bgj lbs41 the force on ,a ,b would become too great1 and fracture would arise from the collapse of this strut4 #dba4 ,when a structure is loaded up to the breaking point of one part1 it is proper for economy that all the other parts should be so designed that they shall be as near as possible to their breaking points4 ,in fact1 since nothing is stronger than its weakest part1 any additional strength which the remaining parts may possess adds no strength to the whole1 and is only so much material wasted4 ,hence our structure would be just as strong1 and would be more properly designed if the section of ,b ,c were reduced to one-fifth1 for the tie would then break when the tension upon it amounted to #djj lbs4 ,when ,w is #bgj lbs4 the compression on ,a ,b is #cjj lbs41 and the tension on ,b ,c is #dje lbs41 so that both tie and strut attain their breaking loads together4 ,the principle of duly apportioning the strength of each piece to the load it has to carry1 involves the essence of sound engineering4 ,in that greatest of mechanical feats1 the construction of a mighty railway bridge across a wide span1 attention to this principle is of vital importance4 ,such a bridge has to bear the occasional load of a passing train1 but it has always to support the far greater load of the bridge materials4 ,there is thus every inducement to make the weight of each part of the bridge as light as may be consistent with safety4 ,a ,,bridge ,,with ,,two ,,struts4 #dbb4 ,we shall next examine the structure of a type of bridge1 shown in ,fig4 #eh4 ,illustration3 ,,fig4 #eh4 ,it consists of two beams1 ,a ,b1 #d' long1 placed parallel to each other at a distance of #ce1 and supported at each end2 they are firmly clamped to the supports1 and a roadway of short pieces is laid upon them4 ,at the points of trisection of the beams ,c1 ,d1 struts ,c ,f and ,d ,e are clamped1 their lower ends being supported by the framework3 these struts are #b' long1 and there are two of them supporting each of the beams4 ,the tray ,g is attached by a chain to a stout piece of wood1 which rests upon the roadway at the centre of the bridge4 #dbc4 ,we shall first determine the strength of this bridge by actual experiment1 and then we shall endeavour to explain the results in accordance with mechanical principles4 ,we can observe the deflection of the bridge by the cathetometer in the manner already described 7,art4 #cfb74 ,by this means we shall ascertain whether the load has permanently injured the elasticity of the structure 7,art4 #cfg74 ,we begin by testing the deflection when a load is distributed uniformly1 as the weights are disposed in the case of ,fig4 #fb4 ,a cross is marked upon one of the beams1 and is viewed in the cathetometer4 ,we arrange #aa stone weights along the bridge1 and the cathetometer shows that the deflection is only #jji3 the elasticity of the bridge remains unaltered1 for when the weights are removed the cross on the beam returns to its original position2 hence the bridge is well able to bear this load4 #dbd4 ,we remove the row of weights from the bridge and suspend the tray from the roadway4 ,i take my place at the cathetometer to note the deflection1 while my assistant places weights ,h ,h on the tray4 #a cwt4 being the load1 ,i see that the deflection amounts to #jb2 with #b cwt4 the deflection reaches #jdc2 and the bridge breaks with #bch lbs4 #dbe4 ,let us endeavour to calculate the additional strength which the struts have imparted to the bridge4 ,by ,table ,,xxiv4 we see that a rod #dj #je #je is broken by a load of #ai lbs43 hence the beams of the bridge would have been broken by a load of #ch lbs4 if their ends had been free4 ,as1 however1 the ends of the beams had been clamped down1 we learn from ,art4 #daa that a double load would be necessary4 ,we may1 however1 be confident that about #hj lbs4 would have broken the unsupported bridge4 ,the strength is1 therefore1 increased threefold by the struts1 for a load of #bch lbs4 was required to produce fracture4 #dbf4 ,we might have anticipated this result1 because the points ,c and ,d being supported by the struts may be considered as almost fixed points2 in fact1 we see that ,c cannot descend1 because the triangle ,a ,c ,f is unalterable1 and for a similar reason ,d remains fixed3 the beam breaks between ,c and ,d1 and the force required must therefore be sufficient to break a beam supported at the points ,c and ,d1 whose ends are secured4 ,but ,c ,d is one-third of ,a ,b1 and we have already seen that the strength of a beam is inversely as its length 7,art4 #chh72 hence the force required to break the beam when supported by the struts is three times as large as would have been necessary to break the unsupported beam4 ,thus the strength of the bridge is explained4 #dbg4 ,as a load of #bch lbs4 applied near the centre is necessary to break this bridge1 it follows from the principle of ,art4 #djh that a load of about double this amount must be placed uniformly on the roadway before it succumbs2 we can1 therefore1 understand how a load of #aa stone was easily borne 7,art #dbc7 without permanent injury to the elasticity of the structure4 ,if we take the factor of safety as #c1 we see that a bridge of the form we have been considering may carry1 as its ordinary working load1 a far greater weight than would have crushed it if unsupported by the struts and with free ends4 #dbh4 ,the strength of the bridge in ,fig4 #eh is greater in some parts than in others4 ,at the points ,c and ,d a maximum load could be borne2 the weakest places on the bridge are in the middle points of the segments ,a ,c1 ,d ,c1 and ,d ,b4 ,the load applied by the tray was principally borne at the middle of ,d ,c1 but owing to the piece of wood which sustained the chain being about #ah long1 the load was to some extent distributed4 ,the thrust upon the struts is not so easy to calculate accurately4 ,that down ,c ,f for example must be less than if the part ,c ,d were removed1 and half the load were suspended from ,c4 ,the force in this case can be determined by principles already explained 7,art4 #dbj74 ,a ,,bridge ,,with ,,four ,,struts4 #dbi4 ,the same principles that we have employed in the construction of the bridge of ,fig4 #eh may be extended further1 as shown in the diagram of ,fig4 #ei4 ,illustration3 ,,fig4 #ei4 ,we have here two horizontal rods1 #dh #je #je1 each end being secured to the supports2 one of these rods is shown in the figure4 ,it is divided into five equal parts in the points ,b1 ,c1 ,c1 ,b4 ,we support the rod in these four points by struts1 the other extremities of which are fastened to the framework4 ,the points ,b1 ,c1 ,c1 ,b are fixed1 as they are sustained by the struts3 hence a weight suspended from ,p1 which is to break the bridge1 must be sufficiently strong to break a piece ,c ,c1 which is secured at the ends2 the rod ,a ,a would have been broken with #ch lbs41 hence #aij lbs4 would be necessary to break ,c ,c4 ,there is a similar beam on the other side of the bridge1 and therefore to break the bridge #chj lbs4 would be necessary1 but this force must be applied exactly at the centre of ,c ,c2 and if the weights be spread over any considerable length1 a heavier load will be necessary4 ,in fact1 if ,i were to distribute the weight uniformly over the distance ,c ,c1 it appears from ,art4 #djh that double the load would be necessary to produce fracture4 #dcj4 ,we shall now break this model4 ,i place #ah stone upon it ranged uniformly1 and the cathetometer tells me that the bridge only deflects #ja1 and that its elasticity is not injured4 ,placing the tray in position1 and loading the bridge by this means1 ,i find with a weight of #b cwt4 that there is a deflection of #jae2 with #d cwt4 the deflection amounts to #j'gb4 ,we therefore infer that the bridge is beginning to yield1 and the clamps give way when the load is increased to #ejj lbs4 ,a ,,bridge ,,with ,,two ,,ties4 #dca4 ,it might happen that circumstances would not make it convenient to obtain points of support below the bridge on which to erect the struts4 ,in such a case1 if suitable positions for ties can be obtained1 a bridge of the form represented in ,fig4 #fj may be used4 ,a ,d is a horizontal rod of pine #dj #je #je2 it is trisected in the points ,b and ,c1 from which points the ties ,b ,e and ,c ,f are secured to the upper parts of the framework4 ,a ,d is then supported in the points ,b and ,c1 which may therefore be regarded as fixed points4 ,hence1 for the reasons we have already explained1 the strength of the bridge should be increased nearly threefold4 ,remembering that the bridge has two beams we know it would require about #gj lbs4 or #hj lbs4 to produce fracture without the ties1 and therefore we might expect that over #bjj lbs4 would be necessary when the beams were supported by the ties4 ,i perform the experiment1 and you see the bridge yields when the load reaches #aid lbs43 this is somewhat less than the amount we had calculated2 the reason being1 ,i think1 that one of the clamps slipped before fracture4 ,illustration3 ,,fig4 #fj4 ,a ,,simple ,,form ,,of ,,truss4 #dcb4 ,it is often not convenient1 or even possible1 to sustain a bridge by the methods we have been considering4 ,it is desirable therefore to inquire whether we cannot arrange some plan of strengthening a beam1 by giving to it what shall be equivalent to an increase of depth4 #dcc4 ,we shall only be able to describe here some very simple methods for doing this4 ,superb examples are to be found in railway bridges all over the country1 but the full investigation of these complex structures is a problem of no little difficulty1 and one into which it would be quite beyond our province to enter4 ,we shall1 however1 show how by a judicious combination of several parts a structure can offer sufficient resistance4 ,the most complex lattice girder is little more than a network of ties and struts4 ,illustration3 ,,fig4 #fa4 #dcd4 ,let ,a ,b 7,fig4 #fa7 be a rod of pine #dj #je #je1 secured at each end4 ,we shall suppose that the load is applied at the two points ,g and ,h1 in the manner shown in the figure4 ,the load which a bridge must bear when a train passes over it is distributed over a distance equal to the length of the train1 and the weight of the bridge itself is of course arranged along the entire span2 hence the load which a bridge bears is at all times more or less distributed and never entirely concentrated at the centre in the manner we have been considering4 ,in the present experiment we shall apply the breaking load at the two points ,g and ,h1 as this will be a variation from the mode we have latterly used4 ,e ,f is an iron bar supported in the loops ,e ,g and ,f ,h4 ,let us first try what weight will break the beam4 ,suspending the tray from ,e ,f1 ,i find that a load of #dh lbs4 is sufficient2 much less would have done had not the ends been clamped4 ,we have already applied a load in this manner in ,art4 #djf4 #dce4 ,you observed that the beam1 as usual1 deflected before it broke2 if we could prevent deflection we might reasonably expect to increase the strength4 ,thus if we support the centre of the beam ,c1 deflection would be prevented4 ,this can be done very simply4 ,we clamp the pieces ,d ,a1 ,d ,b1 ,d ,c1 on a similar beam1 and it is evident that ,c cannot descend so long as the joints at ,a1 ,b1 ,d1 ,c remain firmly secured4 ,we now find that even with a weight of #aab lbs4 in the tray1 the bar is unbroken4 ,an arrangement of this kind is frequently employed in engineering1 for it seems to be able to bear more than double the load which is sufficient to break the unsupported beam4 ,illustration3 ,,fig4 #fb4 #dcf4 ,two frames of this kind1 with a roadway laid between them1 would form a bridge1 or if the frames were turned upside down they would answer equally well1 though of course in this case ,d ,a and ,d ,b would become ties1 and ,d ,c a strut1 but a better arrangement for a bridge will be next described4 ,,the ,,wye ,,bridge4 #dcg4 ,an instructive bridge was erected by the late ,sir ,i4 ,brunel over the ,wye1 for the purpose of carrying a railway4 ,the essential parts of the bridge are represented in the model shown in ,fig4 #fb1 which as before is made of slips of pine clamped together4 #dch4 ,our model is composed of two similar frames1 one of which we shall describe1 ,a ,b is a rod of pine #dh #je #je1 supported at each extremity4 ,this rod is sustained at its points of trisection ,d1 ,c by the uprights ,d ,e and ,c ,f1 while ,e and ,f are supported by the rods ,b ,e1 ,f ,e1 and ,a ,f2 the rectangle ,d ,e ,f ,c is stiffened by the piece ,c ,e1 and it would be proper in an actual structure to have a piece connecting ,d and ,f1 but it has not been introduced into the model4 ,illustration3 ,,fig4 #fc4 #dci4 ,we shall understand the use of the diagonal ,c ,e by an inspection of ,fig4 #fc4 ,suppose the quadrilateral ,a ,b ,c ,d be formed of four pieces of wood hinged at the corners4 ,it is evident that this quadrilateral can be deformed by pressing ,a and ,c together1 or by pulling them asunder4 ,even if there were actual joints at the corners1 it would be almost impossible to make the quadrilateral stiff by the strength of the joints4 ,you see this by the frame which ,i hold in my hand2 the pieces are clamped together at the corners1 but no matter how tightly ,i compress the clamps1 ,i am able with the slightest exertion to deform the figure4 #ddj4 ,we must therefore look for some method of stiffening the frame4 ,i have here a triangle of three pieces1 which have been simply clamped together at the corners2 this triangle is unalterable in form2 in fact1 since it is impossible to make two different triangles with the same three sides1 it is evident the triangle cannot be deformed4 ,this points to a guiding principle in all bridgework4 ,the quadrilateral is not stiff because innumerable different quadrilaterals can be made with the same four sides4 ,but if we draw the diagonal ,a ,c of the quadrilateral it is divided into two triangles1 and hence when we attach to the quadrilateral1 which has been clamped at the four corners1 an additional piece in the direction of one of the diagonals1 it becomes unalterable in shape4 #dda4 ,in ,fig4 #fc we have drawn the two diagonals ,a ,c and ,b ,d3 one would be theoretically sufficient1 but it is desirable to have both1 and for the following reason4 ,if ,i pull ,a and ,c apart1 ,i stretch the diagonal ,a ,c and compress ,b ,d4 ,if ,i compress ,a and ,c together1 ,i compress the line ,a ,c and extend ,b ,d2 hence in one of these cases ,a ,c is a tie1 and in the other it is a strut4 ,it therefore follows that in all cases one of the diagonals is a tie1 and the other a strut4 ,if then we have only one diagonal1 it is called upon to perform alternately the functions of a tie and of a strut4 ,this is not desirable1 because it is evident that a piece which may act perfectly as a tie may be very unsuitable for a strut1 and vice vers4 ,but if we insert both diagonals we may make both of them ties1 or both of them struts1 and the frame must be rigid4 ,thus for example1 ,i might make ,a ,c and ,b ,d slender bars of wrought iron1 which form admirable ties1 though quite incapable of acting as struts4 #ddb4 ,what we have said with reference to the necessity for dividing a quadrilateral figure into triangles applies still more to a polygon with a large number of sides1 and we may lay down the general principle that every such piece of framework should be composed of triangles4 #ddc4 ,returning to ,fig4 #fb1 we see the reason why the rectangle ,e ,d ,c ,f should have one or both of its diagonals introduced4 ,a load placed1 for example1 at ,d would tend to depress the piece ,d ,e1 and thus deform the rectangle1 but when the diagonals are introduced this deformation is impossible4 #ddd4 ,hence one of these frames is almost as strong as a beam supported at the points ,c and ,d1 and therefore1 from the principles of ,art4 #chh1 its strength is three times as great as that of an unsupported beam4 #dde4 ,the two frames placed side by side and carrying a roadway form an admirable bridge1 quite independent of any external support1 except that given by the piers upon which the extremities of the frames rest4 ,it would be proper to connect the frames together by means of braces1 which are not1 however1 shown in the figure4 ,the model is represented as carrying a uniform load in contradistinction to ,fig4 #eh1 where the weight is applied at a single point4 #ddf4 ,with eight stone ranged along it1 the bridge of ,fig4 #fb did not indicate an appreciable deflection4 ,,lecture ,,xiv4 ,,the ,,mechanics ,,of ,a ,,bridge4 ,introduction4--,the ,girder4--,the ,tubular ,bridge4--,the ,suspension ,bridge4 ,,introduction4 #ddg4 ,perhaps it may be thought that the structures we have been lately considering are not those which are most universally used1 and that the bridges which are generally referred to as monuments of engineering skill are of quite a different construction4 ,every one is familiar with the arch1 and most of us have seen suspension bridges and the celebrated ,menai tube4 ,we must therefore allude further to some of these structures1 and this we propose to do in the present lecture4 ,it will only be possible to take a very slight survey of an extensive subject to which elaborate treatises have been devoted4 ,we shall first give a brief account of the use of iron in the arts of construction4 ,we shall then explain simply the principle of the tubular bridge1 and also of the suspension bridge4 ,the more complex forms are beyond our scope4 ,,the ,,girder4 #ddh4 ,a horizontal beam supported at each end1 and perhaps at intermediate points1 and designed to support a heavy load is called a girder4 ,those rods upon which we have performed experiments1 the results of which have been given in ,table ,,xxiv41 are small girders2 but the term is generally understood to relate to structures of iron3 the greatest girders for railway bridges are made of bars or plates of iron riveted together4 #ddi4 ,we shall first consider the application of cast iron to girders1 and show what form they should assume4 #dej4 ,a beam of cast iron1 supposing its section to be rectangular1 has its strength determined by the same laws as the beams of pine4 ,thus1 supposing the section of two beams to be the same1 their strengths are inversely proportional to their lengths1 and the strength of a beam placed edgewise is to its strength placed flatwise in the proportion of the greater dimension of its section to the less dimension4 ,these laws determine the strength of every rectangular beam of cast iron when that of one beam is known1 and we must perform an experiment in order to find the breaking load in a particular case4 #dea4 ,i take here a piece of cast iron1 which is #b' long1 and #je #je in section4 ,i support this beam at each end upon a frame2 the distance between the supports is #bj4 ,i attach the tray to the centre of the beam and load it with weights4 ,the ends of the beam rest freely upon the supports1 but ,i have taken the precaution of tying each end by a piece of wire1 so that they may not fly about when the fracture occurs4 ,loading the tray1 ,i find that with #bhj lbs4 the crash comes4 #deb4 ,let us compare this result with ,no4 #h of ,table ,,xxiv4 7p4 #aij74 ,there we find that a piece of pine1 the same size as the cast iron1 was broken with #cf lbs43 the ratio of #bhj to #cf is nearly #h1 so that the beam of cast iron is about #h times as strong as the piece of pine of the same size4 ,this result is a little larger than we would have inferred from an examination of tables of the strength of large bars of cast iron2 the reason may be that a very small casting1 such as this bar1 is stronger in proportion than a larger one1 owing to the iron not being so uniform throughout the larger mass4 #dec4 ,i hold here a bar of cast iron #ab long and #a #a in section4 ,i have not sufficient weights at hand to break it1 but we can compute how much would be necessary by our former experiment4 #ded4 ,in the first place a bar #ab long1 and #je #je of section1 would require #bj #bhj #ab #dfg lbs4 by the law that the strength is inversely as the length4 ,we also know that one beam #ab #a #a is just as strong as two beams #ab #a #je1 each placed edgewise2 each of these latter beams is twice as strong as #ab #a #je placed flatwise1 because the strength when placed edgewise is to the strength when placed flatwise1 as the depth to the breadth1 that is as #b to #a3 hence the original beam is four times as strong as one beam #ab #a #je placed flatwise3 but this last beam is twice as strong as a beam #ab #je #je1 and hence we see that a beam #ab #a #a must be #h times as strong as a beam of #ab #je #je1 but this last beam would require a load of #dfg lbs4 to break it1 and hence the beam of #ab #a #a would require #dfg #h #cgcf lbs4 to produce fracture4 ,this amounts to more than a ton and a half4 #dee4 ,it is a rule sometimes useful to practical men that a cast iron bar one foot long by one inch square would break with about a ton weight4 ,if the iron be of the same quality as that which we have used1 this result is too small1 but the error is on the safe side2 the real strength will then be generally a little greater than the strength calculated from this rule4 ,what we have said 7,art4 #djc7 with reference to the precaution for safety in bars of wood applies also to cast iron4 ,the load which the beam has to bear in ordinary practice should only be a small fraction of that which would break it4 #def4 ,in making any description of girder it is desirable on very special grounds that as little material as possible be uselessly employed4 ,it will of course be remembered that a girder has to support its own weight1 besides whatever may be placed upon it3 and if the girder be massive1 its own weight is a serious item4 ,of two girders1 each capable of bearing the same total load1 the lighter1 besides employing less material1 will be able to bear a greater weight placed upon it4 ,it is therefore for a double reason desirable to diminish the weight4 ,this remark applies especially to such a material as cast iron1 which can be at once given the form in which it shall be capable of offering the greatest resistance4 #deg4 ,the principles which will guide us in ascertaining the proper form to give a cast iron girder1 are easily deduced from what we have laid down in ,lectures ,,xi4 and ,,xii4 ,we have seen that depth is very desirable for a strong beam4 ,if therefore we strive to attain great depth in a light beam1 the beam must be very thin4 ,now an extremely thin beam will not be safe4 ,in the first place it would be flexible and liable to displacement sideways2 and1 in the second place1 there is a still more fatal difficulty4 ,we have shown that when a beam of wood is supporting a weight1 the fibres at the bottom of the beam are extended1 the tendency being to tear them 7,art4 #cgf74 ,the fibres on the top of the beam are compressed1 while the centre of the beam is in its natural state4 ,the condition of strain in a cast iron beam is precisely similar2 the bottom portions are in a state of extension1 while the top is compressed4 ,if therefore a beam be very thin1 the material at the lower part may not be sufficient to withstand the forces of extension1 and fracture is produced4 ,to obviate this1 we strengthen the bottom of the beam by placing extra material there4 ,thus we are led to the idea of a thin beam with an excess of iron at the bottom4 ,illustration3 ,,fig4 #fd4 #deh4 ,e ,f 7,fig4 #fd7 is the thin iron beam along the bottom of which is the stout flange shown at ,c ,d2 rupture cannot commence at the bottom unless this flange be torn asunder2 for until this happens it is clear that fracture cannot begin to attack the upper and slender part of the beam ,e ,f4 #dei4 ,but the beam is in a state of compression along its upper side1 just as in the wooden beams which we have already considered4 ,if therefore the upper parts were not powerful enough to resist this compression1 they would be crushed1 and the beam would give way4 ,the remedy for this source of weakness is obvious2 a second flange runs along the top of the beam1 as shown at ,a ,b4 ,if this be strong enough to resist the compression1 the stability of the beam is ensured4 #dfj4 ,the upper flange is made very much smaller than the lower one1 in consequence of a property of cast iron4 ,this metal is more capable of resisting forces of compression than forces of extension1 and it is only necessary to use one-sixth of the iron on the upper flange that is required for the lower4 ,when the section has been thus proportioned1 the beam is equally strong at both top and bottom2 adding material to either flange without strengthening the other1 will not benefit the girder1 but will rather prove a source of weakness1 by increasing the weight which has to be supported4 #dfa4 ,i have here a small girder made of what we are familiar with under the name of 8tin10 but which is of course sheet iron thinly covered over with tin4 ,it has the shape shown in ,fig4 #fd1 and it is #ab long4 ,i support it at each end1 and you see it bears two hundred weight without apparent deflection4 ,,the ,,tubular ,,bridge4 #dfb4 ,i shall commence the description of the principle of this bridge by performing some experiments upon a tube1 which ,i hold in my hand4 ,the tube is square1 #a #a in section1 and #ch long4 ,it is made of 8tin10 and weighs rather less than a pound4 #dfc4 ,here is a solid rod of iron of the same length as the tube1 but containing considerably more metal4 ,this is easily verified by weighing the tube and the rod one against the other4 ,i shall regard them as two girders1 and experiment upon their strength1 and we shall find that1 though the tube contains less substance than the rod1 it is much the stronger4 #dfd4 ,i place the rod on a pair of supports about #c' apart2 ,i then attach the tray to the middle of the rod3 #ad lbs4 produce a deflection of #jea1 and #db lbs4 bends down the rod through #cah4 ,this is a large deflection2 and when ,i remove the load1 the rod only returns through #agh1 thus showing that a permanent deflection of #adj is produced4 ,this proves that the rod is greatly injured1 and demonstrates its unsuitability for a girder4 #dfe4 ,next we place the tube upon the same supports1 and treat it in the same manner4 ,a load of #ef lbs4 only produces a deflection of #jji1 and1 when this load is removed1 the tube returns to its original position3 this is shown by the cathetometer1 for a cross is marked on the tube1 and ,i bring the image of it on the horizontal wire of the telescope before the load of #ef lbs4 is placed in the tray4 ,when the load is removed1 ,i see that the cross returns exactly to where it was before1 thus proving that the elasticity of the tube is unimpaired4 ,we double the load1 thus placing #a cwt4 in the tray1 the deflection only reaches #jbf1 and1 when the load is removed1 