372 



THE POPULAR EDUCATOit. 



bon gr6 de ses intentions, quelles quo soient les suites de sa conduite. 

 17. Me garderez-vous IB secret ? 18. Je vous le garderai de bou gre. 

 19. Votre soeur gurde-t-elle le lit de bon gre"? 20. Elle ne garde pas 

 la chiiuibre <le bou gre". 21. Bon gre", ma) gre", il faut qu'elle garde la 

 chambre, quaiul elle est mulade. 22. G ."derez-vous le secret sur ce 

 point ? 23. Je le gaiderai de bou gre. 24. Je vous sais bon gre de 

 vos bouues intentions. 25. Lui savez-vous bou grd de cela? 26. Je 

 lui en sais bou gre. 27. Le juge gardera-t-il son domestique ? 28. II 

 le gardera. 29. Fait-il sou travail a sou gre 1 ? 30. II le fait a son gre. 

 31. M. votre frere est-il oblige de garder la tnaison ? 32. II est oblige* 

 de garder le lit. 33. N'a-t-il pas quitto sa chambre? 34. II n'a pas 

 encore quittd sa chambre ; il est trop malade pour la quitter. 35. 

 Si TOUS vouliez faire cela, je vous en sauraia le meilleur gre du moude, 



Fig. 70. 



MECHANICS. XIII. 



PRINCIPLE OP VIRTUAL VELOCITIES THE THKEE 

 SYSTEMS OP PULLEYS. 



IN our last lesson we noticed the two kinds of single pulleys, 

 the fixed and the movable. In the fixed pulley there is no gain