the tube is found to be permanently deflected by a quantity1 at all events not greater than #jjjd2 hence we learn that the tube bears easily and without injury a load more than twice as great as that which practically destroyed a rod of wrought iron1 containing more iron than the tube4 ,we load the tube still further by placing additional weights in the tray1 and with #adj lbs4 the tube breaks2 the fracture has occurred at a joint which was soldered1 and the real breaking strength of the tube1 had it been continuous1 is doubtless far greater4 ,enough1 however1 has been borne to show the increase of strength obtained by the tubular form4 #dff4 ,we can explain the reason of this remarkable result by means of ,fig4 #fd4 ,were the thin portion of the girder ,e ,f made of two parts placed side by side1 the strength would not be altered4 ,if we then imagine the flange ,a ,b widened to the width of ,c ,d1 and the two parts which form ,e ,f opened out so as to form a tube1 the strength of the girder is still retained in its modified form4 #dfg4 ,a tube of rectangular section has the advantage of greater depth than a solid rod of the same weight2 and if the bottom of the tube be strong enough to resist the extension1 and the top strong enough to resist the compression1 the girder will be stiff and strong4 #dfh4 ,in the ,menai ,tubular ,bridge1 where a gigantic tube supported at each end bridges over a span of four hundred and sixty feet1 special arrangements have been made for strengthening the top4 ,it is formed of cells1 as wrought iron disposed in this way is especially adapted for resisting compression4 #dfi4 ,we have only spoken of rectangular tubes1 but it is equally true for tubes of circular or other sections that when suitably constructed they are stronger than the same quantity of material1 if made into a solid rod4 #dgj4 ,we find this principle in nature2 bones and quills are often found to be hollow in order to combine lightness with strength1 and the stalks of wheat and other plants are tubular for the same reason4 ,,the ,,suspension ,,bridge4 #dga4 ,where a great span is required1 the suspension bridge possesses many advantages4 ,it is lighter than a girder bridge of the same span1 and consequently cheaper1 while its singular elegance contrasts very favourably with the appearance of more solid structures4 ,on the other hand1 a suspension bridge is not so well suited for railway traffic as the lattice girder4 #dgb4 ,the mechanical character of the suspension bridge is simple4 ,if a rope or a chain be suspended from two points to which its ends are attached1 the chain hangs in a certain curve known to mathematicians as the catenary4 ,the form of the catenary varies with the length of the rope1 but it would not be possible to make the chain lie in a straight line between the two points of support1 for reasons pointed out in ,art4 #bj4 ,no matter how great be the force applied1 it will still be concave4 ,when the chain is stretched until the depression in the middle is small compared with the distance between the points of support1 the curve though always a catenary1 has a very close resemblance to the parabola4 #dgc4 ,in ,fig4 #fe a model of a suspension bridge is shown4 ,the two chains are fixed one on each side at the points ,e and ,f2 they then pass over the piers ,a1 ,d1 and bridge a span of nine feet4 ,the vertical line at the centre ,b ,c shows the greatest amount by which the chain has deflected from the horizontal ,a ,d4 ,when the deflection of the middle of the chain is about one-tenth part of ,a ,d1 the curve ,a ,c ,d becomes for all practical purposes a parabola4 ,the roadway is suspended by slender iron rods from the chains1 the lengths of the suspension rods being so regulated as to make it nearly horizontal4 #dgd4 ,the roadway in the model is laden with #h stone weights4 ,we have distributed them in this manner in order to represent the permanent load which a great suspension bridge has to carry4 ,the series of weights thus arranged produces substantially the same effect as if it were actually distributed uniformly along the length4 ,in a real suspension bridge the weight of the chain itself adds greatly to the tension4 #dge4 ,we assume that the chain hangs in the form of a parabola1 and that the load is uniformly ranged along the bridge4 ,the tension upon the chains is greatest at their highest points1 and least at their lowest points1 though the difference is small4 ,the amount of the tension can be calculated when the load1 span1 and deflection are known4 ,we cannot give the steps of the calculation1 but we shall enunciate the result4 ,illustration3 ,,fig4 #fe4 #dgf4 ,the magnitude of the tension at the lowest point ,c of each chain is found by multiplying the total weight 7including chains1 suspension rods1 and roadway7 by the span1 and dividing the product by sixteen times the deflection4 ,the tension of the chain at the highest point ,a exceeds that at the lowest point ,c1 by a weight found by multiplying the total load by the deflection1 and dividing the product by twice the span4 #dgg4 ,the total weight of roadway1 chains1 and load in the model is #abj lbs42 the deflection is #aj1 the span #ajh2 the product of the weight and span is #ab1ifj2 sixteen times the deflection is #afj2 and1 therefore1 the tension at the point ,c is found1 by dividing #ab1ifj by #afj1 to be #ha lbs4 ,to find the tension at the point ,a1 we multiply #abj by #aj1 and divide the product by #baf2 the quotient is nearly #f4 ,this added to #ha lbs4 gives #hg lbs4 for the tension on the chain at ,a4 #dgh4 ,one chain of the model is attached to a spring balance at ,a2 by reference to the scale we see the tension indicated to be #ij lbs43 a sufficiently close approximation to the calculated tension of #hg lbs4 #dgi4 ,a large suspension bridge has its chains strained by an enormous force4 ,it is therefore necessary that the ends of these chains be very firmly secured4 ,a good attachment is obtained by anchoring the chain to a large iron anchor imbedded in solid rock4 #dhj4 ,in ,art4 #de we have pointed out how the dimensions of the tie rod could be determined when the tension was known4 ,similar considerations will enable us to calculate the size of the chain necessary for a suspension bridge when we have ascertained the tension to which it will be subjected4 #dha4 ,we can easily determine by trial what effect is produced on the tension of the chain1 by placing a weight upon the bridge in addition to the permanent load4 ,thus an additional stone weight in the centre raises the tension of the spring balance to #ajj lbs42 of course the tension in the other chain is the same3 and thus we find a weight of #ad lbs4 has produced additional tensions of #aj lbs4 each in the two chains4 ,with a weight of #bh lbs4 at the centre we find a strain of #aaj lbs4 on the chain4 #dhb4 ,these additional weights may be regarded as analogous to the weights of the vehicles which the suspension bridge is required to carry4 ,in a large suspension bridge the tension produced by the passing loads is only a small fraction of the permanent load4 ,,lecture ,,xv4 ,,the ,,motion ,,of ,a ,,falling ,,body4 ,introduction4--,the ,first ,law of ,motion4--,the ,experiment of ,galileo from the ,tower of ,pisa4--,the ,space is proportional to the ,square of the ,time4--,a ,body falls #af' in the ,first ,second4--,the ,action of ,gravity is independent of the ,motion of the ,body4--,how the ,force of ,gravity is defined4--,the ,path of a ,projectile is a ,parabola4 ,,introduction4 #dhc4 ,kinetics is that branch of mechanics which treats of the action of forces in the production of motion4 ,we shall find it rather more difficult than the subjects with which we have been hitherto occupied2 the difficulties in kinetics arise from the introduction of the element of time1 into our calculations4 ,the principles of kinetics were unknown to the ancients4 ,galileo discovered some of its truths in the seventeenth century2 and1 since his time1 the science has grown rapidly4 ,the motion of a falling body was first correctly apprehended by ,galileo2 and with this subject we can appropriately commence4 ,,the ,,first ,,law ,,of ,,motion4 #dhd4 ,velocity1 in ordinary language1 is supposed to convey a notion of rapid motion4 ,such is not precisely the meaning of the word in mechanics4 ,by velocity is merely meant the rate at which a body moves1 whether the rate be fast or be slow4 ,this rate is most conveniently measured by the number of feet moved over in one second4 ,hence when it is said the velocity of a body is #be1 it is meant that if the body continued to move for one second with its velocity unaltered1 it would in that time have moved over #be feet4 ,illustration3 ,,fig4 #ff4 #dhe4 ,the first law of motion may be stated thus4 ,if no force act upon a body1 it will1 if at rest1 remain for ever at rest2 or if in motion1 it will continue for ever to move with a uniform velocity4 ,we know this law to be true1 and yet no one has ever seen it to be true for the simple reason that we cannot realise the condition which it requires4 ,we cannot place a body in the condition of being unacted upon by any forces4 ,but we may convince ourselves of the truth of the law by some such reasoning as the following4 ,if a stone be thrown along the road1 it soon comes to rest4 ,the body leaves the hand with a certain initial velocity and is not further acted upon by it4 ,hence1 if no other force acted on the stone1 we should expect1 if the first law be true1 that it would continue to run on for ever with the original velocity at the moment of leaving the hand4 ,but other forces do act upon the stone2 the attraction of the earth pulls it down2 and1 when it begins to bound and roll upon the ground1 friction comes into operation1 deprives the stone of its velocity1 and brings it to rest4 ,but let the stone be thrown upon a surface of smooth ice2 when it begins to slide1 the force of gravity is counteracted by the reaction of the ice3 there is no other force acting upon the stone except friction1 which is small4 ,hence we find that the stone will run on for a considerable distance4 ,it requires but little effort of the imagination to suppose a lake whose surface is an infinite plane1 perfectly smooth1 and that the stone is perfectly smooth also4 ,in such a case as this the first law of motion amounts to the assertion that the stone would never stop4 #dhf4 ,we may1 in the lecture room1 see the truth of this law verified to a certain extent by ,atwood's machine 7,fig1 #ff74 ,this machine has been devised for the purpose of investigating the laws of motion by actual experiment4 ,it consists principally of a pulley ,c1 mounted so that its axle rests upon two pairs of wheels1 as shown in the figure2 it being the object of this contrivance to enable the wheel to revolve with the utmost freedom4 ,a pair of equal weights ,a1 ,b1 are attached by a silken thread1 which passes over the pulley2 each of the weights is counterbalanced by the other3 so that when the two are in motion1 we may consider either as a body not acted upon by any forces1 and it will be found that it moves uniformly1 as far as the size of the apparatus will permit4 #dhg4 ,if we try to conceive a body free in space1 and not acted upon by any force1 it is more natural to suppose that such a body1 when once started1 should go on moving uniformly for ever1 than that its velocity should be altered4 ,the true proof of the first law of motion is1 that all consequences properly deduced from it1 in combination with other principles1 are found to be verified4 ,astronomy presents us with the best examples4 ,the calculation of the time of an eclipse is based upon laws which in themselves assume the first law of motion2 hence1 when we invariably find that an eclipse occurs precisely at the moment for which it has been predicted1 we have a splendid proof of the sublime truth which the first law of motion expresses4 ,,the ,,experiment ,,of ,,galileo ,,from ,,the ,,tower ,,of ,,pisa4 #dhh4 ,the contrast between heavy bodies and light bodies is so marked that without trial we hardly believe that a heavy body and a light body will fall from the same height in the same time4 ,that they do so ,galileo proved by dropping a heavy ball and a light ball together from the top of the ,leaning ,tower at ,pisa4 ,they were found to reach the ground simultaneously4 ,we shall repeat this experiment on a scale sufficiently reduced to correspond with the dimensions of the lecture room4 ,illustration3 ,,fig4 #fg4 #dhi4 ,the apparatus used is shown in ,fig4 #fg4 ,it consists of a stout framework supporting a pulley ,h at a height of about #bj feet above the ground4 ,this pulley carries a rope2 one end of the rope is attached to a triangular piece of wood1 to which two electro-magnets ,g are fastened4 ,the electro-magnet is a piece of iron in the form of a horse-shoe1 around which is coiled a long wire4 ,the horse-shoe becomes a magnet immediately an electric current passes through the wire2 it remains a magnet as long as the current passes1 and returns to its original condition the moment the current ceases4 ,hence1 if ,i have the means of controlling the current1 ,i have complete control of the magnet2 you see this ball of iron remains attached to the magnet as long as the current passes1 but drops the instant ,i break the current4 ,the same electric circuit includes both the magnets2 each of them will hold up an iron ball ,f when the current passes1 but the moment the current is broken both balls will be released4 ,electricity travels along a wire with prodigious velocity4 ,it would pass over many thousands of miles in a second2 hence the time that it takes to pass through the wires we are employing is quite inappreciable4 ,a piece of thin paper interposed between the magnets and the balls will ensure that they are dropped simultaneously2 when this precaution is not taken one or both balls may hesitate a little before commencing to descend4 ,a long pair of wires ,e1 ,b1 must be attached to the magnets1 the other ends of the wires communicating with the battery ,d2 the triangle and its load is hoisted up by means of the rope and pulley and the magnets thus carry the balls to a height of #bj feet3 the balls we are using weigh about #jbe lb4 and #a lb4 respectively4 #dij4 ,we are now ready to perform the experiment4 ,i break the circuit2 the two balls are disengaged simultaneously2 they fall side by side the whole way1 and reach the ground together1 where it is well to place a cushion to receive them4 ,thus you see the heavy ball and the light one each require the same amount of time to fall from the same height4 #dia4 ,but these balls are both of iron2 let us compare together balls made of different substances1 iron and wood for example4 ,a flat-headed nail is driven into a wooden ball of about #be in diameter1 and by means of the iron in the nail ,i can suspend this ball from one of the magnets2 while either of the iron balls we have already used hangs from the other4 ,i repeat the experiment in the same manner1 and you see they also fall together4 ,finally1 when an iron ball and a cork ball are dropped1 the latter is within two or three inches of its weighty companion when the cushion is reached3 this small difference is due to the greater effect of the resistance of the air on the lighter of the two bodies4 ,there can be no doubt that in a vacuum all bodies of whatever size or material would fall in precisely the same time4 #dib4 ,how is the fact that all bodies fall in the same time to be explained8 ,let us first consider two iron balls4 ,take two equal particles of iron2 it is evident that these fall in the same time2 they would do so if they were very close together1 even if they were touching1 but then they might as well be in one piece3 and thus we should find that a body consisting of two or more iron particles takes the same time to fall as one 7omitting of course the resistance of the air74 ,thus it appears most reasonable that two balls of iron1 even though unequal in size1 should fall in the same time4 #dic4 ,the case of the wooden ball and the iron ball will require more consideration before we realise thoroughly how much ,galileo's experiment proves4 ,we must first explain the meaning of the word mass in mechanics4 #did4 ,it is not correct to define mass by the introduction of the idea of weight1 because the mass of a body is something independent of the existence of the earth1 whereas weight is produced by the attraction of the earth4 ,it is true that weight is a convenient means of measuring mass1 but this is only a consequence of the property of gravity which the experiment proves1 namely1 that the attraction of gravity for a body is proportional to its mass4 #die4 ,let us select as the unit of mass the mass of a piece of platinum which weighs #a lb4 at ,london2 it is then evident that the mass of any other piece of platinum should be expressed by the number of pounds it contains3 but how are we to determine the mass of some other substance1 such as iron8 ,a piece of iron is defined to have the same mass as a piece of platinum1 if the same force acting on either of the bodies for the same time produces the same velocity4 ,this is the proper test of the equality of masses4 ,the mass of any other piece of iron will be represented by the number of times it contains a piece equal to that which we have just compared with the platinum2 similarly of course for other substances4 #dif4 ,the magnitude of a force acting for the time unit is measured by the product of the mass set in motion and the velocity which it has acquired4 ,this is a truth established1 like the first law of motion1 by indirect evidence4 #dig4 ,let us apply these principles to explain the experiment which demonstrated that a ball of wood and a ball of iron fall in the same time4 ,forces act upon the two bodies for the same time1 but the magnitudes of the forces are proportional to the mass of each body multiplied into its velocity1 and1 since the bodies fall simultaneously1 their velocities are equal4 ,the forces acting upon the bodies are therefore proportional to their masses2 but the force acting on each body is the attraction of the earth1 therefore1 the gravitation to the earth of different bodies is proportional to their masses4 #dih4 ,we may here note the contrast between the attraction of gravitation and that of a magnet4 ,a magnet attracts iron powerfully and wood not at all2 but the earth draws all bodies with forces depending on their masses and their distances1 and not on their chemical composition4 ,,the ,,space ,,described ,,by ,a ,,falling ,,body ,,is ,,proportional ,,to ,,the ,,square ,,of ,,the ,,time4 #dii4 ,we have next to discover the law by which we ascertain the distance a body falling from rest will move in a given time2 it is not possible to experiment directly upon this subject1 as in two seconds a body will drop #fd feet and acquire an inconveniently large velocity2 we can1 however1 resort to ,atwood's machine 7,fig4 #ff7 as a means of diminishing the motion4 ,for this purpose we require a clock with a seconds pendulum4 #ejj4 ,on one of the equal cylinders ,a ,i place a slight brass rod1 whose weight gives a preponderance to ,a1 which will consequently descend4 ,i hold the loaded weight in my hand1 and release it simultaneously with the tick of the pendulum4 ,i observe that it descends #e before the next tick4 ,returning the weight to the place from whence it started1 ,i release it again1 and ,i find that at the second tick of the pendulum it has travelled #bj4 ,similarly we find that in three seconds it descends #de4 ,it greatly facilitates these experiments to use a little stage which is capable of being slipped up and down the scale1 and which can be clamped to the scale in any position4 ,by actually placing the stage at the distance of #e1 #bj1 or #de below the point from which the weight starts1 the coincidence of the tick of the pendulum with the tap of the weight on its arrival at the stage is very marked4 #eja4 ,these three distances are in the proportion of #a1 #d1 #i2 that is1 as the squares of the numbers of seconds #a1 #b1 #c4 ,hence we may infer that the distance traversed by a body falling from rest is proportional to the square of the time4 #ejb4 ,the motion of the bodies in ,atwood's machine is much slower than the motion of a body falling freely1 but the law just stated is equally true in both cases so that in a free fall the distance traversed is proportional to the square of the time4 ,atwood's machine cannot directly tell us the distance through which a body falls in one second4 ,if we can find this by other means1 we shall easily be able to calculate the distance through which a body will fall in any number of seconds4 ,a ,,body ,,falls #af' ,,in ,,the ,,first ,,second4 #ejc4 ,the apparatus by which this important truth maybe demonstrated is shown in ,fig4 #fg4 ,a part of it has been already employed in performing the experiment of ,galileo1 but two other parts must now be used which will be briefly explained4 #ejd4 ,at ,a a pendulum is shown which vibrates once every second2 it need not be connected with any clockwork to sustain the motion1 for when once set vibrating it will continue to swing some hundreds of times4 ,when this pendulum is at the middle of its swing1 the bob just touches a slender spring1 and presses it slightly downwards4 ,the electric current which circulates about the magnets ,g 7,art4 #dhi7 passes through this spring when in its natural position2 but when the spring is pressed down by the pendulum1 the current is interrupted4 ,the consequence is that1 as the pendulum swings backwards and forwards1 the current is broken once every second4 ,there is also in the circuit an electric alarm bell ,c1 which is so arranged that1 when the current passes1 the hammer is drawn from the bell2 but1 when the current ceases1 a spring forces the hammer to strike the bell4 ,when the circuit is closed1 the hammer is again drawn back4 ,the pendulum and the bell are in the same circuit1 and thus every vibration of the pendulum produces a stroke of the bell4 ,we may regard the strokes from the bell as the ticks of the pendulum rendered audible to the whole room4 #eje4 ,you will now understand the mode of experimenting4 ,i draw the pendulum aside so that the current passes uninterruptedly4 ,an iron ball is attached to one of the electro-magnets1 and it is then gently hoisted up until the height of the ball from the ground is about #af'4 ,a cushion is placed on the floor in order to receive the falling body4 ,you are to look steadily at the cushion while you listen for the bell4 ,all being ready1 the pendulum1 which has been held at a slight inclination1 is released4 ,the moment the pendulum reaches the middle of its swing it touches the spring1 rings the bell1 breaks the current which circulated around the magnet1 and as there is now nothing to sustain the ball1 it drops down to the cushion2 but just as it arrives there1 the pendulum has a second time broken the electric circuit1 and you observe the falling of the ball upon the cushion to be identical with the second stroke of the bell4 ,as these strokes are repeated at intervals of a second1 it follows that the ball has fallen #af' in one second4 ,if the magnet be raised a few feet higher1 the ball may be seen to reach the cushion after the bell is heard4 ,if the magnet be lowered a few feet1 the ball reaches the cushion before the bell is heard4 #ejf4 ,we have previously shown that the space is proportional to the square of the time4 ,we now see that when the time is one second1 the space is #af feet4 ,hence if the time were two seconds1 the space would be #d #af #fd feet2 and in general the space in feet is equal to #af multiplied by the square of the time in seconds4 #ejg4 ,by the help of this rule we are sometimes enabled to ascertain the height of a perpendicular cliff1 or the depth of a well4 ,for this purpose it is convenient to use a stop-watch1 which will enable us to measure a short interval of time accurately4 ,but an ordinary watch will do nearly as well1 for with a little practice it is easy to count the beats1 which are usually at the rate of five a second4 ,by observing the number of beats from the moment the stone is released till we see or hear its arrival at the bottom1 we determine the time occupied in the act of falling4 ,the square of the number of seconds 7taking account of fractional parts7 multiplied by #af gives the depth of the well or the height of the cliff in feet1 provided it be not high4 ,,the ,,action ,,of ,,gravity ,,is ,,independent ,,of ,,the ,,motion ,,of ,,the ,,body4 #ejh4 ,we have already learned that the effect of gravity does not depend upon the actual chemical composition of the body4 ,we have now to learn that its effect is uninfluenced by any motion which the body may possess4 ,gravity pulls a body down #af' per second1 if the body starts from rest4 ,but suppose a stone be thrown upwards with a velocity of #bj feet1 where will it be at the end of a second8 ,did gravity not act upon the stone1 it would be at a height of #bj feet4 ,the principle we have stated tells us that gravity will draw this stone towards the earth through a distance of #af'1 just as it would have done if the stone had started from rest4 ,since the stone ascends #bj' in consequence of its own velocity1 and is pulled back #af' by gravity1 it will1 at the end of a second1 be found at the height of #d'4 ,if1 instead of being shot up vertically1 the body had been projected in any other direction1 the result would have been the same2 gravity would have brought the body at the end of one second #af' nearer the earth than it would have been had gravity not acted4 ,for example1 if a body had been shot vertically downwards with a velocity of #bj'1 it would in one second have moved through a space of #cf'4 #eji4 ,we shall illustrate this remarkable property by an experiment4 ,the principle of doing so is as follows3--,suppose we take two bodies1 ,a and ,b4 ,if these be held at the same height1 and released together1 of course they reach the ground at the same instant2 but if ,a1 instead of being merely dropped1 be projected with a horizontal velocity at the same moment that ,b is released1 it is still found that a and ,b strike the floor simultaneously4 #eaj4 ,you may very simply try this without special apparatus4 ,in your left hand hold a marble1 and drop it at the same instant that your right hand throws another marble horizontally4 ,it will be seen that the two marbles reach the ground together4 #eaa4 ,a more accurate mode of making the experiment is shown by the contrivance of ,fig4 #fh4 ,illustration3 ,,fig4 #fh4 ,in this we have an arrangement by which we ensure that one ball shall be released just as the other is projected4 ,at ,a ,b is shown a piece of wood about #b thick2 the circular portion 7#b' radius7 on which the ball rests is grooved1 so that the ball only touches the two edges and not the bottom of the groove4 ,each edge of the groove is covered with tinfoil ,c1 but the pieces of tinfoil on the two sides must not communicate4 ,one edge is connected with one pole of the battery ,k1 and the other edge with the other pole1 but the current is unable to pass until a communication by a conductor is opened between the two edges4 ,the ball ,g supplies the bridge2 it is covered with tinfoil1 and therefore1 as long as it rests upon the edges1 the circuit is complete2 the groove is so placed that the tangent to it at the lowest point ,b is horizontal1 and therefore1 when the ball rolls down the curve1 it is projected from the bottom in a horizontal direction4 ,an india-rubber spring is used to propel the ball2 and by drawing it back when embraced by the spring1 ,i can communicate to the missile a velocity which can be varied at pleasure4 ,at ,h we have an electro-magnet1 the wire around which forms part of the circuit we have been considering4 ,this magnet is so placed that a ball suspended from it is precisely at the same height above the floor as the tinned ball is at the moment when it leaves the groove4 #eab4 ,we now understand the mode of experimenting4 ,so long as the tinned ball ,g remains on the curve the bridge is complete1 the current passes1 and the electro-magnet will sustain ,h1 but the moment ,g leaves the curve1 ,h is allowed to fall4 ,we invariably find that whatever be the velocity with which ,g is projected1 it reaches the ground at the same instant as ,h arrives there4 ,various dotted lines in the figure show the different paths which ,g may traverse2 but whether it fall at ,d1 at ,e1 or at ,f1 the time of descent is the same as that taken by ,h4 ,of course1 if ,g were not projected horizontally1 we should not have arrived at this result2 all we assert is that whatever be the motion of a body1 it will 7when possible7 be at the end of a second1 sixteen feet nearer the earth than it would have been if gravity had not acted4 ,if the body be projected horizontally1 its descent is due to gravity alone1 and is neither accelerated nor retarded by the horizontal velocity4 ,what this experiment proves is1 that the mere fact of a body having velocity does not affect the action of gravity thereon4 #eac4 ,though we have only shown that a horizontal velocity does not affect the action of gravity1 yet neither does a velocity in any direction4 ,this is verified1 like the first law of motion1 by the accordance between the consequences deduced from it and the facts of observation4 #ead4 ,we may summarize these results by saying that no matter what be the material of which a particle is composed1 whether it be heavy or light1 moving or at rest1 if no force but gravity act upon the particle for t seconds1 it will then be #aft feet nearer the earth than it would have been had gravity not acted4 ,illustration3 ,,fig4 #fi4 #eae4 ,a proposition which is of some importance may be introduced here4 ,let us suppose a certain velocity and a certain force4 ,let the velocity be such that a point starting from ,a1 ,fig4 #fi1 would in one second move uniformly to ,b4 ,let the force be such that if it acted on a particle originally at rest at ,a1 it would in one second draw the particle to ,d2 if then the force act on a particle having this velocity where will it be at the end of the second8 ,complete the parallelogram ,a ,b ,c ,d1 and the particle will be found at ,c4 ,by what we have stated the force will equally discharge its duty whatever be the initial velocity4 ,the force will therefore make the particle move to a distance equal and parallel to ,a ,d from whatever position the particle would have assumed1 had the force not acted2 but had the force not acted1 the particle would have been found at ,b3 hence1 when the force does act1 the particle must be found at ,c1 since ,b ,c is equal and parallel to ,a ,d4 ,,how ,,the ,,force ,,of ,,gravity ,,is ,,defined4 #eaf4 ,from the formula ,distance #aft1 we learn that a body falls through #fd' in #b seconds2 and as we know that it falls #af' in the first second1 it must fall #dh' in the next second4 ,let us examine this4 ,after falling for one second1 the body acquires a certain velocity1 and with that velocity it commences the next second4 ,now1 according to what we have just seen1 gravity will act during the next second quite independently of whatever velocity the body may have previously had4 ,hence in the second second gravity pulls the body down #af'1 but the body moves altogether through #dh'2 therefore it must move through #cb' in consequence of the velocity which has been impressed upon it by gravity during the first second4 ,we learn by this that when gravity acts for a second1 it produces a velocity such that1 if the body be conceived to move uniformly with the velocity acquired1 the body would in one second move over #cb'4 #eag4 ,in three seconds the body falls #add'1 therefore in the third second it must have fallen #add' - #fd' #hj'2 but of this #hj' only #af' could be due to the action of gravity impressed during that second2 the rest1 #hj' - #af' #fd'1 is due to the velocity with which the body commenced the third second4 #eah4 ,we see therefore that after the lapse of two seconds gravity has communicated to the body a velocity of #fd' per second2 we should similarly find1 that at the end of the third second1 the body has a velocity of #if'1 and in general at the end of t seconds a velocity of #cbt4 ,thus we illustrate the remarkable law that the velocity developed by gravity is proportional to the time4 #eai4 ,this law points out that the most suitable way of measuring gravity is by the velocity acquired by a falling body at the end of one second4 ,hence we are accustomed to say that g 7as gravity is generally designated7 is #cb4 ,we shall afterwards show in the lecture on the pendulum 7,,xviii47 how the value of g can be obtained accurately4 ,from the two equations1 v #cbt and s #aft it is easy to infer another very well known formula1 namely1 v #fds4 ,,the ,,path ,,of ,a ,,projectile ,,is ,a ,,parabola4 #ebj4 ,we have already seen1 in the experiments of ,fig4 #fh1 that a body projected horizontally describes a curved path on its way to the ground1 and we have to determine the geometrical nature of the curve4 ,as the movement is rapid1 it is impossible to follow the projectile with the eye so as to ascertain the shape of its path with accuracy2 we must therefore adopt a special contrivance1 such as that represented in ,fig4 #gj4 ,b ,c is a quadrant of wood #b thick2 it contains a groove1 along which the ball ,b will run when released4 ,a series of cardboard hoops are properly placed on a black board1 and the ball1 when it leaves the quadrant1 will pass through all these hoops without touching any1 and finally fall into a basket placed to receive it4 ,the quadrant must be secured firmly1 and the ball must always start from precisely the same place4 ,the hoops are easily adjusted by trial4 ,letting the ball run down the quadrant two or three times1 we can see how to place the first hoop in its right position1 and secure it by drawing pins2 then by a few more trials the next hoop is to be adjusted1 and so on for the whole eight4 #eba4 ,the curved line from the bottom of the quadrant1 which passes through the centres of the hoops1 is the path in which the ball moves2 this curve is a parabola1 of which ,f is the focus and the line ,a ,a the directrix4 ,illustration3 ,,fig4 #gj4 ,it is a property of the parabola that the distance of any point on the curve from the focus is equal to its perpendicular distance from the directrix4 ,this is shown in the figure4 ,for example1 the dotted line ,f ,d1 drawn from ,f to the centre of the lowest hoop ,d1 is equal in length to the perpendicular ,d ,p let fall from ,d on the directrix ,a ,a4 #ebb4 ,the direction in which the ball is projected is in this case horizontal1 but1 whatever be the direction of projection1 the path is a parabola4 ,this can be proved mathematically as a deduction from the theorem of ,art4 #eae4 ,,lecture ,,xvi4 ,,inertia4 ,inertia4--,the ,hammer4--,the ,storing of ,energy4--,the ,fly-wheel4--,the ,punching ,machine4 ,,inertia4 #ebc4 ,a body unacted upon by force will continue for ever at rest1 or for ever moving uniformly in a straight line4 ,this is asserted by the first law of motion 7,art4 #dhe74 ,it is usual to say that ,inertia is a property of all matter1 by which it is meant that matter cannot of itself change its state of rest or of motion4 ,force is accordingly required for this purpose4 ,in the present chapter we shall discuss some important mechanical considerations connected with the application of force in changing the state of a body from rest or in altering its velocity when in motion4 ,in the next chapter we shall study the application of force in compelling a body to swerve from its motion in a straight line4 #ebd4 ,we have in earlier lectures been concerned with the application of force either to raise a weight or to overcome friction4 ,we have now to consider the application of force to a body1 not for the purpose of raising it1 nor for pushing it along against a frictional resistance1 but merely for the purpose of generating a velocity4 ,unfortunately there is a practical difficulty in the way of making the experiments precisely in the manner we should wish4 ,we want to get a mass isolated both from gravitation and from friction1 but this is just what we cannot do--that is1 we cannot do it perfectly4 ,we have1 however1 a simple appliance which will be sufficiently isolated for our present purpose4 ,here is a heavy weight of iron1 about #be lbs41 suspended by an iron wire from the ceiling about #cb feet above the floor 7see ,fig4 #hb74 ,this weight may be moved to and fro by the hand4 ,it is quite free from friction1 for we need not at present remember the small resistance which the air offers4 ,we may also regard the gravity of the weight as neutralized by the sustaining force of the wire1 and accordingly as the body now hangs at rest it may for our purposes be regarded as a body unacted upon by any force4 #ebe4 ,to give this ball a horizontal velocity ,i feel that ,i must apply force to it4 ,this will be manifest to you all when ,i apply the force through the medium of an india-rubber spring4 ,if ,i pull the spring sharply you notice how much it stretches2 you see therefore that the body will not move quickly unless a considerable force is applied to it4 ,it thus follows that merely to generate motion in this mass force has been required4 #ebf4 ,so1 too1 when the body is in motion as it now is ,i cannot stop it without the exertion of force4 ,see how the spring is stretched and how strong a pull has thus been exerted to deprive the body of motion4 ,notice also that while a small force applied sufficiently long will always restore the body to rest1 yet that to produce rest quickly a large force will be required4 #ebg4 ,it is an universal law of nature that action and reaction are equal and opposite4 ,hence when any agent acts to set a body in motion1 or to modify its motion in any way1 the body reacts on the agent1 and this force has been called the ,kinetic reaction4 #ebh4 ,for example4 ,when a railway train starts1 the locomotive applies force to the carriages1 and the speed generated during one second is added to that produced during the next1 and the pace improves4 ,the kinetic reaction of the train retards the engine from attaining the speed it would acquire if free from the train4 ,,the ,,hammer4 #ebi4 ,the hammer and other tools which give a blow depend for their action upon inertia4 ,a gigantic hammer might force in a nail by the mere weight of the head resting on the nail1 but with the help of inertia we drive the nail by blows from a small hammer4 ,we have here inertia aiding in the production of a mechanical power to overcome the considerable resistance which the wood opposes to the entrance of the nail4 ,to drive in the nail usually requires a direct force of some hundreds of pounds1 and this we are able conveniently to produce by suddenly checking the velocity of a small moving body4 ,illustration3 ,,fig4 #ga4 #ecj4 ,the theory of the hammer is illustrated by the apparatus in ,fig4 #ga4 ,it is a tripod1 at the top of which1 about #i' from the ground1 is a stout pulley ,c2 the rope is about #ae' long1 and to each end of it ,a and ,b are weights attached4 ,these weights are at first each #ad lbs4 ,i raise ,a up to the pulley1 leaving ,b upon the ground2 ,i then let go the rope1 and down falls ,a3 it first pulls the slack rope through1 and then1 when ,a is about #c' from the ground1 the rope becomes tight1 ,b gets a violent chuck and is lifted into the air4 ,what has raised ,b8 ,it cannot be the mere weight of ,a1 because that being equal to ,b1 could only just balance ,b1 and is insufficient to raise it4 ,it must have been a force which raised ,b2 that force must have been something more than the weight of ,a1 which was produced when the motion was checked4 ,a was not stopped completely2 it only lost some of its velocity1 but it could not lose any velocity without being acted upon by a force4 ,this force must have been applied by the rope by which ,a was held back1 and the tension thus arising was sufficient to pull up ,b4 #eca4 ,let us remove the #ad lb4 weight from ,b1 and attach there a weight of #bh lbs41 ,a remaining the same as before 7#ad lbs474 ,i raise ,a to the pulley2 ,i allow it to fall4 ,you observe that ,b1 though double the weight of ,a1 is again chucked up after the rope has become tight4 ,we can only explain this by the supposition that the tension in the rope exerted in checking the motion of ,a is at least #bh lbs4 #ecb4 ,finally1 let us remove the #bh lbs4 from ,b1 put on #ef lbs41 and perform the experiment again2 you see that even the #ef lbs4 is raised up several inches4 ,here a tension in the rope has been generated sufficient to overcome a weight four times as heavy as ,a4 ,we have then1 by the help of inertia1 been able to produce a mechanical power1 for a small force has overcome a greater4 #ecc4 ,after ,b is raised by the chuck to a certain height it descends again1 if heavier than ,a1 and raises ,a4 ,the height to which ,b is raised is of course the same as the height through which ,a descends4 ,you noticed that the height through which #bh lbs4 was raised was considerably greater than that through which the #ef lbs4 was raised4 ,hence we may draw the inference1 that when ,a was deprived of its velocity while passing through a short space1 it required to be opposed by a greater force than when it was gradually deprived of its velocity through a longer space4 ,this is a most important point4 ,supposing ,i were to put a hundredweight at ,b1 ,i have little doubt1 if the rope were strong enough to bear the strain1 that though ,a only weighs #ad lbs41 ,b would yet be raised a little3 here ,a would be deprived of its motion in a very short space1 but the force required to arrest it would be very great4 #ecd4 ,it is clear that matters would not be much altered if ,a were to be stopped by some force1 exerted from below rather than above2 in fact1 we may conceive the rope omitted1 and suppose ,a to be a hammer-head falling upon a nail in a piece of wood4 ,the blow would force the nail to penetrate a small distance1 and the entire velocity of ,a would have to be destroyed while moving through that small distance3 consequently the force between the head of the nail and the hammer would be a very large one4 ,this explains the effect of a blow4 #ece4 ,in the case that we have supposed1 the weight merely drops upon the nail3 this is actually the principle of the hammer used in pile-driving machines4 ,a pile is a large piece of timber1 pointed and shod with iron at one end3 this end is driven down into the ground4 ,piles are required for various purposes in engineering operations4 ,they are often intended to support the foundations of buildings2 they are therefore driven until the resistance with which the ground opposes their further entrance affords a guarantee that they shall be able to bear what is required4 #ecf4 ,the machine for driving piles consists essentially of a heavy mass of iron1 which is raised to a height1 and allowed to fall upon the pile4 ,the resistance to be overcome depends upon the depth and nature of the soil3 a pile may be driven two or three inches with each blow1 but the less the distance the pile enters each time1 the greater is the actual pressure with which the blow forces it downwards4 ,in the ordinary hammer1 the power of the arm imparts velocity to the hammer-head1 in addition to that which is due to the fall2 the effect produced is merely the same as if the hammer had fallen from a greater height4 #ecg4 ,another point may be mentioned here4 ,a nail will only enter a piece of wood when the nail and the wood are pressed together with sufficient force4 ,the nail is urged by the hammer4 ,if the wood be lying on the ground1 the reaction of the ground prevents the wood from getting away and the nail will enter4 ,in other cases the element of time is all-important4 ,if the wood be massive less force will make the nail penetrate than would suffice to move the wood quickly enough4 ,if the wood be thin and unsupported1 less force may be required to make it yield than to make the nail penetrate4 ,the usual remedy is obvious4 ,hold a heavy mass close at the back of the wood4 ,the nail will then enter because the augmented mass cannot now escape as rapidly as before4 ,,the ,,storing ,,of ,,energy4 #ech4 ,our study of the subject will be facilitated by some considerations founded on the principles of energy4 ,in the experiment of ,fig4 #ga let ,a be #ad lbs41 and ,b1 on the ground1 be #ef lbs4 ,since the rope is #ae' long1 ,a is #c' from the ground1 and therefore #f' from the pulley4 ,i raise ,a to the pulley1 and1 in doing so1 expend #f #ad #hd units of energy4 ,energy is never lost1 and therefore ,i shall expect to recover this amount4 ,i allow ,a to fall2 when it has fallen #f'1 it is then precisely in the same condition as it was before being raised1 except that it has a considerable velocity of descent4 ,in fact1 the #hd units of energy have been expended in giving velocity to ,a4 ,b is then lifted to a maximum height ,x1 in which #ef ,x units of energy have been consumed4 ,at the instant when ,b is at the summit ,x1 ,a must be at a distance of #f ,x feet from the pulley2 hence the quantity of work performed by ,a is #ad 7#f ,x74 ,but the work done by ,a must be equal to that done upon ,b1 and therefore #ad7f ,x7 #ef ,x1 whence ,x #b4 ,if there were no loss by friction1 ,b would therefore be raised #b'2 but owing to friction1 and doubtless also to the imperfect flexibility of the rope1 the effect is not so great4 ,we may regard the work done in raising ,a as so much energy stored up1 and when ,a is allowed to fall1 the energy is reproduced in a modified form4 #eci4 ,let us apply the principle of energy to the pile-driving engine to which we have referred 7,art4 #ecf72 we shall then be able to find the magnitude of the force developed in producing the blow4 ,suppose the 8monkey10 that is the heavy hammer-head1 weighs #efj lbs4 7a quarter of a ton74 ,a couple of men raise this by means of a small winch to a height of #ae'4 ,it takes them a few minutes to do so2 their energy is then saved up1 and they have accumulated a store of #efj #ae #h1djj units4 ,when the monkey falls upon the top of the pile it transfers thereto nearly the whole of the #h1djj units of energy1 and this is expended in forcing the pile into the ground4 ,suppose the pile to enter one inch1 the reaction of the pile upon the monkey must be so great that the number of units of energy consumed in one inch is #h1djj4 ,hence this reaction must be #h1djj #ab #ajj1hjj lbs4 ,if the reaction did not reach this amount1 the monkey could not be brought to rest in so short a distance4 ,the reaction of the pile upon the monkey1 and therefore the action of the monkey upon the pile1 is about #de tons4 ,this is the actual pressure exerted4 #edj4 ,if the soil which the pile is penetrating be more resisting than that which we have supposed1--for example1 if the pile require a direct pressure of #ajj tons to force it in1--the same monkey with the same fall would still be sufficient1 but the pile would not be driven so far with each blow4 ,the pressure required is #bbd1jjj lbs43 this exerted over a space of #jde would be #h1djj units of energy2 hence the pile would be driven #jde4 ,the more the resistance1 the less the penetration produced by each blow4 ,a pile intended to bear a very heavy load permanently must be driven until it enters but little with each blow4 #eda4 ,we may compare the pile-driver with the mechanical powers in one respect1 and contrast it in another4 ,in each1 we have machines which receive energy and restore it modified into a greater power exerted through a smaller distance2 but while the mechanical powers restore the energy at one end of the machine1 simultaneously with their reception of it at the other1 the pile-driver is a reservoir for keeping energy which will restore it in the form wanted4 #edb4 ,we have1 then1 a class of mechanical powers1 of which a hammer may be taken as the type1 which depend upon the storage of energy2 the power of the arm is accumulated in the hammer throughout its descent1 to be instantly transferred to the nail in the blow4 ,inertia is the property of matter which qualifies it for this purpose4 ,energy is developed by the explosion of gunpowder in a cannon4 ,this energy is transferred to the ball1 from which it is again in large part passed on to do work against the object which is struck4 ,here we see energy stored in a rapidly moving body1 a case to which we shall presently return4 #edc4 ,but energy can be stored in many ways2 we might almost say that gunpowder is itself energy in a compact and storable form4 ,the efforts which we make in forcing air into an air-cane are preserved as energy there stored to be reproduced in the discharge of a number of bullets4 ,during the few seconds occupied in winding a watch1 a small charge of energy is given to the spring which it expends economically over the next twenty-four hours4 ,in using a bow my energy is stored up from the moment ,i begin to pull the string until ,i release the arrow4 #edd4 ,many machines in extensive use depend upon these principles4 ,in the clock or watch the demand for energy to sustain the motion is constant1 while the supply is only occasional2 in other cases the supply is constant1 while the demand is only intermittent4 ,we may mention an illustration of the latter4 ,suppose it be required occasionally to hoist heavy weights rapidly up to a height4 ,if an engine sufficiently powerful to raise the weights be employed1 the engine will be idle except when the weights are being raised2 and if the machinery were to have much idle time1 the waste of fuel in keeping up the fire during the intervals would often make the arrangement uneconomical4 ,it would be a far better plan to have a smaller engine2 and even though this were not able to raise the weight directly with sufficient speed1 yet by keeping the engine continually working and storing up its energy1 we might produce enough in the twenty-four hours to raise all the weights which it would be necessary to lift in the same time4 #ede4 ,let us suppose we want to raise slates from the bottom of a quarry to the surface4 ,a large pulley is mounted at the top of the quarry1 and over this a rope is passed3 to each end of the rope a bucket is attached1 so that when one of these is at the bottom the other is at the top1 and their sizes and that of the pulley are so arranged that they pass each other with safety4 ,a reservoir is established at the top of the quarry on a level with the pulley1 and an engine is set to work constantly pumping up water from the bottom of the quarry into the reservoir4 ,each of the buckets is partly composed of a large tank1 which can be quickly filled or emptied4 ,the lower bucket is loaded with slates1 and when ready for work1 the man at the top fills the tank of the upper bucket with water3 this accordingly becomes so heavy that it descends and raises the slates4 ,when the heavier one reaches the bottom1 the water from its tank is let out into the lower reservoir1 from which the engine pumps1 and the slates are removed from the bucket which has been raised4 ,all is then ready for a repetition of the same operation4 ,if the slates be raised at intervals of ten minutes1 the energy of the engine will be sufficient when in ten minutes' work it can pump up enough water to fill one tank2 by the aid of this contrivance we are therefore able to accumulate for one effort the whole power of the engine for ten minutes4 ,,the ,,fly-,,wheel4 #edf4 ,one of the best means of storing energy is by setting a heavy body in rapid motion4 ,this has already been referred to in the case of the cannon-ball4 ,in order to render this method practically available for the purposes of machinery1 the heavy body we use is a fly-wheel1 and the rapid motion imparted to it is that of rotation about its axis4 ,a very large amount of energy can by this means be stored in a manageable form4 #edg4 ,we shall illustrate the principle by the apparatus of ,fig4 #gb4 ,this represents an iron fly-wheel ,b3 its diameter is #ah1 and its weight is #bf lbs42 the fly is carried upon a shaft 7,a7 of wrought iron in diameter4 ,we shall store up a quantity of energy in this wheel1 by setting it in rapid motion1 and then we shall see how we can recover from it the energy we have imparted4 #edh4 ,a rope is coiled around the shaft2 by pulling this rope the wheel is made to turn round3 thus the rope is the medium by which my energy shall be imparted to the wheel4 ,to measure the operation accurately1 ,i attach the rope to the hook of the spring balance 7,fig4 #i72 and by taking the ring of the balance in my hand1 ,i learn from the index the amount of the force ,i am exerting4 ,i find that when ,i walk backwards as quickly as is convenient1 pulling the rope all the time1 the scale shows a tension of about #ej lbs4 ,to set the wheel rapidly in motion1 ,i pull about #bj' of rope from the axle1 so that ,i have imparted to the wheel somewhere about #ej #bj #a1jjj units of energy4 ,the rope is fastened to the shaft1 so that1 after it has been all unwound1 the wheel now rapidly rotating winds it in4 ,by measuring the time in which the wheel made a certain number of coils of the rope around the shaft1 ,i find that it makes about #fjj revolutions per minute4 ,illustration3 ,,fig4 #gb4 #edi4 ,let us see how the stored-up energy can be exhibited4 ,a piece of pine #bd #a #a of which both ends are supported1 requires a force of #cjj lbs4 applied to its centre to produce fracture 7,see p4 #aij74 ,i arrange such a piece of pine near the wheel4 ,as the shaft is winding in the rope1 a tremendous chuck would be given to anything which tried to stop the motion4 ,if ,i tied the end of the rope to the piece of pine1 the chuck would break the rope2 therefore ,i have fastened one end of a #aj' length of chain to the rope1 and the other has been tied round the middle of the wooden bar4 ,the wheel first winds in the rope1 then the chain takes a few turns before it tightens1 and crack goes the rod of pine4 ,the wheel had no choice2 it must either stop or break the rod3 but nature forbids it to be stopped1 unless by a great force1 which the rod was not strong enough to apply4 ,here ,i never exerted a force greater than #ej lbs4 in setting the wheel in motion4 ,the wheel stored up and modified my energy so as to produce a force of #cjj lbs41 which had1 however1 only to be exerted over a very small distance4 #eej4 ,but we may show the experiment in another way1 which is that represented in the figure 7#gb74 ,we see the chain is there attached to two #ef lb4 weights4 ,the mode of proceeding is that already described4 ,the rope is first wound round the shaft1 then by pulling the rope the wheel is made to revolve2 the wheel then begins to wind in the rope again1 and when the chain tightens the two #ef lbs4 are raised up to a height of #c or #d feet4 ,here1 again1 the energy has been stored and recovered4 ,but though the fly-wheel will thus preserve energy1 it does so at some cost3 the store is continually being frittered away by friction and the resistance of the air2 in fact1 the energy would altogether disappear in a little time1 and the wheel would come to rest2 it is therefore economical to make the wheel yield up what it has received as soon as possible4 #eea4 ,these principles are illustrated by the function of the fly-wheel in a steam-engine4 ,the pressure of the steam upon the piston varies according to the different parts of the stroke3 and the fly-wheel obviates the inconvenience which would arise from such irregularity4 ,its great inertia makes its velocity but little augmented by the exuberant action of the piston when the pressure is greatest1 while it also sustains the motion when the piston is giving no assistance4 ,the fly-wheel is a vast reservoir into which the engine pours its energy1 sudden floods alternating with droughts2 but these succeed each other so rapidly1 and the area of the reservoir is so vast1 that its level remains sensibly uniform1 and the supplies sent out to the consumers are regular and unvaried4 ,the consumers of the energy stored in the fly-wheel of an engine are the machines in the mill2 they are supplied by shafts which traverse the building1 conveying1 by their rotation1 the energy originally condensed within the coal from which combustion has set it free4 ,,the ,,punching ,,machine4 #eeb4 ,when energy has been stored in a fly-wheel1 it can be withdrawn either as a small force acting over a great distance1 or as a large force over a small distance4 ,in the latter case the fly-wheel acts as a mechanical power1 and in this form it is used in the very important machine to be next described4 ,a model of the punching machine is shown in ,fig4 #gc4 ,the punching machine is usually worked by a steam-engine1 a handle will move our small model4 ,the handle turns a shaft on which the fly-wheel ,f is mounted4 ,on the shaft is a small pinion ,d of #dj teeth3 this works into a large wheel ,e of #bjj teeth1 so that1 when the fly and the pinion have turned round #e times1 ,e will have turned round once1 ,c is a circular piece of wood called a cam1 which has a hole bored through it1 between the centre and circumference2 by means of this hole1 the cam is mounted on the same axle as ,e1 to which it is rigidly fastened1 so that the two must revolve together4 ,a is a lever of the first order1 whose fulcrum is at ,a3 the remote end of this lever rests upon the cam ,c2 the other end ,b contains the punch4 ,as the wheel ,e revolves it carries with it the cam3 this raises the lever and forces the punch down a hole in a die1 into which it fits exactly4 ,the metal to be operated on is placed under the punch before it is depressed by the cam1 and the pressure drives the punch through1 cutting out a cylindrical piece of metal from the plate3 this model will1 as you see1 punch through ordinary tin4 ,illustration3 ,,fig4 #gc4 #eec4 ,let us examine the mode of action4 ,the machine being made to rotate rapidly1 the punch is depressed once for every #e revolutions of the fly2 the resistance which the metal opposes to being punched is no doubt very great1 but the lever acts at a twelve-fold advantage4 ,when the punch comes down on the surface of the metal1 one of three things must happen3 either the motion must stop suddenly1 or the machine must be strained and injured1 or the metal must be punched4 ,but the motion cannot be stopped suddenly1 because1 before this could happen1 an infinite force would be developed1 which must make something yield4 ,if therefore we make the structure sufficiently massive to prevent yielding1 the metal must be punched4 ,such machines are necessarily built strong enough to make the punching of the metal easier than breaking the machine4 #eed4 ,we shall be able to calculate1 from what we have already seen in ,art4 #bdh1 the magnitude of the force required for punching4 ,we there learned that about #bbe tons of pressure was necessary to shear a bar of iron one square inch in section4 ,punching is so far analogous to shearing that in each case a certain area of surface has to be cut2 the area in punching is measured by the cylinder of iron which is removed4 #eee4 ,suppose it be required to punch a hole #je in diameter through a plate #jh thick1 the area of iron that has to be cut across is _/ _/ #abf square inches3 and as #bbe tons per square inch are required for shearing1 this hole will require #bbe #abf #bhd tons4 ,a force of this amount must therefore be exerted upon the punch3 which will require from the cam a force of more than #b tons upon its end of the lever4 ,though the iron must be pierced to a depth of #jh1 yet it is obvious that almost immediately after the punch has penetrated the surface of the iron1 the cylinder must be entirely cut and begin to emerge from the other side of the plate4 ,we shall certainly be correct in supposing that the punching is completed before the punch has penetrated to a depth of #jb1 and that for not more than this distance has the great pressure of #bh tons been exerted2 for a small pressure is afterwards sufficient to overcome the friction which opposes the motion of the cylinder of iron4 ,hence1 though so great a pressure has been required1 yet the number of units of energy consumed is not very large2 it is _/ #bbdj #bhd #ajfb4 ,the energy actually required to punch a hole of half an inch diameter through a plate eight-tenths of an inch thick is therefore less than that which would be expended in raising #a cwt4 to a height of ten feet4 #eef4 ,the fly-wheel may be likened to the reservoir in ,art4 #ede4 ,the time that is actually occupied in the punching is extremely small1 and the sudden expenditure of #ajfb units is gradually reimbursed by the engine4 ,if the rotating fly-wheel contain #ej1jjj units of energy1 the abstraction of #acfb units will not perceptibly affect its velocity4 ,there is therefore an advantage in having a heavy fly sustained at a high speed for the working of a punching machine4 ,,lecture ,,xvii4 ,,circular ,,motion4 ,the ,nature of ,circular ,motion4--,circular motion in ,liquids4--,the ,applications of ,circular ,motion4--,the ,permanent ,axes4 ,,the ,,nature ,,of ,,circular ,,motion4 #eeg4 ,to compel a body to swerve from motion in a straight line force must be exercised4 ,in this chapter we shall study the comparatively simple case of a body revolving in a circle4 #eeh4 ,when a body moves round uniformly in a circle force must be continuously applied1 and the first question for us to examine is1 as to the direction of that force4 ,we have to demonstrate the important fact1 that it constantly tends towards the centre4 #eei4 ,the direction of the force can be exhibited by actual experiment1 and its magnitude will be at the same time clearly indicated by the extent to which a spring is stretched4 ,the apparatus we use is shown in ,fig4 #gd4 ,the essential parts of the machine consist of two balls ,a1 ,b1 each #b in diameter3 these are thin hollow spheres of silvered brass4 ,the balls are supported on arms ,p ,a1 ,q ,b1 which are attached to a piece of wood1 ,p ,q1 capable of turning on a socket at ,c4 ,the arm ,a ,p is rigidly fixed to ,p ,q2 the other arm1 ,b ,q1 is capable of turning round a pin at ,q4 ,an india-rubber door-spring is shown at ,f2 one end of this is secured to ,p ,q1 the other end to the movable arm1 ,q ,b4 ,if the arm ,q ,b be turned so as to move ,b away from ,c1 the spring ,f must be stretched4 ,illustration3 ,,fig4 #gd4 ,a small toothed wheel is mounted on the same socket as ,c2 this is behind ,p ,q1 and is therefore not seen in the figure3 the whole is made to revolve rapidly by the large wheel ,e1 which is turned by the handle ,d4 #efj4 ,the room being darkened1 a beam from the lime-light is allowed to fall on the apparatus3 the reflections of the light are seen in the two silvered balls as two bright points4 ,when ,d is turned1 the balls move round rapidly1 and you see the points of light reflected from them describe circles4 ,the ball ,b when at rest is #d from ,c1 while ,a is #h from ,c2 hence the circle described by ,b is smaller than that described by ,a4 ,the appearance presented is that of two concentric luminous circles4 ,as the speed increases1 the inner circle enlarges till the two circles blend into one4 ,by increasing the speed still more1 you see the circle whose diameter is enlarging actually exceeding the fixed circle1 and its size continues to increase until the highest velocity which it is safe to employ has been communicated to the machine4 #efa4 ,what is the explanation of this8 ,the arm ,a is fixed and the distance ,a ,c cannot alter1 hence ,a describes the fixed circle4 ,b1 on the other hand1 is not fixed2 it can recede from ,c1 and we find that the quicker the speed the further it recedes4 ,the larger the circle described by ,b the more is the spring stretched1 and the greater is the force with which ,b is attracted towards the centre4 ,this experiment proves that the force necessary to retain a body in a circular path must be increased when the speed is increased4 #efb4 ,thus we see that uniform motion of a body in a circle can only be produced by an uniform force directed to the centre4 ,if the motion1 even though circular1 have variable speed the law of the force is not so simple4 #efc4 ,we can measure the magnitude of this force by the same apparatus4 ,the ball ,b weighs #ja lb4 ,i find that ,i must pull it with a force of #c lbs4 in order to draw it to a distance of #h from ,c2 that is1 to the same distance as ,a is from ,c4 ,hence1 when the diameters of the circles in which the balls move are equal1 the central force must be #c lbs42 that is1 it must be nearly thirty times as great as gravity4 #efd4 ,the necessity for the central force is thus shown3 ,let us conceive a weight attached to a string to be swung round in a circle1 a portion of which is shown in ,fig4 #ge4 ,illustration3 ,,fig4 #ge4 ,suppose the weight be at ,s and moving towards ,p1 and let a tangent to the circle be drawn at ,p4 ,take two points on the circle1 ,a and ,b1 very near ,p2 the small arc ,a ,b does not differ perceptibly from the part ,a ,b on the tangent line2 hence1 when the particle arrives at ,a1 it is a matter of indifference whether it travels in the arc ,a ,b1 or along the line ,a ,b4 ,let us suppose it to move along the line4 ,by the first law of motion1 a particle moving in the line ,a ,b would continue to do so2 hence1 if the particle be allowed1 it will move on to ,q3 but the particle is not allowed to move to ,q2 it is found at ,r4 ,hence it must have been withdrawn by some force4 #efe4 ,this force is supplied by the string to which the weight is attached4 ,the incessant change from the rectilinear motion of the weight requires a constantly applied force1 and this is always directed to the centre4 ,should the string be released1 the body flies off in the direction of the tangent to the circle at the point which the body occupied at the instant of release4 #eff4 ,the central force increases in proportion to the square of the velocity4 ,if ,i double the speed with which the weight is whirled round in the circle1 ,i quadruple the force which the string must exert on the body4 ,if the speed be trebled1 the force is increased ninefold1 and so on4 ,when the speeds with which two equal masses are revolving in two circles are equal1 the central force in the smaller circle is greater than that of the larger circle1 in the proportion of the radius of the larger circle to that of the smaller4 ,,the ,,action ,,of ,,circular ,,motion ,,upon ,,liquids4 #efg4 ,i have here a small bucket nearly filled with water3 to the handle a piece of string is attached4 ,if ,i whirl the bucket round in a vertical plane sufficiently fast1 you see no water escapes1 although the bucket is turned upside down once in every revolution4 ,this is because the water has not time to fall out during such a brief interval4 ,a body would not fall half an inch from rest in the twentieth of a second4 #efh4 ,the action of circular motion upon liquids is illustrated by the experiment which is represented in ,fig4 #gf4 ,illustration3 ,,fig4 #gf4 ,a glass beaker about half full of water is mounted so that it can be spun round rapidly4 ,the motion is given by means of a large wheel turned by a handle1 as shown in the figure4 ,when the rotation commences1 the water is seen to rise up against the glass sides and form a hollow in the centre4 #efi4 ,in order to demonstrate this clearly1 ,i turn upon the vessel a beam from the lime-light4 ,i have previously dissolved a little quinine in the water4 ,the light from the lamp is transmitted through a piece of dense blue glass4 ,when the light thus coloured falls on the water1 the presence of the quinine makes the entire liquid glow with a bluish hue4 ,this remarkable property of quinine1 which is known as fluorescence1 enables you to see distinctly the hollow form caused by the rotation4 #egj4 ,you observe that as the speed becomes greater the depth of the hollow increases1 and that if ,i turn the wheel sufficiently fast the water is actually driven out of the glass4 ,the shape of the curve which the water assumes is that which would be produced by the revolution of a parabola about its axis4 #ega4 ,the explanation is simple4 ,as soon as the glass begins to revolve1 the friction of its sides speedily imparts a revolving motion to the water2 but in this case there is nothing to keep the particles near the centre like the string in the revolving weight1 so the liquid rises at the sides of the glass4 #egb4 ,but you may ask why all the particles of the water should not go to the circumference1 and thus line the inside of the glass with a hollow cylinder of water instead of the parabola4 ,such an arrangement could not exist in a liquid acted on by gravity4 ,the lower parts of the cylinder must bear the pressure of the water above1 and therefore have more tendency to flatten out than the upper portions4 ,this tendency could not be overcome by any consequences of the movement1 for such must be alike on all parts at the same distance from the axis4 #egc4 ,a very beautiful experiment was devised by ,plateau for the purpose of studying the revolution of a liquid removed from the action of gravity4 ,the apparatus employed is represented in ,fig4 #gg4 ,a glass vessel #i cube is filled with a mixture of alcohol and water4 ,the relative quantities ought to be so proportioned that the fluid has the same specific gravity as olive oil1 which is heavier than alcohol and lighter than water4 ,in practice1 however1 it is found so difficult to adjust the composition exactly that the best plan is to make two alcoholic mixtures so that olive oil will just float on one of them1 and just sink in the other4 ,the lower half of the glass is to be filled with the denser mixture1 and the upper half with the lighter4 ,if1 then1 the oil be carefully introduced with a funnel it will form a beautiful sphere in the middle of the vessel1 as shown in the figure4 ,we thus see that a liquid mass freed from the action of terrestrial gravity1 forms a sphere by the mutual attraction of its particles4 ,illustration3 ,,fig4 #gg4 ,through the liquid a vertical spindle passes4 ,on this there is a small disk at the middle of its length1 about which the sphere of oil arranges itself symmetrically4 ,to the end of the spindle a handle is attached4 ,when the handle is turned round slowly1 the friction of the disk and spindle communicates a motion of rotation to the sphere of oil4 ,we have thus a liquid spheroidal mass endowed with a movement of rotation2 and we can study the effect of the motion upon its form4 ,we first see the sphere flatten down at its poles1 and bulge at its equator4 ,in order to show the phenomenon to those who may not be near to the table1 the sphere can be projected on the screen by the help of the lime-light lamp and a lens4 ,it first appears as a yellow circle1 and then1 as the rotation begins1 the circle gradually transforms into an ellipse4 ,but a very remarkable modification takes place when the handle is turned somewhat rapidly4 ,the ellipsoid gradually flattens down until1 when a certain velocity has been attained1 the surface actually becomes indented at the poles1 and flies from the axis altogether4 ,ultimately the liquid assumes the form of a beautiful ring1 and the appearance on the screen is shown in ,fig4 #gh4 ,illustration3 ,,fig4 #gh4 #egd4 ,the explanation of the development of the ring involves some additional principles3 as the sphere of oil spins round in the liquid1 its surface is retarded by friction2 so that when the velocity attains a certain amount1 the internal portions of the sphere1 which are in the neighbourhood of the spindle1 are driven from the centre into the outer portions1 but the full account of the phenomenon cannot be given here4 #ege4 ,the earth was1 we believe1 originally in a fluid condition4 ,it had then1 as it has now1 a diurnal rotation1 and one of the consequences of this rotation has been to cause the form to be slightly protuberant at the equator1 just as we have seen the sphere of oil to bulge out under similar conditions4 #egf4 ,bodies lying on the earth are whirled around in a great circle every day4 ,hence1 if there were not some force drawing them to the centre1 they would fly off at a tangent4 ,a part of the earth's attraction goes for this purpose1 and the remainder1 which is the apparent weight1 is thus diminished by a quantity increasing from the pole to the equator 7,art4 #hf74 ,,the ,,applications ,,of ,,circular ,,motion4 #egg4 ,these principles have many applications in the mechanical arts2 we shall mention two of them4 ,the first is to the governor-balls of a steam-engine2 the second is to the process of sugar refining4 ,an engine which turns a number of machines in a factory should work uniformly4 ,irregularities of motion may be productive of loss and various inconveniences4 ,an engine would work irregularly either from variation in the production of steam1 or from the demands upon the power being lessened or increased4 ,even if the first of these sources of irregularity could be avoided by care1 it is clear that the second could not4 ,some machines in the mill are occasionally stopped1 others occasionally set in motion1 and the engine generally tends to go faster the less it has to do4 ,it is therefore necessary to provide means by which the speed shall be restrained within narrow limits1 and it is obviously desirable that the contrivance used for this purpose should be self-acting4 ,we must1 therefore1 have some arrangement which shall admit more steam to the cylinder when the engine is moving too slowly1 and less steam when it is moving too quickly4 ,the valve which is to regulate this must be worked by some agent which depends upon the velocity of the engine2 this at once points to circular motion because the force acting on the revolving body depends upon its velocity4 ,such was the train of reasoning which led to the happy invention of the governor-balls3 these are shown in ,fig4 #gi4 ,illustration3 ,,fig4 #gi4 ,a ,b is a vertical spindle which is turned by the engine4 ,p ,p is a piece firmly attached to the spindle and turning with it4 ,p ,w1 ,p ,w are arms terminating in weights ,w ,w2 these are balls of iron1 generally very massive3 the arms are free to turn round pins at ,p ,p4 ,at ,q ,q links are placed1 attached to another piece ,r ,r1 which is free to slide up and down the spindle4 ,when ,a ,b rotates1 ,w and ,w are carried round1 and therefore fly outwards from the spindle2 to do this they must evidently pull the piece ,r ,r up the shaft4 ,we can easily imagine an arrangement by which ,r ,r shall be made to shut or open the steam-valve according as it ascends or descends4 ,the problem is then solved1 for if the engine begin to go too rapidly1 the balls fly out further just as they did in ,fig4 #gd3 this movement raises the piece ,r ,r1 which diminishes the supply of steam1 and consequently checks the speed4 ,on the other hand1 when the engine works too slowly1 the balls fall in towards the spindle1 the piece ,r ,r descends1 the valve is opened1 and a greater supply of steam is admitted4 ,the objection to this governor is that though it moderates1 it does not completely check irregularity4 ,there are other governors occasionally employed which depend also on circular motion2 some of these are more sensitive than the governor-balls2 but they are elaborate machines1 only to be employed under exceptional circumstances4 #egh4 ,the application of circular motion to sugar refining is a very beautiful invention4 ,to explain it ,i must briefly describe the process of refining4 ,the raw sugar is dissolved in water1 and the solution is purified by straining and by filtration through animal charcoal4 ,the syrup is then boiled4 ,in order to preserve the colour of the sugar1 and to prevent loss1 this boiling is conducted in vacuo1 as by this means the temperature required is much less than would be necessary with the ordinary atmospheric pressure4 ,the evaporation having been completed1 crystals of sugar form throughout the mass of syrup4 ,to separate these crystals from the liquor which surrounds them1 the aid of circular motion force is called in4 ,a mass of the mixture is placed in a large iron tub1 the sides of which are perforated with small holes4 ,the tub is then made to rotate with prodigious velocity2 its contents instantly fly off to the circumference1 the liquid portions find an exit through the perforations in the sides1 but the crystals are left behind4 ,a little clear syrup is then sprinkled over the sugar while still rotating3 this washes from the crystals the last traces of the coloured liquid1 and passes out through the holes2 when the motion ceases1 the inside of the tub contains a layer of perfectly pure white sugar1 some inches in thickness1 ready for the market4 #egi4 ,circular motion is peculiarly fitted for this purpose2 each particle of liquid strives to get as far away from the axis as possible4 ,the action on the sugar is very different from what it would have been had the mass been subjected to pressure by a screw-press or similar contrivance2 the particles immediately acted on would then have to transmit the pressure to those within2 and the consequence would be that while the crystals of sugar on the outside would be crushed and destroyed1 the water would only be very imperfectly driven from the interior3 for it could lurk in the interstices of the sugar1 which remain notwithstanding the pressure4 #ehj4 ,but with circular motion the water must go1 not because it is pushed by the crystals1 but because of its own inertia2 and it can be perfectly expelled by a velocity of rotation less than that which would be necessary to produce sufficient pressure to make the crystals injure each other4 ,,the ,,permanent ,,axes4 #eha4 ,there are some curious properties of circular motion which remain to be considered4 ,these we shall investigate by means of the apparatus of ,fig4 #hj4 ,this consists of a pair of wheels ,b ,c1 by which a considerable velocity can be given to a horizontal shaft4 ,this shaft is connected by a pair of bevelled wheels ,d with a vertical spindle ,f4 ,the machine is worked by a handle ,a1 and the object to be experimented upon is suspended from the spindle4 #ehb4 ,i first take a disk of wood #ah in diameter2 a hole is bored in the margin of this disk2 through this hole a rope is fastened1 by means of which the disk is suspended from the spindle4 ,the disk hangs of course in a vertical plane4 ,illustration3 ,,fig4 #hj4 #ehc4 ,i now begin to turn the handle round gently1 and you see the disk begins to rotate about the vertical diameter2 but1 as the speed increases1 the motion becomes a little unsteady2 and finally1 when ,i turn the handle very rapidly1 the disk springs up into a horizontal plane1 and you see it like the surface of a small table3 the rope swings round and round in a cone1 so rapidly that it is hardly seen4 #ehd4 ,we may repeat the experiment in a different manner4 ,i take a piece of iron chain about #b' long1 ,g2 ,i pass the rope through the two last links of its extremities1 and suspend the rope from the spindle4 ,when ,i commence to turn the handle1 you see the chain gradually opens out into a loop ,h2 and as the speed increases1 the loop becomes a complete ring4 ,still increasing the speed1 ,i find the ring becomes unsteady1 till finally it rises into a horizontal plane4 ,the ring of chain in the horizontal plane is shown at ,i4 ,when the motion is further increased1 the ring swings about violently1 and so ,i cease turning the handle4 ,illustration3 ,,fig4 #ha4 #ehe4 ,the principles already enunciated will explain these remarkable results2 we shall only describe that of the chain1 as the same explanation will include that of the disk of wood4 ,we shall begin with the chain hanging vertically from the spindle3 the moment rotation commences1 the chain begins to spin about a vertical axis2 the parts of the chain fly outwards from this axis just as the ball flies outwards in ,fig4 #gd2 this is the cause of the looped form ,h which the chain assumes4 ,as the speed is increased the loop gradually opens more and more1 just as the diameter of the circle ,fig4 #gd increases with the velocity4 ,but we have also to inquire into the cause of the remarkable change of position which the ring undergoes2 instead of continuing to rotate about a vertical diameter1 it comes into a horizontal plane4 ,this will be easily understood with the help of ,fig4 #ha4 ,let ,o ,p represent the rope attached to the ring1 and ,o ,c be the vertical axis4 ,suppose the ring to be spinning about the axis ,o ,c1 when ,o ,c was a diameter2 if then1 from any cause1 the ring be slightly displaced1 we can show that the circular motion will tend to drive the ring further from the vertical plane1 and force it into the horizontal plane4 ,let the ring be in the position represented in the figure2 then1 since it revolves about the vertical line ,o ,c1 the tendency of ,p ,p and ,q ,q is to move outwards in the directions of the arrows1 thus evidently tending to bring the ring into the horizontal plane4 #ehf4 ,in ,art4 #ajc we have explained what is meant by stable and unstable equilibrium2 we have here found a precisely analogous phenomenon in motion4 ,the rotation of the ring about its diameter is unstable1 for the minutest deviation of the ring from this position is fatal2 circumstances immediately combine to augment the deviation more and more1 until finally the ring is raised into the horizontal plane4 ,once in the horizontal plane1 the motion there is stable1 for if the ring be displaced the tendency is to restore it to the horizontal4 #ehg4 ,the ring1 when in a horizontal plane1 rotates permanently about the vertical axis through its centre2 this axis is called permanent1 to distinguish it from all other directions1 as being the only axis about which the motion is stable4 #ehh4 ,we may show another experiment with the chain3 if instead of passing the rope through the links at its ends1 ,i pass the rope through the centre of the chain1 and allow the ends of the chain to hang downwards4 ,i now turn the handle2 instantly the parts of the chain fly outwards in a curved form2 and by increasing the velocity1 the parts of the chain at length come to lie almost in a straight line4 ,,lecture ,,xviii4 ,,the ,,simple ,,pendulum4 ,introduction4--,the ,circular ,pendulum4--,law connecting the ,time of ,vibration with the ,length4--,the ,force of ,gravity determined by the ,pendulum4--,the ,cycloid4 ,,introduction4 #ehi4 ,if a weight be attached to a piece of string1 the other end of which hangs from a fixed point1 we have what is called a simple pendulum4 ,the pendulum is of the utmost importance in science1 as well as for its practical applications as a time-keeper4 ,in this lecture and the next we shall treat of its general properties2 and the last will be devoted to the practical applications4 ,we shall commence with the simple pendulum1 as already defined1 and prove1 by experiment1 the remarkable property which was discovered by ,galileo4 ,the simple pendulum is often called the circular pendulum4 ,,the ,,circular ,,pendulum4 ,illustration3 ,,fig4 #hb4 #eij4 ,we first experiment with a pendulum on a large scale4 ,our lecture theatre is #cb feet high1 and there is a wire suspended from the ceiling #bg' long2 to the end of this a ball of cast iron weighing #be lbs4 is attached4 ,this wire when at rest hangs vertically in the direction ,o ,c 7,fig4 #hb74 ,i draw the ball from its position of rest to ,a2 when released1 it slowly returns to ,c1 its original position2 it then moves on the other side to ,b1 and back again to my hand at ,a4 ,the ball--or to speak more precisely1 the centre of the ball--moves in a circle1 the centre being the point ,o in the ceiling from which the wire is suspended4 #eia4 ,what causes the motion of the pendulum when the weight is released8 ,it is the force of gravity2 for by moving the ball to ,a ,i raise it a little1 and therefore1 when ,i release it gravity compels it to return to ,c it being the only manner in which the mode of suspension will allow it to fall4 ,but when it has reached its original position at ,c1 why does it continue its motion8--for gravity must be acting against the ball during the journey from ,c to ,b4 ,the first law of motion explains this4 7,art4 #dhe74 ,in travelling from ,a to ,c the ball has acquired a certain velocity1 hence it has a tendency to go on1 and only by the time it has arrived at ,b will gravity have arrested the velocity1 and begin to make it descend4 #eib4 ,you see1 the ball continues moving to and fro--oscillating1 as it is called--for a long time4 ,the fact is that it would oscillate for ever1 were it not for the resistance of the air1 and for some loss of energy at the point of suspension4 #eic4 ,by the time of an oscillation is meant the time of going from ,a to ,b1 but not back again4 ,the time of our long pendulum is nearly three seconds4 #eid4 ,with reference to the time of oscillation ,galileo made a great discovery4 ,he found that whether the pendulum were swinging through the arc ,a ,b1 or whether it had been brought to the point ,a1 and was thus describing the arc ,a ,b1 the time of oscillation remained nearly the same4 ,the arc through which the pendulum oscillates is called its amplitude1 so that we may enunciate this truth by saying that the time of oscillation is nearly independent of the amplitude4 ,the means by which ,galileo proved this would hardly be adopted in modern days4 ,he allowed a pendulum to perform a certain number of vibrations1 say #ajj1 through the arc ,a ,b1 and he counted his pulse during the time2 he then counted the number of pulsations while the pendulum vibrated #ajj times in the arc ,a ,b1 and he found the number of pulsations in the two cases to be equal4 ,assuming1 what is probably true1 that ,galileo's pulse remained uniform throughout the experiment1 this result showed that the pendulum took the same time to perform #ajj vibrations1 whether it swung through the arc ,a ,b1 or through the arc ,a ,b4 ,this discovery it was which first suggested the employment of the pendulum as a means of keeping time4 #eie4 ,we shall adopt a different method to show that the time does not depend upon the amplitude4 ,i have here an arrangement which is represented in ,fig4 #hc4 ,it consists of two pendulums ,a ,d and ,b ,c1 each #ab' long1 and suspended from two points ,a ,b1 about #a' apart1 in the same horizontal line4 ,each of these pendulums carries a weight of the same size3 they are in fact identical4 ,illustration3 ,,fig4 #hc4 #eif4 ,i take one of the balls in each hand4 ,if ,i withdraw each of them from its position of rest through equal distances and then release them1 both balls return to my hands at the same instant4 ,this might have been expected from the identity of the circumstances4 #eig4 ,i next withdraw the weight ,c in my right hand to a distance of #a'1 and the weight ,d in my left hand to a distance of #b'1 and release them simultaneously4 ,what happens8 ,i keep my hands steadily in the same position1 and ,i find that the two weights return to them at the same instant4 ,hence1 though one of the weights moved through an amplitude of #b' 7,c ,e7 while the other moved through an amplitude of #d' 7,d ,f71 the times occupied by each in making two oscillations are identical4 ,if ,i draw the right-hand ball away #c'1 while ,i draw the left hand only #a' from their respective positions of rest1 ,i still observe the same result4 #eih4 ,in two oscillations we can see no effect on the time produced by the amplitude1 and we are correct in saying that1 when the amplitude is only a small fraction of the length of the pendulum1 its effect is inappreciable4 ,but if the amplitude of one pendulum were very large1 we should find that its time of oscillation is slightly greater than that of the other1 though to detect the difference would require a delicate test4 ,one consequence of what is here remarked will be noticed at a later page4 7,art4 #fee47 #eii4 ,we next inquire whether the weight which is attached to the pendulum has any influence upon the time of vibration4 ,using the #ab' pendulums of ,fig4 #hc1 ,i place a weight of #ab lbs4 on one hook and one of #f lbs4 on the other4 ,i withdraw one in each hand2 ,i release them2 they return to my hand at the same moment4 ,whether ,i withdraw the weights through long arcs or short arcs1 equal or unequal1 they invariably return together1 and both therefore have the same time of vibration4 ,with other iron weights the same law is confirmed1 and hence we learn that1 besides being independent of the amplitude1 the time of vibration is also independent of the weight4 #fjj4 ,finally1 let us see if the material of the pendulum can influence its time of vibration4 ,i place a ball of wood on one wire and a ball of iron on the other2 ,i swing them as before3 the vibrations are still performed in equal times4 ,a ball of lead is found to swing in the same time as a ball of brass1 and both in the same time as a ball of iron or of wood4 #fja4 ,in this we may be reminded of the experiments on gravity 7,art4 #dia71 where we showed that all bodies fall to the ground in equal times1 whatever be their sizes or their materials4 ,from both cases the inference is drawn that the force of gravity upon different bodies is proportional to their masses1 though the bodies be made of various substances4 ,it was indeed by means of experiments with the pendulum that ,newton proved that gravity had this property1 which is one of the most remarkable truths in nature4 ,,law ,,connecting ,,the ,,time ,,of ,,vibration ,,with ,,the ,,length4 #fjb4 ,we have seen that the time of vibration of a pendulum depends neither upon its amplitude1 material1 nor weight2 we have now to learn on what the time does depend4 ,it depends upon the length of the pendulum4 ,the shorter a pendulum the less is its time of vibration4 ,we shall find by experiment the relation between the time and the length of the cord by which the weight is suspended4 ,illustration3 ,,fig4 #hd4 #fjc4 ,i have here 7,fig4 #hd7 two pendulums ,a ,d1 ,b ,c1 one of which is #ab' long and the other #c'2 they are mounted side by side1 and the weights are at the same distance from the floor4 ,i take one of the weights in each hand1 and withdraw them to the same distance from the position of rest4 ,i release the balls simultaneously2 ,c moves off rapidly1 arrives at the end ,c while ,d has only reached ,d1 and returns to my hand just as ,d has completed one oscillation4 ,i do not seize ,c3 it goes off again1 only to return at the same moment when ,d reaches my hand4 ,thus ,c has performed four oscillations while ,d has made no more than two4 ,this proves that when one of two pendulums is a quarter the length of the other1 the time of vibration of the shorter one is half that of the other4 #fjd4 ,we shall repeat the experiment with another pendulum #bg' long1 suspended from the ceiling1 and compare its vibrations with those of a pendulum #c' long4 ,i draw the weights to one side and release them as before2 and you see that the small pendulum returns twice to my hand while the long pendulum is still absent2 but that1 keeping my hands steadily in the same place throughout the experiment1 the long pendulum at last returns simultaneously with the third arrival of the short one4 ,hence we learn that a pendulum #bg' long takes three times as much time for a single vibration as a #c' pendulum4 #fje4 ,the lengths of the three pendulums on which we have experimented 7#bg'1 #ab'1 #c'71 are in the proportions of the numbers #i1 #d1 #a2 and the times of the oscillations are proportional to #c1 #b1 #a3 hence we learn that the period of oscillation of a pendulum is proportional to the square root of its length4 #fjf4 ,but the time of vibration must also depend upon gravity2 for it is only owing to gravity that the pendulum vibrates at all4 ,it is evident that1 if gravity were increased1 all bodies would fall to the earth more than #af' in the first second4 ,the effect on the pendulum would be to draw the ball more quickly from ,d to ,d 7,fig4 #hd71 and thus the time of vibration would be diminished4 ,it is found by calculation1 and the result is confirmed by experiment1 that the time of vibration is represented by the expression1 #cadaf 7 ,length _/ ,force of gravity74 #fjg4 ,the accurate value of the force of gravity in ,london is #cbaijh1 so that the time of vibration of a pendulum there is #jeecg length3 the length of the seconds pendulum is #c'bfaf4 ,,the ,,force ,,of ,,gravity ,,determined ,,by ,,the ,,pendulum4 #fjh4 ,the pendulum affords the proper means of measuring the force of gravity at any place on the earth4 ,we have seen that the time of vibration can be expressed in terms of the length and the force of gravity2 so conversely1 when the length and the time of vibration are known1 the force of gravity can be determined and the expression for it is-- ,length 7#cadaf _/ ,time74 #fji4 ,it is impossible to observe the time of a single vibration with the necessary degree of accuracy2 but supposing we consider a large number of vibrations1 say #ajj1 and find the time taken to perform them1 we shall then learn the time of one oscillation by dividing the entire period by #ajj4 ,the amplitudes of the oscillations may diminish1 but they are still performed in the same time2 and hence1 if we are sure that we have not made a mistake of more than one second in the whole time1 there cannot be an error of more than #jja second1 in the time of one oscillation4 ,by taking a still larger number the time may be determined with the utmost precision1 so that this part of the inquiry presents little difficulty4 #faj4 ,but the length of the pendulum has also to be ascertained1 and this is a rather baffling problem4 ,the ideal pendulum whose length is required1 is supposed to be composed of a very fine1 perfectly flexible cord1 at the end of which a particle without appreciable size is attached2 but this is very different from the pendulum which we must employ4 ,we are not sure of the exact position of the point of suspension1 and1 even if we had a perfect sphere for the weight1 the distance between its centre and the point of suspension is not the same thing as the length of the simple pendulum that would vibrate in the same time4 ,owing to these circumstances1 the measurement of the pendulum is embarrassed by considerable difficulties1 which have however been overcome by ingenious contrivances to be described in the next chapter4 #faa4 ,but we shall perform1 in a very simple way1 an experiment for determining the force of gravity4 ,i have here a silken thread which is fastened by being clamped between two pieces of wood4 ,a cast iron ball #bed in diameter is suspended by this piece of silk4 ,the distance from the point of suspension of the silk to the ball is #bdjg1 as well as it can be measured4 ,the length of the ideal pendulum which would vibrate isochronously with this pendulum is #becg1 being about #jjc greater than the distance from the point of suspension to the centre of the sphere4 #fab4 ,the length having been ascertained1 the next element to be determined is the time of vibration4 ,for this purpose ,i use a stop-watch1 which can be started or stopped instantaneously by touching a little stud3 this watch will indicate time accurately to one-fifth of a second4 ,it is necessary that the pendulum should swing in a small arc1 as otherwise the oscillations are not strictly isochronous4 ,quite sufficient amplitude is obtained by allowing the ball to rotate to and fro through a few tenths of an inch4 #fac4 ,in order to observe the movement easily1 ,i have mounted a little telescope1 through which ,i can view the top of the ball4 ,in the eye-piece of the telescope a vertical wire is fastened1 and ,i count each vibration just as the silken thread passes the vertical wire4 ,taking my seat with the stop-watch in my hand1 ,i write down the position of the hands of the stop-watch1 and then look through the telescope4 ,i see the pendulum slowly moving to and fro1 crossing the vertical wire at every vibration2 on one occasion1 just as it passes the wire1 ,i touch the stud and start the watch4 ,i allow the pendulum to make #cjj vibrations1 and as the silk arrives at the vertical wire for the #cjjth time1 ,i promptly stop the watch2 on reference ,i find that #bdaf seconds have elapsed since the time the watch was started4 ,to avoid error1 ,i repeat this experiment1 with precisely the same result3 #bdaf seconds are again required for the completion of #cjj vibrations4 #fad4 ,it is desirable to reckon the vibrations from the instant when the pendulum is at the middle of its stroke1 rather than when it arrives at the end of the swing4 ,in the former case the pendulum is moving with the greatest rapidity1 and therefore the time of coincidence between the thread and the vertical wire can be observed with especial definiteness4 #fae4 ,the time of a single vibration is found1 by dividing #bdaf by #cjj1 to be #jhje second4 ,this is certainly correct to within a thousandth part of a second4 ,we conclude that a pendulum whose length is #becg #baad1 vibrates in #jhje second2 and from this we find that gravity at ,dublin is #b'aad 7#cadaf _/ #j4hje7 #cbaif4 ,this result agrees with one which has been determined by measurement made with every precaution4 ,another method of measuring gravity by the pendulum will be described in the next lecture 7,art4 #fcg74 ,,the ,,cycloid4 #faf4 ,if the amplitude of the vibration of a circular pendulum bear a large proportion to the radius1 the time of oscillation is slightly greater than if the amplitude be very small4 ,the isochronism of the pendulum is only true for small arcs4 #fag4 ,but there is a curve in which a weight may be made to move where the time of vibration is precisely the same1 whatever be the amplitude4 ,this curve is called a cycloid4 ,it is the path described by a nail in the circumference of a wheel1 as the wheel rolls along the ground4 ,thus1 if a circle be rolled underneath the line ,a ,b 7,fig4 #he71 a point on its circumference describes the cycloid ,a ,d ,c ,p ,b4 ,the lower part of this curve does not differ very much from a circle whose centre is a certain point ,o above the curve4 #fah4 ,suppose we had a piece of wire carefully shaped to the cycloidal curve ,a ,d ,c ,p ,b1 and that a ring could slide along it without friction1 it would be found that1 whether the ring be allowed to drop from ,c1 ,p or ,b1 it would fall to ,d precisely in the same time1 and would then run up the wire to a distance from ,d on the other side equal to that from which it had originally started4 ,in the oscillations on the cycloid1 the amplitude is absolutely without effect upon the time4 #fai4 ,as a frictionless wire is impossible1 we cannot adopt this method1 but we can nevertheless construct a cycloidal pendulum in another way1 by utilizing a property of the curve1 ,o ,a 7,fig4 #he7 as a half cycloid2 in fact1 ,o ,a is just the same curve as ,b ,d1 but placed in a different position1 so also is ,o ,b4 ,if a string of length ,o ,d be suspended from the point ,o1 and have a weight attached to it1 the weight will describe the cycloid1 provided that the string wrap itself along the arcs ,o ,a and ,o ,b2 thus when the weight has moved from ,d to ,p1 the string is wrapped along the curve through the space ,o ,t1 the part ,t ,p only being free4 ,this arrangement will always force the point ,p to move in the cycloidal arc4 ,illustration3 ,,fig4 #he4 #fbj4 ,we are now in a condition to ascertain experimentally1 whether the time of oscillation in the cycloid is independent of the amplitude4 ,we use for this purpose the apparatus shown in ,fig4 #hf4 ,d ,c ,e is the arc of the cycloid2 two strings are attached at ,o1 and equal weights ,a1 ,b are suspended from them2 ,c is the lowest point of the curve4 ,the time ,a will take to fall through the arc ,a ,c is of course half the time of its oscillation4 ,if1 therefore1 ,i can show that ,a and ,b both take the same time to fall down to ,c1 ,i shall have proved that the vibrations are isochronous4 ,illustration3 ,,fig4 #hf4 #fba4 ,holding1 as shown in the figure1 ,a in one hand and ,b in the other1 ,i release them simultaneously1 and you see the result1--they both meet at ,c3 even if ,i bring ,a up to ,e1 and bring ,b down close to ,c1 the result is the same4 ,the motion of ,a is so rapid that it arrives at ,c just at the same instant as ,b4 ,when ,i bring the two balls on the same side of ,c1 and release them simultaneously1 ,a overtakes ,b just at the moment when it is passing ,c4 ,hence1 under all circumstances1 the times of descent are equal4 #fbb4 ,it will be noticed that the string attached to the ball ,b1 in the position shown in the figure1 is almost as free as if it were merely suspended from ,o1 for it is only when the ball is some distance from the lowest point that the side arcs produce any appreciable effect in curving the string4 ,the ball swings from ,b to ,c nearly in a circle of which the centre is at ,o4 ,hence1 in the circular pendulum1 the vibrations when small are isochronous1 for in that case the cycloid and the circle become indistinguishable4 ,,lecture ,,xix4 ,,the ,,compound ,,pendulum ,,and ,,the ,,composition ,,of ,,vibrations4 ,the ,compound ,pendulum4--,the ,centre of ,oscillation4--,the ,centre of ,percussion4--,the ,conical ,pendulum4--,the ,composition of ,vibrations4 ,,the ,,compound ,,pendulum4 ,illustration3 ,,fig4 #hg4 #fbc4 ,pendulous motion must now be studied in other forms besides that of the simple pendulum1 which consists of a weight and a cord4 ,any body rotating about an axis may be made to oscillate by gravity4 ,a body thus vibrating is called a compound pendulum4 ,the ideal form1 which consists of an indefinitely small weight attached to a perfectly flexible and imponderable string1 is an abstraction which can only be approximately imitated in nature4 ,it follows that every pendulum used in our experiments is strictly speaking compound4 #fbd4 ,the first pendulum of this class which we shall notice is that used in the common clock 7,fig4 #hg74 ,this consists of a wooden or steel rod ,a ,e1 to which a brass or leaden bob ,b is attached4 ,this pendulum is suspended by means of a steel spring ,c ,a1 which being very flexible1 allows the vibration to be performed with considerable freedom4 ,the use of the screw at the end will be explained in ,art4 #ffd4 ,a pendulum like this vibrates isochronously1 when the amplitude is small1 but it is not easy to see precisely what is the length of the simple pendulum which would oscillate in the same time4 ,in the first place1 we are uncertain as to the virtual position of the point of suspension1 for the spring1 though flexible1 will not yield at the point ,c to the same extent as a string2 thus the effective point of suspension must be somewhat lower than ,c4 ,the other extremity is still more uncertain1 for the weight1 so far from being a single point1 is not exclusively in the neighbourhood of the bob1 inasmuch as the rod of the pendulum has a mass that is appreciable4 ,this form of pendulum cannot therefore be used where it is necessary to determine the length with accuracy4 #fbe4 ,when the length of a pendulum is to be measured1 we must adopt other means of supporting it than that of suspension by a spring1 as otherwise we cannot have a definite point from which to measure4 ,to illustrate the mode that is to be adopted1 ,i take here an iron bar #f' long and #a square1 which weighs #ai lbs4 ,i wish to support this at one end so that it can vibrate freely1 and at the same time have a definite point of suspension4 ,i have here two small prisms of steel ,e 7,fig4 #hh7 fastened to a brass frame2 the faces of the prisms meet at about an angle of #fj and form the edges about which the oscillation takes place3 this frame and the edges can be placed on the end of the bar1 and can be fixed there by tightening two nuts4 ,the object of having the edges on a sliding frame is that they may be applicable to different parts of the bar with facility4 ,in some instruments used in experiments requiring extreme delicacy1 the edges which are attached to the pendulum are supported upon plates of agate2 they are to be adjusted on the same horizontal line1 and the pendulum really vibrates about this line1 as about an axis4 ,for our purpose it will be sufficient to support the edges upon small pieces of steel4 ,a ,b1 ,fig4 #hh1 represents one side of the top of the iron bar2 ,e is the edge projecting from it1 with its edge perpendicular to the bar4 ,c ,d is a steel plate bearing a knife edge on its upper surface2 this piece of steel is firmly secured to the framework4 ,there is of course a similar piece on the other side1 supporting the other edge4 ,the bar1 thus delicately poised1 will1 when once started1 vibrate backwards and forwards for an hour1 as there is very little friction between the edges and the pieces which support them4 ,illustration3 ,,fig4 #hh4 #fbf4 ,the general appearance of the apparatus1 when mounted1 is shown in ,fig4 #hi4 ,a ,b is the bar3 at ,a the two edges are shown1 and also the pieces of steel which support them4 ,the whole is carried by a horizontal beam bolted to two uprights2 and a glance at the figure will explain the arrangements made to secure the steadiness of the apparatus2 the second pair of edges shown at ,b will be referred to presently 7,art4 #fce74 #fbg4 ,this bar1 as you see1 vibrates to and fro2 and we shall determine the length of a simple pendulum which would vibrate in the same period of time4 ,the length might be deduced by finding the time of vibration1 and then calculating from ,art4 #fjf4 ,this would be the most accurate mode of proceeding1 but ,i have preferred to adopt a direct method which does not require calculation4 ,a simple pendulum1 consisting of a fine cord and a small iron sphere ,c1 is mounted behind the edge1 ,fig4 #hi4 ,the point from which the cord is suspended lies exactly in the line of the two edges1 and there is an adjustment for lengthening or shortening the cord at pleasure4 ,illustration3 ,,fig4 #hi4 #fbh4 ,we first try with #f' of cord1 so that the simple pendulum shall have the same length as the bar4 ,taking the ball in one hand and the bar in the other1 ,i draw them aside1 and you see1 when they are released1 that the bar performs two vibrations and returns to my hand before the ball4 ,hence the length of the isochronous simple pendulum is certainly less than the length of the bar2 for a pendulum of that length is too slow4 #fbi4 ,i now shorten the cord until it is only half the length of the bar2 and1 repeating the experiment1 we find that the ball returns before the bar1 and therefore the simple pendulum is too short4 ,hence we learn that the isochronous pendulum is greater than half the length of the bar1 and less than the whole length4 #fcj4 ,let us finally try a simple pendulum two-thirds of the length of the bar4 ,i make the experiment1 and find that the ball and the bar return to my hand precisely at the same instant4 ,therefore two-thirds of the length of the bar is the length of the isochronous simple pendulum4 ,we may state generally that the time of vibration of a uniform bar about one end equals that of a simple pendulum whose length is two-thirds of the bar2 no doubt the bar we have used is not strictly uniform1 because of the edges2 but in the positions they occupy1 their influence on the time of vibrations is imperceptible4 #fcb4 ,for this rule to be verified1 it is essentially necessary that the edges be properly situated on the bar2 to illustrate this we may examine the oscillations of the small rod1 shown at ,d 7,fig4 #hi74 ,this rod is also of iron #bd #je #je1 and it is suspended from a point near the centre by a pair of edges2 if the edges could be placed so that the centre of gravity of the whole lay in the line of the edges1 it is evident that the bar would rest indifferently however it were placed1 and would not oscillate4 ,if then the edges be very near the centre of gravity1 we can easily understand that the oscillations may be very slow1 and this is actually the case in the bar ,d4 ,by the aid of the stop-watch1 ,i find that one hundred vibrations are performed in #bdh seconds1 and that therefore each vibration occupies #bdh seconds4 ,the length of the simple pendulum which has #bdh seconds for its period of oscillation1 is about #bj'4 ,had the edges been at one end1 the length of the simple pendulum would have been #bd #af4 ,a bar #gb long will vibrate in a shorter time when the edge is #aeb from one end than when it has any other position4 ,the length of the corresponding simple pendulum is #daf4 ,,the ,,centre ,,of ,,oscillation4 #fcc4 ,it appears that corresponding to each compound pendulum we have a specific length equal to that of the isochronous simple pendulum4 ,to take as example the #f' bar already described 7,art4 #fbe71 this length is #d'4 ,if ,i measure off from the edges a distance of #d'1 and mark this point upon the bar1 the point is called the centre of oscillation4 ,more generally the centre of oscillation is found by drawing a line equal to the isochronous simple pendulum from the centre of oscillation through the centre of gravity4 #fcd4 ,in the bar ,d the centre of oscillation would be at a distance of #bj' below the edges2 and in general the position will vary with the position of the edges4 #fce4 ,in the #f' bar ,b is the centre of oscillation4 ,i take another pair of edges and place them on the bar1 so that the line of the edges passes through ,b4 ,i now lift the bar carefully and turn it upside down1 so that the edges ,b rest upon the steel plates4 ,in this position one-third of the bar is above the axis of suspension1 and the remaining two-thirds below4 ,a is of course now at the bottom of the bar1 and is on a level with the ball1 ,c3 the pendulum is made to oscillate about the edges ,b1 and the time of its vibration may be approximately determined by direct comparison with ,c1 as already explained4 ,i find that1 when ,i allow ,c and the bar to swing together1 they both vibrate precisely in the same time4 ,you will remember1 that when the ball was suspended by a string #d' long1 its vibrations were isochronous with those of the bar when suspended from the edges ,a4 ,without having altered ,c1 but having made the bar to vibrate about ,b1 ,i find that the time of oscillation of the bar is still equal to that of ,c4 ,therefore1 the period of oscillation about ,a is equal to that about ,b4 ,hence1 when the bar is vibrating about ,b1 its centre of oscillation must be #d' from ,b1 that is1 it must be at ,a3 so that when the bar is suspended from ,a1 the centre of oscillation is ,b2 while1 when the bar is suspended from ,b1 the centre of oscillation is ,a4 ,this is an interesting dynamical theorem4 ,it may be more concisely expressed by saying that the centre of oscillation and the centre of suspension are reciprocal4 #fcf4 ,though the proof that we have given of this curious law applies only to a uniform bar1 yet the law is itself true in general1 whatever be the nature of the compound pendulum4 #fcg4 ,we alluded in the last lecture 7,art4 #faj7 to the difficulty of measuring with accuracy the length of a simple pendulum2 but the reciprocity of the centres of oscillation and suspension1 suggested to the ingenious ,captain ,kater a method by which this difficulty could be evaded4 ,we shall explain the principle4 ,let one pair of edges be at ,a4 ,let the other pair of edges1 ,b1 be moved as near as possible to the centre of oscillation4 ,we can test whether ,b has been placed correctly3 for the time taken by the pendulum to perform #ajj vibrations about ,a should be equal to the time taken to perform #ajj vibrations about ,b4 ,if the times are not quite equal1 ,b must be moved slightly until the times are properly brought to equality4 ,the length of the isochronous simple pendulum is then equal to the distance between the edges ,a and ,b2 and this distance1 from one edge to the other edge1 presents none of the difficulties in its exact measurement which we had before to contend with3 it can be found with precision4 ,hence1 knowing the length of the pendulum and its time of oscillation1 gravity can be found in the manner already explained 7,art4 #fjh74 #fch4 ,i have adjusted the two edges of the #f' bar as nearly as ,i could at the centres of oscillation and suspension1 and we shall proceed to test the correctness of the positions4 ,mounting the bar first by the edges at ,a1 ,i set it vibrating4 ,i take the stop-watch already referred to 7,art4 #fab71 and record the position of its hands4 ,i then place my finger on the stud1 and1 just at the moment when the bar is at the middle of one of its vibrations1 ,i start the watch4 ,i count a hundred vibrations2 and when the pendulum is again at the middle of its stroke1 ,i stop the watch1 and find it records an interval of #aajd seconds4 ,thus the time of one vibration is #aajd seconds4 ,reversing the bar1 so that it vibrates about its centre of oscillation ,b1 ,i now find that #aajj is the time occupied by one hundred vibrations counted in the same manner as before2 hence #aajj seconds is the time of one vibration about ,b3 thus1 the periods of the vibrations are very nearly equal1 as they differ only by _/th part of a second4 #fci4 ,it would be difficult to render the times of oscillation exactly equal by merely altering the position of ,b4 ,in ,kater's pendulum the two knife-edges are first placed so that the periods are as nearly equal as possible4 ,the final adjustments are given by moving a small sliding-piece on the bar until it is found that the times of vibration about the two edges are identical4 ,we shall not1 however1 use this refinement in a lecture experiment2 ,i shall adopt the mean value of #aajb seconds4 ,the distance of the knife-edges is about #c'iib2 hence gravity may be found from the expression 7,art4 #fjh7 #c'iib 7#cadaf _/ #a4ajb74 ,the value thus deduced is #cb'd1 which is within a small fraction of the true value4 #fdj4 ,with suitable precautions ,kater's pendulum can be made to give a very accurate result4 ,it is to be adjusted so that there shall be no perceptible difference in the number of vibrations in twenty-four hours1 whichever edge be the axis of suspension3 the distance between the edges is then to be measured with the last degree of precision by comparison with a proper standard4 ,,the ,,centre ,,of ,,percussion4 ,illustration3 ,,fig4 #ij4 #fda4 ,the centre of oscillation in a body free to rotate about a fixed axis is identical with another remarkable point1 called the centre of percussion4 ,we proceed to examine some of the properties of a body thus suspended with reference to the effects of a blow4 ,for the purposes of these experiments the method of suspension by edges is however quite unsuited4 #fdb4 ,we shall first use a rod suspended from a pin about which the rod can rotate4 ,a ,b1 ,fig4 #ij1 is a pine rod #dh #a #a1 free to turn round ,b4 ,suppose this rod be hanging vertically at rest4 ,i take a stick in my hand1 and1 giving the rod a blow1 an impulsive shock will instantly be communicated to the pin at ,b2 but the actual effect upon ,b will be very different according to the position at which the blow is given4 ,if ,i strike the upper part of the rod at ,d1 the action of ,a ,b upon the pin is a pressure to the left4 ,if ,i strike the lower part at ,a1 the pressure is to the right4 ,but if ,i strike the point ,c1 which is distant from ,b by two-thirds of the length of the rod1 there is no pressure upon the pin4 ,concisely1 for a blow below ,c1 the pressure is to the right2 for one above ,c1 it is to the left2 for one at ,c it is nothing4 #fdc4 ,we can easily verify this by holding one extremity of a rod between the finger and thumb of the left hand1 and striking it in different places with a stick held in the right hand2 the pressure of the rod1 when struck1 will be so felt that the circumstances already stated can be verified4 ,illustration3 ,,fig4 #ia4 #fdd4 ,a more visible way of investigating the subject is shown in ,fig4 #ia4 ,f ,b is a rod of wood1 suspended from a beam by the string ,f ,g4 ,a piece of paper is fastened to the rod at ,f by means of a small slip of wood clamped firmly to the rod2 the other ends of this piece of paper are similarly clamped at ,p and ,q4 #fde4 ,when the rod receives a blow on the right-hand side at ,a1 we find that the piece of paper is broken across at ,e1 because the end ,f has been driven by the blow towards ,q1 and consequently caused the fracture of the paper at a place1 ,e1 where it had been specially narrowed4 ,i remove the pieces of paper1 and replace them by a new piece precisely similar4 ,i now strike the rod at ,b1--a smart tap is all that is necessary1--and the piece of paper breaks at ,d4 ,finally replacing the pieces of paper by a third piece1 ,i find that when ,i give the rod a tap 7not a violent blow7 at ,c1 neither ,d nor ,e are broken4 #fdf4 ,this point ,c1 where the rod can receive a blow without producing a shock upon the axis of suspension is the centre of percussion4 ,we see1 from its being two-thirds of the length of the rod distant from ,f1 that it is identical with the centre of oscillation of the rod1 if vibrating about knife-edges at ,f4 ,it is true in general1 whatever be the shape of the body1 that the centre of oscillation is identical with the centre of percussion4 #fdg4 ,the principle embodied in the property of the centre of percussion has many practical applications4 ,every cricketer well knows that there is a part of his bat from which the ball flies without giving his hands any unpleasant feeling4 ,the explanation is simple4 ,the bat is a body suspended from the hands of the batman2 and if the ball be struck with the centre of percussion of the bat1 there is no shock experienced4 ,the centre of percussion in a hammer lies in its head1 consequently a nail can receive a violent blow with perfect comfort to the hand which holds the handle4 ,,the ,,conical ,,pendulum4 #fdh4 ,i have here a tripod 7,fig4 #ib7 which supports a heavy ball of cast iron by a string #f' long4 ,if ,i withdraw the ball from its position of rest1 and merely release it1 the ball vibrates to and fro1 the string continues in the same plane1 and the motion is that already discussed in the circular pendulum4 ,if at the same instant that ,i release the ball1 ,i impart to it a slight push in a direction not passing through the position of rest1 the ball describes a curved path1 returning to the point from which it started4 ,this motion is that of the conical pendulum1 because the string supporting the ball describes a cone4 #fdi4 ,in order to examine the nature of the motion1 we can make the ball depict its own path4 ,at the opposite point of the ball to that from which it is suspended1 a hole is drilled1 and in this ,i have fitted a camel's hair paint-brush filled with ink4 ,i bring a sheet of paper on a drawing-board under the vibrating ball2 and you see the brush traces an ellipse upon the paper1 which ,i quickly withdraw4 #fej4 ,by starting the ball in different ways1 ,i can make it describe very different ellipses3 here is one that is extremely long and narrow1 and here another almost circular4 ,when the magnitude of the initial velocity is properly adjusted1 and its direction is perpendicular to the radius1 ,i can make the string describe a right cone1 and the ball a horizontal circle1 but it requires some care and several trials in order to succeed in this4 ,the ellipse may also become very narrow1 so that we pass by insensible gradations to the circular pendulum1 in which the brush traces a straight line4 ,illustration3 ,,fig4 #ib4 #fea4 ,when the ball is moving in a circle1 its velocity is uniform2 when moving in an ellipse1 its velocity is greatest at the extremities of the least axes of this ellipse1 and least at the extremities of the greatest axes2 but1 when the ball is vibrating to and fro1 as in the ordinary circular pendulum1 the velocity is greatest at the middle of each vibration1 and vanishes of course each time the pendulum attains the extremity of its swing4 ,it is worthy of notice that under all circumstances the brush traces an ellipse upon the paper2 for the circle and the straight line are only extreme cases1 the one being a very round ellipse and the other a very thin one4 ,if1 however1 the arc of vibration be large the movement is by no means so simple4 #feb4 ,how are we to account for the elliptic movement8 ,to do so fully would require more calculation than can be admitted here1 but we may give a general account of the phenomenon4 ,let us suppose that the ellipse ,a ,c ,b ,d1 ,fig4 #ic1 is the path described by a particle when suspended by a string from a point vertically above ,o1 the centre of the ellipse4 ,to produce this motion ,i withdraw the particle from its position of rest at ,o to ,a4 ,if merely released1 the particle would swing over to ,b1 and back again to ,a2 but ,i do not simply release it1 ,i impart a velocity impelling it in the direction ,a ,t4 ,through ,o draw ,c ,d parallel to ,a ,t4 ,if ,i had taken the particle at ,o1 and1 without withdrawing it from its position of rest1 had started it off in the direction ,o ,d1 the particle would continue for ever to vibrate backwards and forwards from ,c to ,d4 ,hence1 when ,i release the particle at ,a1 and give it a velocity in the direction ,a ,t1 the particle commences to move under the action of two distinct vibrations1 one parallel to ,a ,b1 the other parallel to ,c ,d1 and we have to find the effect of these two vibrations impressed simultaneously upon the same particle4 ,they are performed in the same time1 since all vibrations are isochronous4 ,we must conceive one motion starting from ,a ,a towards ,o at the same moment that the other commences to start from ,o towards ,d4 ,after the lapse of a short time1 the body has moved through ,a ,y in its oscillation towards ,o1 and in the same time through ,o ,z in its oscillation towards ,d2 it is therefore found at ,x4 ,now1 when the particle has moved through a distance equal and parallel to ,a ,o1 it must be found at the point ,d1 because the motion from ,o to ,d takes the same time as from ,a to ,o4 ,similarly the body must pass through ,b1 because the time occupied by going from ,a to ,b1 would have been sufficient for the journey from ,o to ,d1 and back again4 ,the particle is found at ,p1 because1 after the vibration returning from ,b has arrived at ,q1 the movement from ,d to ,o has travelled on to ,r4 ,in this way the particle may be traced completely round its path by the composition of the two motions4 ,it can be proved that for small motions the path is an ellipse1 by reasoning founded upon the fact that the vibrations are isochronous4 ,illustration3 ,,fig4 #ic4 #fec4 ,close examination reveals a very interesting circumstance connected with this experiment4 ,it may be observed that the ellipse described by the body is not quite fixed in position1 but that it gradually moves round in its plane4 ,thus1 in ,fig4 #ib1 the ellipse which is being traced out by the brush will gradually change its position to the dotted line shown on the board4 ,the axis of the ellipse revolves in the same direction as that in which the ball is moving4 ,this phenomenon is more marked with an ellipse whose dimensions are considerable in proportion to the length of the string4 ,in fact1 if the ellipse be very small1 the change of position is imperceptible4 ,the cause of this change is to be found in the fact already mentioned 7,art4 #eih71 that though the vibrations of a pendulum are very nearly isochronous1 yet they are not absolutely so2 the vibrations through a long arc taking a minute portion of time longer than those through a short arc4 ,this difference only becomes appreciable when the larger arc is of considerable magnitude with reference to the length of the pendulum4 ,illustration3 ,,fig4 #id4 #fed4 ,how this causes displacement of the ellipse may be explained by ,fig4 #id4 ,the particle is describing the figure ,a ,d ,c ,b in the direction shown by the arrows4 ,this motion may be conceived to be compounded of vibrations ,a ,c and ,b ,d1 if we imagine the particle to have been started from ,a with the right velocity perpendicular to ,o ,a4 ,at the point ,a1 the entire motion is for the instant perpendicular to ,o ,a2 in fact1 the motion is then exclusively due to the vibration ,b ,d1 and there is no movement parallel to ,o ,a4 ,we may define the extremity of the major axis of the ellipse to be the position of the particle1 when the motion parallel to that axis vanishes4 ,of course this applies equally to the other extremity of the axis ,c1 and similarly at the points ,b or ,d there is no motion of the particle parallel to ,b ,d4 #fee4 ,let us follow the particle1 starting from ,a until it returns there again4 ,the movement is compounded of two vibrations1 one from ,a to ,c and back again1 the other along ,b ,d2 from ,o to ,d1 then from ,d to ,b1 then from ,b to ,o1 taking exactly double the time of one vibration from ,d to ,b4 ,if the time of vibration along ,a ,c were exactly equal to that along ,b ,d1 these two vibrations would bring the particle back to ,a precisely under the original circumstances4 ,but they do not take place in the same time2 the motion along ,a ,c takes a shade longer1 so that1 when the motion parallel to ,a ,c has ceased1 the motion along ,d ,b has gone past ,o to a point ,q1 very near ,o4 ,let ,a ,p ,o ,q1 and when the motion parallel to ,a ,c has vanished1 the particle will be found at ,p2 hence ,p must be the extremity of the major axis of the ellipse4 ,in the next revolution1 the extremity of the axis will advance a little more1 and thus the ellipse moves round gradually4 ,,the ,,composition ,,of ,,vibrations4 #fef4 ,we have learned to regard the elliptic motion in the conical pendulum as compounded of two vibrations4 ,the importance of the composition of vibrations justifies us in examining this subject experimentally in another way4 ,the apparatus which we shall employ is represented in ,fig4 #ie4 ,a is a ball of cast iron weighing #be lbs41 suspended from the tripod by a cord3 this ball itself forms the support of another pendulum1 ,b4 ,the second pendulum is very light1 being merely a globe of glass filled with sand4 ,through a hole at the bottom of the glass the sand runs out upon a drawing-board placed underneath to receive it4 ,illustration3 ,,fig4 #ie4 ,thus the little stream of sand depicts its own journey upon the drawing-board1 and the curves traced out thus indicate the path in which the bob of the second pendulum has moved4 ,illustration3 ,,fig4 #if4 #feg4 ,if the lengths of the two pendulums be equal1 and their vibrations be in different planes1 the curve described is an ellipse1 passing at one extreme into a circle1 and at the other into a straight line4 ,this is what we might have expected1 for the two vibrations are each performed in the same time1 and therefore the case is analogous to that of the conical pendulum of ,art4 #fdh4 #feh4 ,but the curve is of a very different character when the cords are unequal4 ,let us study in particular the case in which the second pendulum is only one-fourth the length of the cord supporting the iron ball4 ,this is the experiment represented in ,fig4 #ie4 ,the form of the path delineated by the sand is shown in ,fig4 #if4 ,the arrowheads placed upon the curve show the direction in which it is traced4 ,let us suppose that the formation of the figure commences at ,a2 it then goes on to ,b1 to ,o1 to ,c1 to ,d1 and back to ,a3 this shows us that the bob of the lower pendulum must have performed two vibrations up and down1 while that of the upper has made one right and left4 ,the motion is thus compounded of two vibrations at right angles1 and the time of one is half that of the other4 ,the time of vibration is proportional to the square root of the length2 and1 since the lower pendulum is one-fourth the length of the upper1 its time of vibration is one-half that of the upper4 ,in this experiment1 therefore1 we have a confirmation of the law of ,art #fje4 ,,lecture ,,xx4 ,,the ,,mechanical ,,principles ,,of ,a ,,clock4 ,introduction4--,the ,compensating ,pendulum4--,the ,escapement4--,the ,train of ,wheels4--,the ,hands4--,the ,striking ,parts4 ,,introduction4 #fei4 ,we come now to the most important practical application of the pendulum4 ,the vibrations being always isochronous1 it follows that1 if we count the number of vibrations in a certain time1 we shall ascertain the duration of that time4 ,it is simply the product of the number of vibrations with the period of a single one4 ,let us take a pendulum #ciaci inches long2 which will vibrate exactly once a second in ,london1 and is therefore called a seconds pendulum 7,see ,art4 #fjg74 ,if ,i set one of these pendulums vibrating1 and contrive mechanism by which the number of its vibrations shall be recorded1 ,i have a means of measuring time4 ,this is of course the principle of the common clock3 the pendulum vibrates once a second and the number of vibrations made from one epoch to another epoch is shown by the hands of the clock4 ,for example1 when the clock tells me that #ae minutes have elapsed1 what it really shows is that the pendulum has made #fj #ae #ijj vibrations1 each of which has occupied one second4 #ffj4 ,one duty of the clock is therefore to count and record the number of vibrations1 but the wheels and works have another part to discharge1 and that is to sustain the motion of the pendulum4 ,the friction of the air and the resistance experienced at the point of suspension are forces tending to bring the pendulum to rest2 and to counteract the effect of these forces1 the machine must be continually invigorated with fresh energy4 ,this supply is communicated by the works of the clock1 which will be sufficiently described presently4 #ffa4 ,when the weight driving the clock is wound up1 a store of energy is communicated which is doled out to the pendulum in a very small impulse1 at every vibration4 ,the clock-weight is just large enough to be able to counterbalance the retarding forces when the pendulum has a proper amplitude of vibration4 ,in all machines there is some energy lost in maintaining the parts in motion in opposition to friction and other resistances2 in clocks this represents the whole amount of the force1 as there is no external work to be performed4 ,,the ,,compensating ,,pendulum4 #ffb4 ,the actual length of the pendulum used1 depends upon the purposes for which the clock is intended1 but it is essential for correct performance that the pendulum should vibrate at a constant rate2 a small irregularity in this respect may appreciably affect the indications of the clock4 ,if the pendulum vibrates in #ajja seconds instead of in one second1 the clock loses one thousandth of a second at each beat2 and1 since there are #hf1djj seconds in a day1 it follows that the pendulum will make only #hf1djj - #hfc vibrations in a day1 and therefore the clock will lose #hfc seconds1 or nearly a minute and a half daily4 #ffc4 ,for accurate time-keeping it is therefore essential that the time of vibration shall remain constant4 ,now the time of vibration depends upon the length1 and therefore it is necessary that the length of the pendulum be absolutely unalterable4 ,if the length of the pendulum be changed even by one-tenth of an inch1 the clock will lose or gain nearly two minutes daily1 according to whether the pendulum has been made longer or shorter4 ,in general we may say that1 if the alteration in the length amount to k thousandths of an inch1 the number of seconds gained or lost per day is #aajc k with a seconds pendulum4 #ffd4 ,this explains the practice of raising the bob of the pendulum when the clock is going too slow or lowering it when going too fast4 ,if the thread of the screw used in doing this have twenty threads to the inch2 then one complete revolution of the screw will raise the bob through #ej thousandths of an inch1 and therefore the effect on the rate will be #aajc #ej #ee nearly4 ,thus1 the rate of the clock will be altered by about #ee seconds daily4 ,whatever be the screw1 its effect can be calculated by the simple rule expressed as follows4 ,divide #aajc by the number of threads to the inch2 the quotient is the number of seconds that the clock can be made to gain or lose daily by one revolution of the screw on the bob of the pendulum4 #ffe4 ,let us suppose that the length of the pendulum has been properly adjusted so that the clock keeps accurate time4 ,it is necessary that the pendulum should not alter in length4 ,but there is an ever-present cause tending to change it4 ,that cause is the variation of temperature4 ,we shall first illustrate by actual experiment the well known law that bodies expand under the action of heat2 then we shall consider the irregularities thus introduced into the motion of the pendulum2 and1 finally1 we shall point out means by which these irregularities may be effectually counteracted4 ,illustration3 ,,fig4 #ig4 #fff4 ,we have here a brass bar a yard long2 it is at present at the temperature of the room4 ,if we heat the bar over a lamp1 it becomes longer2 but upon cooling1 it returns to its original dimensions4 ,these alterations of length are very small1 indeed too small to be perceived except by careful measurement2 but we shall be able to demonstrate in a simple way that elongation is the consequence of increased temperature4 ,i place the bar ,a ,d in the supports shown in ,fig4 #ig4 ,it is firmly secured at ,b by means of a binding screw1 and passes quite freely through ,c2 if the bar elongate when it is heated by the lamp1 the point ,d must approach nearer to ,e4 ,at ,h is an electric battery1 and ,g is a bell rung by an electric current4 ,one wire of the battery connects ,h and ,g1 another connects ,g with ,e1 and a third connects ,h with the end of the brass rod ,b4 ,until the electric current becomes completed1 the bell remains dumb1 the current is not closed until the point touches ,e3 when this is the case1 the current rushes from the battery along the bar1 then from ,d to ,e1 from that through the bell1 and so back to the battery4 ,at present the point is not touching ,e1 though extremely close thereto4 ,indeed if ,i press ,e towards the point1 you hear the bell1 showing that the circuit is complete2 removing my finger1 the bell again becomes silent1 because ,e springs back1 and the current is interrupted4 #ffg4 ,i place the lamp under the bar3 which begins to heat and to elongate2 and as it is firmly held at ,b1 the point gradually approaches ,e3 it has now touched ,e2 the circuit is complete1 and the bell rings4 ,if ,i withdraw the lamp1 the bar cools4 ,i can accelerate the cooling by touching the bar with a damp sponge2 the bar contracts1 breaks the circuit1 and the bell stops3 heating the bar again with the lamp1 the bell again rings1 to be again stopped by an application of the sponge4 ,though you have not been able to see the process1 your ears have informed you that heat must have elongated the bar1 and that cold has produced contraction4 #ffh4 ,what we have proved with respect to a bar of brass1 is true for a bar of any material2 and thus1 whatever be the substance of which a pendulum is made1 a simple uncompensated rod must be longer in hot weather than in cold weather3 hence a clock will generally have a tendency to go faster in winter than in summer4 #ffi4 ,the amount of change thus produced is1 it is true1 very small4 ,for a pendulum with a steel rod1 the difference of temperature between summer and winter would cause a variation in the rate of five seconds daily1 or about half a minute in the week4 ,the amount of error thus introduced is of no great consequence in clocks which are only intended for ordinary use2 but in astronomical clocks1 where seconds or even portions of a second are of importance1 inaccuracies of this magnitude would be quite inadmissible4 ,illustration3 ,,fig4 #ih4 #fgj4 ,there are1 it is true1 some substances--for example1 ordinary timber--in which the rate of expansion is less than that of steel2 consequently1 the irregularities introduced by employing a pendulum with a wooden rod are less than those of the steel pendulum we have mentioned2 but no substance is known which would not originate greater variations than are admissible in the performance of an astronomical clock4 ,we must1 therefore1 devise some means by which the effect of temperature on the length of a pendulum can be avoided4 ,various means have been proposed1 and we shall describe one of the best and simplest4 #fga4 ,the mercurial pendulum 7,fig4 #ih7 is frequently used in clocks intended to serve as standard time-keepers4 ,the rod by which the pendulum is suspended is made of steel2 and the bob consists of a glass jar of mercury4 ,the distance of the centre of gravity of the mercury from the point of suspension may practically be considered as the length of the pendulum4 ,the rate of expansion of mercury is about sixteen times that of steel3 hence1 if the bob be formed of a column of mercury one-eighth part of the length of the steel rod1 the compensation would be complete4 ,for1 suppose the temperature of the pendulum be raised1 the steel rod would be lengthened1 and therefore the vase of mercury would be lowered2 on the other hand1 the column of mercury would expand by an amount double that of the steel rod3 thus the centre of the column of mercury would be elevated as much as the steel was elongated2 hence the centre of the mercury is raised by its own expansion as much as it is lowered by the expansion of the steel1 and therefore the effective length of the pendulum remains unaltered4 ,by this contrivance the time of oscillation of the pendulum is rendered independent of the temperature4 ,the bob of the mercurial pendulum is shown in ,fig4 #ih4 ,the screw is for the purpose of raising or lowering the entire vessel of mercury in order to make the rate correct in the first instance4 ,,the ,,escapement4 #fgb4 ,practical skill as well as some theoretical investigation has been expended upon that part of a clock which is called the escapement1 the excellence of which is essential to the correct performance of a timepiece4 ,the pendulum must have its motion sustained by receiving an impulse at every vibration3 at the same time it is desirable that the vibration should be hampered as little as possible by mechanical connection4 ,the isochronism on which the time-keeping depends is in strictness only a characteristic of oscillations performed with a total freedom from constraint of every description2 hence we must endeavour to approximate the clock pendulum as nearly as possible to one which is swinging quite freely4 ,to effect this1 and at the same time to maintain the arc of vibration tolerably constant1 is the property of a good escapement4 ,illustration3 ,,fig4 #ii4 #fgc4 ,a common form of escapement is shown in ,fig4 #ii4 ,the arrangement is no doubt different from that actually found in a clock2 but ,i have constructed the machine in this way in order to show clearly the action of the different parts4 ,g is called the escapement-wheel3 it is surrounded by thirty teeth1 and turns round once when the pendulum has performed sixty vibrations1--that is1 once a minute4 ,i represents the escapement2 it vibrates about an axis and carries a fork at ,k which projects behind1 and the rod of the pendulum hangs between its prongs4 ,the pendulum is itself suspended from a point ,o4 ,at ,n1 ,h are a pair of polished surfaces called the pallets3 these fulfil a very important function4 #fgd4 ,the escapement-wheel is constantly urged to turn round by the action of the weight and train of wheels1 of which we shall speak presently2 but the action of the pallets regulates the rate at which the wheel can revolve4 ,when a tooth of the wheel falls upon the pallet ,n1 the latter is gently pressed away3 this pressure is transmitted by the fork to the pendulum2 as ,n moves away from the wheel1 the other pallet ,h approaches the wheel2 and by the time ,n has receded so far that the tooth slips from it1 ,h has advanced sufficiently far to catch the tooth which immediately drops upon ,h4 ,in fact1 the moment the tooth is free from ,n1 the wheel begins to revolve in consequence of the driving weight2 but it is quickly stopped by another tooth falling on ,h3 and the noise of this collision is the well known tick of the clock4 ,the pendulum is still swinging to the left when the tooth falls on ,h4 ,the pressure of the tooth then tends to push ,h outwards1 but the inertia of the pendulum in forcing ,h inwards is at first sufficient to overcome the outward pressure arising from the wheel2 the consequence is that1 after the tooth has dropped1 the escapement-wheel moves back a little1 or 8recoils10 as it is called4 ,if you look at any ordinary clock1 which has a second-hand1 you will notice that after each second is completed the hand recoils before starting for the next second4 ,the reason of this is1 that the second-hand is turned directly by the escapement-wheel1 and that the inertia of the pendulum causes the escapement-wheel to recoil4 ,but the constant pressure of the tooth soon overcomes the inertia of the pendulum1 and ,h is gradually pushed out until the tooth is able to 8escape02 the moment it does so the wheel begins to turn round1 but is quickly brought up by another tooth falling on ,n1 which has moved sufficiently inwards4 ,the process we have just described then recurs over again4 ,each tooth escapes at each pallet1 and the escapements take place once a second2 hence the escapement-wheel with thirty teeth will turn round once in a minute4 #fge4 ,when the tooth is pushing ,n1 the pendulum is being urged to the left2 the instant this tooth escapes1 another tooth falls on ,h1 and the pendulum1 ere it has accomplished its swing to the left1 has a force exerted upon it to bring it to the right4 ,when this force and gravity combined have stopped the pendulum1 and caused it to move to the right1 the tooth soon escapes at ,h1 and another tooth falls on ,n1 then retarding the pendulum4 ,hence1 except during the very minute portion of time that the wheel turns after one escapement1 and before the next tick1 the pendulum is never free2 it is urged forwards when its velocity is great1 but before it comes to the end of its vibration it is urged backwards2 this escapement does not therefore possess the characteristics which we pointed out 7,art4 #fgb7 as necessary for a really good instrument4 ,but for ordinary purposes of time-keeping1 the recoil escapement works sufficiently well1 as the force which acts upon the pendulum is in reality extremely small4 ,for the refined applications of the astronomical clock1 the performance of a recoil escapement is inadequate4 ,the obvious defect in the recoil is that the pendulum is retarded during a portion of its vibration2 the impulse forward is of course necessary1 but the retarding force is useless and injurious4 #fgf4 ,the 8dead-beat0 escapement was devised by the celebrated clockmaker ,graham1 in order to avoid this difficulty4 ,if you observe the second-hand of a clock1 controlled by this escapement1 you will understand why it is called the dead beat3 there is no recoil2 the second-hand moves quickly over each second1 and remains there fixed until it starts for the next second4 ,the wheel and escapement by which this effect is produced is shown in ,fig4 #ajj4 ,a and ,b are the pallets1 by the action of the teeth on which the motion is given to the crutch1 which turns about the centre ,o2 from the axis through this centre the fork descends1 so that as the crutch is made to vibrate to and fro by the wheel1 the fork is also made to vibrate1 and thus sustain the motion of the pendulum4 ,the essential feature in which the dead-beat escapement differs from the recoil escapement is that when the tooth escapes from the pallet ,a1 the wheel turns3 but the tooth which in the recoil escapement would have fallen on the other pallet1 now falls on a surface ,d1 and not on the pallet ,b4 ,d is part of a circle with its centre at ,o1 the centre of motion2 consequently1 the tooth remains almost entirely inert so long as it remains on the circular arc ,d4 #fgg4 ,there is thus no recoil1 and the pendulum is allowed to reach the extremity of its swing to the right unretarded2 but when the pendulum is returning1 the crutch moves until the tooth passes from the circular arc ,d on to the pallet ,b3 instantly the tooth slides down the pallet1 giving the crutch an impulse1 and escaping when the point has traversed ,b4 ,the next tooth that comes into action falls upon the circular arc ,c1 of which the centre is also at ,o2 this tooth likewise remains at rest until the pendulum has finished its swing1 and has commenced its return2 then the tooth slides down ,a1 and the process recommences as before4 ,illustration3 ,,fig4 #ajj4 #fgh4 ,the operations are so timed that the pendulum receives its impulse 7which takes place when a tooth slides down a pallet7 precisely when the oscillation is at the point of greatest velocity2 the pendulum is then unacted upon till it reaches a similar position in the next vibration4 ,this impulse at the middle of the swing does not affect the time of vibration4 #fgi4 ,there is still a small frictional force acting to retard the pendulum4 ,this arises from the pressure of the teeth upon the circular arcs1 for there is a certain amount of friction1 no matter how carefully the surfaces may be polished4 ,it is not however found practically to be a source of appreciable irregularity4 ,in a clock furnished with a dead-beat escapement and a mercurial pendulum1 we have a superb time-keeper4 ,,the ,,train ,,of ,,wheels4 #fhj4 ,we have next to consider the manner in which the supply of energy is communicated to the escapement-wheel1 and also the mode in which the vibrations of the pendulum are counted4 ,a train of wheels for this purpose is shown in ,fig4 #ii4 ,the same remark may be made about this train that we have already made about the escapement1--namely1 that it is more designed to explain the principle clearly than to show the actual construction of a clock4 #fha4 ,the weight ,a which animates the whole machine is attached to a rope1 which is wound around a barrel ,b2 the process of winding up the clock consists in raising this weight4 ,on the same axle as the barrel ,b is a large tooth-wheel ,c2 this wheel contains #bjj teeth4 ,the wheel ,c works into a pinion ,d1 containing #bj teeth2 consequently1 when the wheel ,c has turned round once1 the pinion ,d has turned round ten times4 ,the large wheel ,e is on the same axle with the pinion ,d1 and turns with ,d2 the wheel ,e contains #ahj teeth1 and works into the pinion ,f1 containing #cj teeth3 consequently when ,e has gone round once1 ,f will have turned round six times2 and therefore1 when the wheel ,c and the barrel ,b have made one revolution1 the pinion ,f will have gone round sixty times2 but the wheel ,g is on the same shaft as the pinion ,f1 and therefore1 for every sixty revolutions of the escapement-wheel1 the wheel ,c will have gone round once4 ,we have already shown that the escapement-wheel goes round once a minute1 and hence the wheel ,c must go round once in an hour4 ,if therefore a hand be placed on the same axle with ,c1 in front of a clock dial1 the hand will go completely round once an hour2 that is1 it will be the minute-hand of the clock4 #fhb4 ,the train of wheels serves to transmit the power of the descending weight and thus supply energy to the pendulum4 ,in the clock model you see before you1 the weight sustaining the motion is #ef lbs4 ,the diameter of the escapement-wheel is about double that of the barrel1 and the wheel turns round sixty times as fast as the barrel2 therefore for every inch the weight descends1 the circumference of the escapement-wheel must move through #abj inches4 ,from the principle of work it follows that the energy applied at one end of a machine equals that obtained from the other1 friction being neglected4 ,the force of #ef lbs4 is therefore1 reduced to the one hundred-and-twentieth part of its amount at the circumference of the escapement-wheel4 ,and as the friction is considerable2 the actual force with which each tooth acts upon the pallet is only a few ounces4 #fhc4 ,in a good clock an extremely minute force need only be supplied to the pendulum1 so that1 notwithstanding #hf1djj vibrations have to be performed daily1 one winding of the clock will supply sufficient energy to sustain the motion for a week4 ,,the ,,hands4 #fhd4 ,we shall explain by the model shown in ,fig4 #aja1 how the hour-hand and the minute-hand are made to revolve with different velocities about the same dial4 ,illustration3 ,,fig4 #aja4 ,g is a handle by which ,i can turn round the shaft which carries the wheel ,f1 and the hand ,b4 ,there are #bj teeth in ,f1 and it gears into another wheel1 ,e1 containing #hj teeth2 the shaft which is turned by ,e carries a third wheel ,d1 containing #be teeth1 and ,d works with a fourth ,c1 containing #ge teeth1 ,c is capable of turning freely round the shaft1 so that the motion of the shaft does not affect it1 except through the intervention of the wheels ,e1 ,f1 and ,d4 ,to ,c another hand ,a is attached1 which therefore turns round simultaneously with ,c4 ,let us compare the motion of the two hands ,a and ,b4 ,we suppose that the handle ,g is turned twelve times2 then1 of course1 the hand ,b1 since it is on the shaft1 will turn twelve times4 ,the wheel ,f also turns twelve times1 but ,e has four times the number of teeth that ,a has1 and therefore1 when ,f has gone round four times1 ,e will only have gone round once3 hence1 when ,f has revolved twelve times1 ,e will have gone round three times4 ,d turns with ,e1 and therefore the twelve revolutions of the handle will have turned ,d round three times1 but since ,c has #ge teeth and ,d #be teeth1 ,c will have only made one revolution1 while ,d has made three revolutions2 hence the hand ,a will have made only one revolution1 while the hand ,b has made twelve revolutions4 ,we have already seen 7,art4 #fha7 how1 by a train of wheels1 one wheel can be made to revolve once in an hour4 ,if that wheel be upon the shaft instead of the handle ,g1 the hand ,b will be the minute-hand of the clock1 and the hand ,a the hour-hand4 #fhe4 ,the adjustment of the numbers of teeth is important1 and the choice of wheels which would answer is limited4 ,for since the shafts are parallel1 the distance between the centres of ,f and ,e must equal that between the centres of ,c and of ,d4 ,but it is evident that the distance from the centre of ,f to the centre of ,e is equal to the sum of the radii of the wheels ,f and ,e4 ,hence the sum of the radii of the wheels ,f and ,e must be equal to the sum of the radii of ,c and ,d4 ,but the circumferences of circles are proportional to their radii1 and hence the sum of the circumferences of ,f and ,e must equal that of ,c and ,d2 it follows that the sum of the teeth in ,e and ,f must be equal to the sum of the teeth in ,c and ,d4 ,in the present case each of these sums is one hundred4 #fhf4 ,other arrangements of wheels might have been devised1 which would give the required motion2 for example1 if ,f were #bj1 as before1 and ,e #bdj1 and if ,c and ,d were each equal to #acj1 the sum of the teeth in each pair would be #bfj4 ,e would only turn once for every twelve revolutions of ,f1 and ,c and ,d would turn with the same velocity as ,e2 hence the motion of the hand ,a would be one-twelfth that of ,b4 ,this plan requires larger wheels than the train already proposed4 ,,the ,,striking ,,parts4 #fhg4 ,we have examined the essential features of the going parts of the clock2 to complete our sketch of this instrument we shall describe the beautiful mechanism by which the striking is arranged4 ,the model which we represent in ,fig4 #ajb is1 as usual1 rather intended to illustrate the principles of the contrivance than to be an exact counter-part of the arrangement found in clocks4 ,some of the details are not reproduced in the model2 but enough is shown to explain the principle1 and to enable the model to work4 #fhh4 ,when the hour-hand reaches certain points on the dial1 the striking is to commence2 and a certain number of strokes must be delivered4 ,the striking apparatus has both to initiate the striking and to control the number of strokes2 the latter is by far the more difficult duty4 ,two contrivances are in common use2 we shall describe that which is used in the best clocks4 #fhi4 ,an essential feature of the striking mechanism in the repeating clock is the snail1 which is shown at ,b4 ,this piece must revolve once in twelve hours1 and is1 therefore1 attached to an axle which performs its revolution in exactly the same time as the hour-hand of the clock4 ,in the model1 the striking gear is shown detached from the going parts1 but it is easy to imagine how the snail can receive this motion4 ,the margin of the snail is marked with twelve steps1 numbered from one to twelve4 ,the portions of the margin between each pair of steps is a part of the circumference of a circle1 of which the axis of the snail is the centre4 ,the correct figuring of the snail is of the utmost importance to the correct performance of the clock4 ,above the snail is a portion of a toothed wheel1 ,f1 called the rack2 this contains about fourteen or fifteen teeth4 ,when this wheel is free1 it falls down until a pin comes in contact with the snail at ,b4 ,illustration3 ,,fig4 #ajb4 #fij4 ,the distance through which the rack falls depends upon the position of the snail2 if the pin come in contact with the part marked ,i41 as it does in the figure1 the rack will descend but a small distance1 while1 if the pin fall on the part marked ,,vii41 the rack will have a longer fall3 hence as the snail changes its position with the successive hours1 so the distance through which the rack falls changes also4 ,the snail is so contrived that at each hour the rack falls on a lower step than it does in the preceding hour2 for example1 during the hour of three o'clock1 the rack would1 if allowed to fall1 always drop upon the part of the snail marked ,,iii41 but1 when four o'clock has arrived1 the rack would fall on the part marked ,,iv42 it is to insure that this shall happen correctly that such attention must be paid to the form of the snail4 #fia4 ,a is a small piece called the 8gathering pallet02 it is so placed with reference to the rack that1 at each revolution of ,a1 the pallet raises the rack one tooth4 ,thus1 after the rack has fallen1 the gathering pallet gradually raises it4 #fib4 ,on the same axle as the gathering pallet1 and turning with it1 is another piece ,c1 the object of which is to arrest the motion when the rack has been raised1 sufficiently4 ,on the rack is a projecting pin2 the piece ,c passes free of this pin until the rack has been lifted to its original height1 when ,c is caught by the pin1 and the mechanism is stopped4 ,the magnitude of the teeth in the rack is so arranged with reference to the snail1 that the number of lifts which the pallet must make in raising the rack is equal to the number marked upon the step of the snail upon which the rack had fallen2 hence the snail has the effect of controlling the number of revolutions which the gathering pallet can make4 ,the rack is retained by a detent ,f1 after being raised each tooth4 #fic4 ,the gathering pallet is turned by a small pinion of #bg teeth1 and the pinion is worked by the wheel ,c1 of #ahj teeth4 ,this wheel carries a barrel1 to which a movement of rotation is given by a weight1 the arrangement of which is evident3 a second pinion of #bg teeth on the same axle with ,d is also turned by the large wheel ,c4 ,since these pinions are equal1 they revolve with equal velocities4 ,over ,d the bell ,i is placed2 its hammer ,e is so arranged that a pin attached to ,d strikes the bell once in every revolution of ,d4 ,the action will now be easily understood4 ,when the hour-hand reaches the hour1 a simple arrangement raises the detent ,f2 the rack then drops2 the moment the rack drops1 the gathering pallet commences to revolve and raises up the rack2 as each tooth is raised a stroke is given to the bell1 and thus the bell strikes until the piece ,c is brought to rest against the pin4 #fid4 ,the object of the fan ,h is to control the rapidity of the motion3 when its blades are placed more or less obliquely1 the velocity is lessened or increased4 ,,appendix ,i4 ,the formul in the tables on p4 #gc and after can be deduced by two methods1--one that of graphical construction1 the other that of least squares4 ,the first method is the more simple and requires but little calculation2 though neatness and care are necessary in constructing the diagrams4 ,the second method will be described for the benefit of those who possess the requisite mathematical knowledge4 ,the formul used in the preparation of the tables have been generally deduced from the method of least squares1 as the results are to a slight1 though insignificant1 extent more accurate than those of the method of graphical construction4 ,this remark will explain why the figures in some of the formul are carried to a greater number of places of decimals than could be obtained by the other method4 ,we shall confine the numerical examples to ,tables ,,iii4 and ,,iv41 and show how the formul of these tables have been deduced by the two different methods4 ,tables ,v41 ,,xiv41 ,,xvi41 ,,xxi41 are to be found in the same manner as ,table ,,iii42 and ,tables ,,vi41 ,,ix41 ,x41 ,,xi41 ,,xv41 ,,xvii41 ,,xviii41 ,,xix41 ,,xx41 ,,xxi41 ,,xxii41 in the same manner as ,table ,,iv4 ,,the ,,method ,,of ,,graphical ,,construction4 ,,table ,,iii4 ,a horizontal line ,,aps1 shown on a diminished scale in ,fig4 #ajc1 is to be neatly drawn upon a piece of cardboard about #ad #f4 ,a scale which reads to the hundredth of an inch is to be used in the construction of the figure4 ,a pocket lens will be found convenient in reading the small divisions4 ,by means of a pair of compasses and the scale1 points are to be marked upon the line ,,aps1 at distances #ad1 #bh1 #db1 #ef1 #gj1 #hd1 #ih1 #aab from the origin ,a4 ,these distances correspond to the magnitudes of the loads placed upon the slide on the scale of #ja to #a lb4 ,perpendiculars to ,,aps are to be erected at the points marked1 and distances ,f1 ,f1 ,f1 c4 set off upon these perpendiculars4 ,these distances are to be equal1 on the adopted scale1 to the frictions for the corresponding loads4 ,for example1 we see from ,table ,,iii41 ,experiment #c1 that when the load upon the slide is #db lbs41 the friction is #abb lbs42 hence the point ,f is found by measuring a distance #db from ,a1 and erecting a perpendicular #abb4 ,thus1 for each of the loads a point is determined4 ,the positions of these points should be indicated by making each of them the centre of a small circle #ja diameter4 ,these circles1 besides neatly defining the points1 will be useful in a subsequent part of the process4 ,illustration3 ,,fig4 #ajc4 ,it will be found that the points ,f1 ,f1 c4 are very nearly in a straight line4 ,we assume that1 if the apparatus and observations were perfect1 the points would lie exactly in a straight line4 ,the object of the construction is to determine the straight line1 which on the whole is most close to all the points4 ,if it be true that the friction is proportional to the pressure1 this line should pass through the origin ,a1 for then the perpendicular which represents the friction is proportional to the line cut off from ,a1 which represents the load4 ,it will be found that a line ,,at can be drawn through the origin ,a1 so that all the points are in the immediate vicinity of this line1 if not actually upon it4 ,a string of fine black silk about #ae long1 stretched by a bow of wire or whalebone1 is a convenient straight-edge for finding the required line4 ,the circles described about the points ,f1 ,f1 c4 will facilitate the placing of the silk line as nearly as possible through all the points4 ,it will not be found possible to draw a line through ,a1 which shall intersect all the circles2 the best line passes below but very near to the circles round ,f1 ,f1 ,f1 ,f1 touches the circle about ,f1 intersects the circles about ,f and ,f1 and passes above the circle round ,f4 ,the line should be so placed that its depth below the point which is most above it1 is equal to the height at which it passes above the point which is most below it4 ,from ,a measure ,,as1 a length of #aj1 and erect the perpendicular ,,st4 ,we find by measurement that ,,st is #bg4 ,if1 then1 we suppose that the friction for any load is really represented by the distance cut off by the line ,,at upon the perpendicular1 it follows that ,f 3 ,r 33 #bg 3 #aj4 or ,f #jbg ,r4 ,this is the formula from which ,table ,,iii4 has been constructed4 ,,table ,,iv4 ,by a careful application of the silk bow-string1 ,x ,y ,q can be drawn1 which1 itself in close proximity to ,a1 passes more nearly through ,f1 ,f1 c4 than is possible for any line which passes exactly through ,a4 ,x ,y ,q will be found not only to intersect all the small circles1 but to cut off a considerable arc from each4 ,measure off ,x ,p a distance of #aj1 and erect the perpendicular ,p ,q2 then1 if ,r be the load1 and ,f the corresponding friction1 we must have from similar triangles-- ,,ay ,f - -------- #a lb4 #ja ,,pq ---------------------------------- ------------4 ,r ,,px ,by measurement it is found that ,,ay #jad1 and ,,pq #bec4 ,we have1 therefore1 ,f #ad #jbec ,r4 ,this is practically the same formula as ,f #add #jbeb ,r1 from which the table has been constructed4 ,in fact1 the column of calculated values of the friction might have been computed from the former1 without appreciably differing from what is found in the table4 ,,the ,,method ,,of ,,least ,,squares4 ,,table ,,iii4 ,let k be the coefficient of friction4 ,it is impossible to find any value for k which will satisfy the equation1 ,f - k ,r #j1 for all the observed pairs of values of ,f and ,r4 ,we have then to find the value for k which1 upon the whole1 best represents the experiments4 ,f - k ,r is to be as near zero as possible for each pair of values of ,f and ,r4 ,in accordance with the principle of least squares1 it is well known to mathematicians1 the best value of k is that which makes 7,f - k ,r7 7,f - k ,r7 c4 7,f - k ,r7 a minimum where ,f and ,r1 ,f and ,r c4 are the simultaneous values of ,f and ,r in the several experiments4 ,in fact1 it is easy to see that1 if this quantity be small1 each of the essentially positive elements1 7,f - k ,r71 c4 of which it is composed1 must be small also1 and that therefore ,f - k ,r must always be nearly zero4 ,differentiating the sum of squares and equating the differential coefficient to zero1 we have according to the usual notation1 ,r 7,f - k ,r7 #j2 ,r ,f whence k --------------------------------4 ,r ,the calculation of k becomes simplified when 7as is generally the case in the tables7 the loads ,r1 ,r1 c41 ,r are of the form1 ,n1 #b,n1 #c,n c41 m ,n4 ,in this case1 ,r ,f ,n 7,f #b;,f #c;,f c4 m,f74 ,r ,n 7#a #b c4 m7 m7m #a7 7#bm #a7 ,n ---------------------------------------- ------ #f 7,f #b;,f c4 m,f7 k #f ---------------------------------------- ------------------ 4 ,nm7m #a7 7#bm #a7 ,in the case of ,table ,,iii4 m #h1 ,n #ad1 ,f #b;,f #c;,f m,f #ggji2 whence k #jbg4 ,thus the formula ,f #jbg ,r is deduced both by the method of least squares1 and by the method of graphical construction4 ,,table ,,iv4 ,the formula for this table is to be deduced from the following considerations4 ,no values exist for x and y1 so that the equation ,f x y ,r shall be satisfied for all pairs of values of ,f and ,r1 but the best values for x and y are those which make 7,f - x - y ,r7 7,f - x - y ,r7 c4 7,f - x - y ,r7 a minimum4 ,differentiating with respect to x and y1 and equating the differential coefficients to zero1 we have 7,f - x - y ,r7 #j1 ,r7,f - x - y ,r7 #j4 ,this gives two equations for the determination of x and y4 ,suppose1 as is usually the case1 the loads be of the form1 ,n1 #b,n1 #c,n1 #d,n c4 m,n1 and making ,a ,f ,f ,f c4 ,f ,b ,f #b;,f #c;,f c4 m,f1 we have the equations m7m #a7 ,a - mx - -------------------------- ,n y #j1 #b m7m #a7 m7m #a7 7#bm #a7 ,b - -------------------------- x - ---------------------------------------- ------ ,n y #j4 #b #f ,solving these1 we find #b #dm #f x ---------------------- ,a - ------------------ ,b1 m - m m - m #ab ,b #f ,a y ------------------ ---------- - ------------------- ---------- 4 m - m ,n m - m ,n ,in the present case1 m #h1 ,n #ad1 ,a #achd1 ,b #ggji2 whence x #add y #jbeb1 and we have the formula1 ,f #add #jbeb ,r4 ,,appendix ,,ii4 ,,details ,,of ,,the ,,willis ,,apparatus ,,used ,,in ,,illustrating ,,the ,,foregoing ,,lectures4 ,the ultimate parts of the various contrivances figured in this volume are mainly those invented by the late ,professor ,willis of ,cambridge4 ,they are minutely described and illustrated in a work written by him for the purpose under the title ,system of ,apparatus for the use of ,lecturers and ,experimenters in ,mechanical ,philosophy1 ,london1 ,weale ,co41 #ahea4 ,this work has long been out of print4 ,it may therefore be convenient if ,i give here a brief account of those parts of this admirable apparatus that ,i have found especially useful ,the illustrations have been copied from the plates in ,professor ,willis' book4#b #b ,i ought to acknowledge the kindness with which ,mr4 ,j4 ,willis ,clark1 of ,cambridge1 the literary executor of ,professor ,willis1 has responded to my queries1 while ,i am also under obligations to the courtesy of ,messrs4 ,crosby1 ,lockwood1 ,co4 ,the ,willis system provides the means for putting versatile framework together with or without revolving gear for the purpose of mechanical illustration4 ,many parts which enter into the construction of the machine used at the lecture to-day will reappear to-morrow as essential parts of some totally different contrivance4 ,the parts are sufficiently substantial to work thoroughly well4 ,the scantlings and dimensions generally have been so chosen as to produce models readily visible to a large class4 ,it will of course be understood that every model contains some one or more special parts such as the punch and die in ,fig4 #gc1 or the spring balance in ,fig4 #ag1 or the pulley-block in ,fig4 #cc4 ,but for the due exhibition of the operation of the machine a further quantity of ordinary framework and of moving mechanism is usually necessary4 ,this material1 which may be regarded as of a general type1 it is the function of the ,willis system to provide4 ,,the ,,bolts4--,the system mainly owes its versatility and its steadiness to the use of the iron screw bolt for all attachments4 ,the bolts used are diameter2 the shape of the head is hemispherical and the shank must be square for a short distance from the head so that the bolt cannot turn round when passed through the slits of the brackets or rectangles4 ,when the head of the bolt bears on a slit in one of the wooden pieces a circular iron washer #b in diameter1 or a square washer #b on each side1 is necessary to protect the wood from crushing4 ,there is to be a square hole1 in the washer to receive the square shank of the bolt and the thickness of the washers should be 4 ,the nut is square or hexagonal1 and should always have a washer underneath when screwed home with a spanner or screw-wrench4 ,the most useful lengths are #b1 #d1 #f4 ,the proper kind are known commercially as coach-bolts1 and they should be chosen with easy screws1 for facility in erecting or modifying apparatus4 ,at least two dozen of the intermediate size and a dozen of each of the others are required4 ,for elaborate contrivances many more will be necessary4 ,illustration3 ,,the ,,bed4 ,,fig4 #ajd4 ,,the ,,beds4--,the simplest as well as the longest parts of the framework are called 8beds0 7,fig4 #ajd74 ,each bed is made of two wooden bars4 ,these bars are united by strong screws passing through small blocks of hard wood so as to keep the bars full asunder1 and thus allow the shanks of the bolts to pass freely through the slit ,the scantling of each bar is #b #a1 and the beds are of various lengths from #a' to #aj' or even longer4 ,the beds can be attached together in any required position by bolts #f long4 ,the rectangles and the brackets are attached to the beds by #d bolts4 ,in one conjunction or another the beds will be found represented in almost every figure in the book4 ,we may specially refer to ,figs4 #bj1 #dd1 #dh1 #di1 #ej1 #fe1 #hc4 ,illustration3 ,,the ,,stool4 ,,fig4 #aje4 ,,the ,,stool4--,most of the larger pieces of apparatus have the stool as their foundation 7see ,figs4 #aa1 #ci1 #ajb74 ,it is often convenient as in ,fig4 #fe to employ a pair of stools1 while one stool superposed on another gives the convenient stand in ,fig4 #hj4 ,the stool is a stout wooden frame1 providing a choice of slits to which beds or other pieces may be attached by bolts4 ,the structure of the frame is shown in ,fig4 #aje4 ,it is #b' #f high and its extreme horizontal dimensions are #b' #f #a' #i of which the greater is ,a ,e4 ,in other words1 the longer sides of the stool are those open at the top4 ,each top corner is strengthened by an iron plate of which a separate sketch is shown4 ,the scantlings of the parts of the stool are as follows3--,the legs and horizontal top rails are #c #b4 ,two of these rails with the intervening slit make the top and legs to be #d wide4 ,the bottom front rail ,i is #c wide and #d deep4 ,the double side rails ,d1 ,h are #a wide and #b deep1 being made thinner than the legs into which they are mortised in order to allow the washers of the bolts to pass behind them4 ,the slits are to be full wide throughout4 ,beech or birch are very suitable materials1 but softer woods will answer if large washers are invariably used4 ,illustration3 ,,the ,,rectangle4 ,,fig4 #ajf4 ,,the ,,rectangle4--,the useful element of the ,willis system known by this name is of iron cast in one piece 7,fig4 #ajf74 ,the rectangles are used in the attachment of beds to each other under special conditions1 or they are often attached to the stools or to brackets4 ,indeed their uses are multifarious1 see for examples ,figs4 #ab1 #eh1 #fb1 #hi1 #ig1 #ajb and many others4 ,the faces of the rectangle are #b broad4 ,the outside dimensions are #f and #i1 and the thickness of metal is 4 ,each side of the rectangle has the usual bolt slit clear4 ,rectangles of a larger size are often found useful1 their weight makes them effective stands 7see ,figs4 #ce1 #dc1 #eb1 #fe74 ,,the ,,toothed ,,wheel4--,the most convenient type of toothed wheel for our present purpose is that known as the cast iron ten-pitch4 ,in all such wheels the number of teeth is simply ten times the number of inches in the diameter4 ,for example a wheel with #abj teeth is #ab inches in diameter4 ,a number of ten-pitch wheels large and small must be provided4 ,the actual assortment that will be necessary depends upon circumstances4 ,for most purposes it will be sufficient to have the multiples of #e from #be upwards to #abj1 and then a few larger sizes such as #aej1 #ahj1 #bjj4 ,duplicates of the constantly recurring numbers such as #cj1 #fj1 #abj are convenient4 ,arm wheels are always preferable to plate wheels in lightness and appearance as well as in price4 ,all wheels are to be #a thick at the boss which is faced in the latter at each side1 and bored with a hole full #a diameter1 in which a key groove is cut4 ,a pair of mitre wheels such as are used in ,fig4 #hj are sometimes useful4 ,illustration3 ,,the ,,stud ,,socket4 ,,fig4 #ajg4 ,,the ,,pulley4--,we have frequent occasion to use the pulley for conveying a cord1 and a somewhat varied stock is convenient ,thus light brass pulleys are used in the apparatus shown in ,fig4 #c1 and a stout pulley in ,fig4 #ga4 ,a cast iron pulley about #aj in diameter is seen in ,figs4 #cb and #cd4 ,it is bored #a in diameter with a key groove1 and the boss is #a thick4 ,some small pulley-blocks similar to those used on yachts are often very useful4 ,,the ,,stud ,,socket4--,for mounting toothed wheels on the larger pulleys or for almost any rotating or oscillating pieces the stud socket is used 7see ,fig4 #ajg74 ,the socket ,a ,b may be made of brass or of cast iron4 ,it is #a in diameter so as to pass through the bosses of the wheels that have been bored to #a with this object3--,the socket is provided with a shoulder at one end 7,a7 which is #a diameter1 and with a strong screw ,b and octagonal nut at the other end4 ,the extreme length of the socket is #c1 and the plain part of the #a cylinder is #a long4 ,when two wheels are placed on the socket each of which has a boss #a thick1 the tightening of the nut will secure the wheels against the shoulder4 ,a feather is screwed on the plain part which enters the key grooves in the wheels1 and thus ensures that the wheels shall turn together4 ,this feather should be small enough to slip easily into the key groove4 ,if only a single wheel or if any peculiar piece such as a wooden cam or a disk of sheet iron has to be mounted1 then collars or large thick washers must be placed on the socket so as permit the screw to bind the whole together4 ,the socket revolves upon a stout iron stud ,c ,d1 which is in diameter4 ,it bears a shoulder or flange ,c at the back of the same diameter as the base of the socket ,the stud bears on the other side of the shoulder a strong screw and nut which project #a so as to allow the stud to be secured in a hole #a deep in one of the brackets 7to be presently described74 ,the plain part of this screw near the shoulder must be diameter4 ,the front end of the stud is pierced with a hole to receive a spring pin to keep the socket from sliding off the stud4 ,among the many applications of the stud socket we may mention those shown in ,figs4 #cj1 #gc1 #gd4 ,illustration3 ,,bracket ,,no4 ,i4 ,,fig4 #ajh4 ,illustration3 ,,bracket ,,no4 ,,ii4 ,,fig4 #aji4 ,illustration3 ,,bracket ,,no4 ,,iii4 ,,fig4 #aaj4 ,,the ,,bracket4--,there are six different forms of cast iron brackets represented in the adjoining figures 7,figs4 #ajh-aac74 ,the brackets are primarily intended as the supports of the stud sockets4 ,for this purpose each has a head #a thick bored with a hole diameter1 and thus fitted to receive the screw on any of the studs4 ,each bracket stands on a base or sole with a slit full wide for the bolts4 ,the thickness of the sole is 4 ,the larger of the brackets ,i41 ,,ii41 and ,,iv4 have also slits in their vertical faces4 ,brackets can be fastened either to the stool or to the beds or rectangles1 and the variety of their forms enables the wheel-work carried on the stud sockets to be disposed in any desired fashion4 ,brackets avail for many other purposes besides those of supporting rotating mechanism4 7,look at ,figs4 #aa1 #ab1 #ag1 #bj1 #cc1 #ch1 #ci1 #gc and many others47 ,illustration3 ,,bracket ,,no4 ,,iv4 ,,fig4 #aaa4 ,illustration3 ,,bracket ,,no4 ,v4 ,,fig4 #aab4 ,illustration3 ,,bracket ,,no4 ,,vi4 ,,fig4 #aac4 ,illustration3 ,,the ,,tube ,,fitting4 ,,fig4 #aad4 ,,the ,,shafts ,,and ,,tube-,,fittings4--,the stud sockets will not provide for every case in which wheels have to be mounted and driven4 ,we must often employ shafts 7see for instance ,figs4 #cj1 #dg1 #aja74 ,the shafts we use are turned iron rods in diameter1 and of various lengths from #f up to #d'4 ,to support the shafts we use for bearings the tube fitting 7,fig4 #aad74 ,this is a brass casting which consists of a tube ,m ,n #b long1 and #a in external diameter1 bored so as to fit the shaft4 ,the back of this tube is a flat surface parallel to the bore1 and from it projects a screw diameter1 and #a long with a nut which is however omitted in the drawing4 ,this screw may be of the same size as that of the studs1 and it is intended for the same purpose1 namely to attach the bearing to the hole in a bracket4 ,the tube may of course be fixed at any desired angle in the plane parallel to the face of the bracket4 ,to prevent the endlong motion of the shaft cast iron or brass rings are employed 7,fig4 #aae74 ,these are bored 1 and furnished with a binding screw by which they may be tightened on the shaft in any position4 ,to avoid injury to the shaft it is well to have a narrow flat surface filed along it to receive the end of the binding screw4 ,the use of the rings is shown in ,fig4 #dg4 ,if as often happens 7see for example ,fig4 #ajb7 a barrel has to be set in motion by a shaft the required attachment can be made by simply slipping on the barrel1 and then putting at each end of it two of the pinned rings 7,fig4 #aae74 ,the pins enter holes bored into the barrel for their reception so that when the rings are bound to the shaft by their screws the barrel must revolve with the shaft4 ,illustration3 ,,the ,,pinned ,,ring4 ,,fig4 #aae4 ,illustration3 ,,the ,,adapter4 ,,fig4 #aaf4 ,,the ,,adapter4--,for the attachment of wheels or other rotating pieces to the shaft an adapter is employed 7,fig4 #aaf74 ,it is bored with a hole to fit the shaft1 and the external diameter is #a4 ,at one end is a shoulder through which the binding screw is tapped1 and there is a nut and screw at the opposite end4 ,a feather will prevent the wheel from turning round on the adapter which is itself made to revolve with the shaft by screwing the binding screw down on the shaft4 ,some adapters are only large enough for a single wheel #a thick in the boss1 but it is useful to have others that will take two wheels4 ,adapters are shown in use in ,figs4 #df and #aja4 ,illustration3 ,,the ,,lever ,,arm4 ,fig4 #aag4 ,,the ,,lever ,,arm4--,to give motion to the mechanism a lever arm with a handle is frequently required 7,fig4 #aag74 ,it is bored #a and has a key groove1 and the hole is #a long1 so that the lever arm can be fixed on a stud socket like a wheel4 ,by the aid of an adapter the lever arm is attached to a shaft4 ,for the use of the handle see ,figs4 #cj and #aja4 ,there are however many other uses to which the lever arm is occasionally put4 ,it can be used as a crank1 and in linkage arrangements a pair of lever arms are very convenient4 ,studs ,a or ,c can replace the handle when necessary4 ,such are the parts of the ,willis apparatus which are adapted for our present purpose4 ,it remains to add that the fits should be very easy1 and the parts should be readily interchangeable4 ,,index4 ,a4 ,accident1 risk of1 #cb ,action1 #f ,adapter1 ,willis apparatus1 #ceb ,angle of friction1 #gh of statical friction1 #hj ,apparatus for centre of gravity1 #fb for equilibrium of three forces1 #g to show friction1 #fe1 #gh the ,willis1 #cde ,appendix ,i41 #cci ,atwood's machine1 #bcb ,axes1 permanent1 #bgi ,b4 ,balance1 defective1 #dh spring1 #af ,bar1 equilibrium of a1 #ch ,bat1 cricket1 #cji ,beam1 breadth of1 #aic breaking load of1 #aic1 #aif cast iron1 #bbb collapse of1 #ahf deflection of1 #agi elasticity of1 #ahd load on1 #aig placed edgewise1 #aic strained1 #agh strength of1 #aij uniformly loaded1 #aih with both ends secured1 #bjj with one end secured1 #bja ,beds in ,willis apparatus1 #cdf ,bob1 raising or lowering the1 #cbj ,bolts1 use of1 in ,willis apparatus1 #cdf ,bracket1 ,willis apparatus1 #cej ,brass1 specific gravity of1 #ef ,breaking load1 #agg ,bridge1 deflection of1 #bjh mechanics of1 #bah ,menai1 #bah suspension1 #bbe the ,wye1 #bae tubular1 #bbc with four struts1 #baj two struts1 #bjf two ties1 #baa ,brunei1 ,sir ,j41 the ,wye bridge1 #bae ,c4 ,capstan1 #aea ,cast iron beam1 #bbb ,catenary1 #bbf ,cathetometer1 #ahj ,centre of gravity1 #eg of a wheel1 #fa position of1 #ei oscillation1 #cjd percussion1 #cjg ,circular motion1 #bfg action of1 #bga applications of1 #bgf cause of1 #bgj in governor-balls1 #bgf in sugar refining1 #bgf nature of1 #bfg on liquids1 #bga on the earth1 #bgf ,circular pendulum1 #bhd ,clamps1 #bjc ,clock pendulum1 #bii principles of1 #cah rate of1 #cbb ,coefficient of friction1 #gd1 #hb ,collapse of a beam1 #ahf ,compensating pendulum1 #cai ,composition of forces1 #a1 #i parallel forces1 #ce1 #cg1 #db vibrations1 #bii1 #cae ,conical pendulum1 #caj ,couple1 #dd ,crane1 #bi1 #afb friction in1 #aff mechanical efficiency of1 #afe ,table ,,xxi4 #afe ,,xxii4 #aff velocity1 ratio of1 #afc ,cricket bat1 #cji ,crowbar1 #abc ,cycloid1 #bie ,d4 ,dead-beat escapement1 #cbh ,definition of force1 #b ,deflection of a beam1 ,table ,,xxiii4 #ahb ,differential pulley1 #aab ,table ,,xi4 #aad ,direction of a force1 #e ,e4 ,eade1 ,mr41 epicycloidal pulley-block1 #aaf ,easter ,island1 #ajj ,elasticity of beam1 #ahd ,energy1 #he1 #id storage of1 #bef1 #beh unit of1 #ie ,engine1 locomotive1 #hc ,epicycloidal pulley-block1 #hj1 #aaf ,table ,,xii4 #aah ,equilibrium1 neutral1 #fa of a bar1 #ch1 #da three forces1 #f two forces1 #f stable1 #ei unstable1 #ei ,escapement1 #cbd dead-beat1 #cbh recoil1 #cbh ,expansion of bodies1 #cba ,experiment by ,m4 ,plateau1 #bgc ,f4 ,fall in a second1 #bci ,falling body1 motion of1 #bcj ,feet1 how represented1 #g ,fibres in state of compression1 #ahd tension1 #ahd ,first law of motion1 #bcj ,fly-wheel1 #bfj in steam-engine1 #bfb ,foot-pound1 #ie ,force1 a small1 and two larger1 #ab definition of1 #b destroying motion1 #c direction of a1 #e magnitude of a1 #d measurement of1 #d of friction1 #fe gravity1 #ej one1 resolved into three1 #bf two1 #ag representation of1 #e standard of1 #d ,forces1 composition of1 #a1 #i equilibrium of three1 #f two1 #f illustrations of1 #c in inclined plane1 #acf parallel1 #cd parallelogram of1 #aj resolution of1 #af ,formula for pulley-block1 #aji1 #aad ,framework1 #bjc1 #cde ,friction1 #fe accurate law of1 #ge a force1 #ff and pressure1 #gb angle of1 #gh angle of statical1 #hj apparatus to show1 #fe1 #fh1 #gh caused by roughness1 #ff coefficient of1 #gd1 #hb diminished1 #ff excessive1 #aae experimenting on1 #ff in crane1 #aff differential pulley-block1 #aac inclined plane1 #acb lever1 #abc pulleys1 #hi law of1 #ia rope and iron bar1 #hg wheel and axle1 #aec wheel and barrel1 #aeh laws of1 #gc1 #ha1 #hb mean1 #ge motion impeded by1 #gj nature of1 #fe overcoming1 #ic ,table ,i4 #fi ,,ii4 #ga ,,iii4 #gd ,,iv4 #gf ,v4 #gh ,,vi4 #ha ,,vii4 #ha ,,viii4 #ha upon axle1 #aee wheels1 #ic ,g4 ,galileo and falling bodies1 #bce kinetics1 #bcj the pendulum1 #bhd tower of ,pisa1 #bcc ,gathering pallet1 #ccf ,girder1 #bai as slight as possible1 #bba ,governor-balls1 #bgf ,graham1 dead-beat escapement1 #cbh ,graphical construction1 #cci ,gravity1 #ej action of1 #bdc and the pendulum1 #bib and weight1 #eb centre of1 #eg defined1 #bdf different effects of1 #ec independent of motion1 #bda in ,london1 #bib specific1 #ec ,grindstone1 treadle of1 #abh ,h4 ,hammer1 #beb theory of the1 #beb ,hands of a clock1 #cca ,horse-power1 #if ,i4 ,illustration of parallelogram of forces1 #aj ,illustrations of forces1 #aj resolution1 #ai ,inches1 how represented1 #g ,inclination of thread1 #adj ,inclined plane1 #aca forces on1 #acf friction in1 #acb mechanical efficiency of1 #aci ,table ,,xiii4 #acd ,,xiv4 #acg ,,xv4 #ach velocity1 ratio of1 #aci ,inertia1 #bej inherent in matter1 #beb ,iron girders1 #bai specific gravity of1 #ee ,isochronous simple pendulum1 #cjc ,ivory1 specific gravity of1 #ef ,j4 ,jib1 #bi1 #afc ,k4 ,kater1 ,captain1 #cje ,kinetics1 #bcj ,l4 ,large wheels1 advantages of1 #ic ,law of falling bodies1 #bch friction in pulleys1 #ia lever of first order1 #abb pressure1 #cg ,laws of friction1 #gc1 #ha1 #hb ,lead1 specific gravity of1 #ef ,leaning tower of ,pisa1 #bcc ,level1 #ef ,lever1 #aai and friction1 #abc applications of1 #abc arm1 ,willis apparatus1 #ceb laws of1 #acj of first order1 #aai law of1 #abb of second order1 #abd of third order1 #abh weight of1 #aba ,lifting crane1 #bi ,line and plummet1 #ef ,load1 breaking1 #agg ,locomotive engine1 #hc ,m4 ,machine1 ,atwood's1 #bcb punching1 #bfc ,machines1 pile-driving1 #bee ,magnitude of a force1 #d ,margin of safety1 #cc ,mass1 #bcf ,mean frictions1 #ge ,measurement of force1 #d ,mechanical powers1 #he1 #ajj apparatus1 ,willis1 #cde ,menai ,bridge1 #bah ,method of least squares1 #cdb ,moment1 #acj ,monkey1 #beg ,motion1 first law of1 #bcj of falling body1 #bcj ,n4 ,neutral equilibrium1 #fa ,newton and gravity1 #bhi ,nut1 #adj ,o4 ,oscillation1 centre of1 #cjd ,p4 ,pair of scales1 #dh testing1 #dh ,parabola1 #bbf ,parallel forces1 #cd composition of1 #ce1 #cg1 #db opposite1 #dd resultant of1 #dc ,parallelogram of forces1 #aj ,path of a projectile1 #bdg ,pendulum and gravity1 #bib circular1 #bhd compensating1 #cai compound1 #bii1 #cja conical1 #caj formula for1 #bib ,galileo and the1 #bhf isochronous simple1 #cjc length of the seconds1 #bib1 #cah motion of the1 #bhe of a clock1 #bii simple1 #bhd time of oscillation1 #bhf1 #bhi ,percussion1 centre of1 #cjg1 #cji ,permanent axes1 #bgi ,pile-driving machines1 #bee ,plateau1 ,m41 experiment by1 #bgc ,plummet1 #ef ,powers1 mechanical1 #he ,pressure and friction1 #gb law of1 #cg of a loaded beam1 #ce1 #cg ,principles of framework1 #bjc ,projectile1 path of1 #bdg ,pulley-block1 #ii differential1 #aaj epicycloidal1 #hj three-sheave1 #ajf velocity1 ratio of1 #aab ,pulley1 ordinary form of1 #hf single movable1 #aja fixed1 #hf use of1 #hh velocity1 ratio of1 #ajc ,pulleys1 friction in1 #hi in windows1 #hf in ,willis apparatus1 #cdi ,punching machine1 #bfc force of1 #bfe ,r4 ,rack1 #ccd ,reaction1 #f ,recoil escapement1 #cbh ,rectangle in ,willis apparatus1 #cdh ,representation of a force1 #e ,resistance to compression1 #agb1 #age extension1 #agb ,resolution of forces1 #af one force into three1 #bf two1 #ag ,resultant1 #i of parallel forces1 #dc ,rings in ,willis apparatus1 #ceb ,risk of accident1 #cb ,s4 ,safety1 margin of1 #cc ,sailing1 #ba against the wind1 #bd ,scales1 #df ,screw1 #aci and wheel and axle1 #afg form of1 #aci ,table ,,xvi4 #adb velocity1 ratio of1 #adc ,screw bolt and nut1 #adh jack1 #aca1 #ade ,table ,,xvii4 #adf ,second1 fall in a1 #bci ,seconds1 pendulum1 #cah ,shafts1 ,willis apparatus1 #cea ,shears1 #abf ,simple pendulum1 #bhd ,single movable pulley1 ,table ,,ix4 #ajd ,snail1 #ccd ,specific gravity1 #ec of brass1 #ef iron1 #ee ivory1 #ef lead1 #ef ,spirit-level1 #ef ,spring balance1 #af ,stable equilibrium1 #ei1 #bhb ,standard of force1 #d ,statical friction1 angle of1 #hj ,stool in ,willis apparatus #cdg ,storage of energy1 #bef1 #beh ,stored-up energy exhibited1 #bfa ,strength of a beam1 #aij ,striking parts1 #ccc ,structures1 #afi ,strut1 #bh ,stud socket in ,willis apparatus1 #cdi ,sugar refining1 #bgf ,suspension bridge1 #bbe mechanics of1 #bbe tension in1 #bbh ,t4 ,table ,i4 #fi ,,ii4 #ga ,,iii4 #gd ,,iv4 #gf ,v4 #gh ,,vi4 #gh ,,vii4 #ha ,,viii4 #ha ,,ix4 #ajd ,x4 #ajh ,,xi4 #aad ,,xii4 #aah ,,xiii4 #acd ,,xiv4 #acg ,,xv4 #ach ,,xvi4 #adb ,,xvii4 #adf ,,xviii4 #aed ,,xix4 #aei ,,xx4 #afb ,,xxi4 #afe ,,xxii4 #aff ,,xxiii4 #ahb ,,xxiv4 #aij ,tacking1 #be ,tension along a cord1 #ag ,three-sheave pulley-block1 #ajf ,tie1 #bh1 #age rod1 #bi1 #cb ,timber1 bending1 #aga compression of1 #agb extension of1 #agb properties of1 #agj rings in1 #aga seasoning1 #aga warping1 #aga ,tin1 #bbc ,toothed wheels1 #afj ,tower of ,pisa1 #bcc ,train of wheels1 #ccj ,transverse strain1 #aha ,treadle of a grindstone1 #abh ,tripod1 #bh strength of1 #bh ,truss1 simple form of1 #bab ,tube fitting1 ,willis apparatus1 #cea ,tubular bridge1 #bbc ,u4 ,unstable equilibrium1 #ei1 #bhb ,v4 ,velocity1 #bca ratio of inclined plane1 #aci pulley1 #ajc pulley-block1 #aab screw1 #adc wheel and axle1 #aeb wheel and pinion1 #afa ,vibrations1 composition of1 #bii1 #cae ,w4 ,wedge1 #aci ,weighing machines1 #abc scales1 #df1 #dh ,weight caused by gravity1 #eb of water1 #ed ,wheel and axle1 #adi and differential pulley1 #afg screw1 #afg experiments on1 #aeb formula for1 #aed friction in1 #aec ,table ,,xviii4 #aed velocity1 ratio of1 #aeb ,wheel and barrel1 #aeh formula for1 #afj friction in1 #aeh ,table ,,xix4 #aei ,wheel and pinion1 #afj efficiency of1 #afa ,table ,,xx4 #afb velocity1 ratio of1 #afa ,wheel1 centre of gravity of1 #fa ,wheels1 #ib friction1 #ic ,wheels in ,willis apparatus1 #cdh ,willis system of apparatus1 #cde ,winch1 #aea ,wind1 direction of1 #bb ,work1 #he1 #id ,wye ,bridge1 #bae ,,the ,,end4 ,,richard ,,clay ,,and ,,sons1 ,,limited1 ,,london ,,and ,,bungay4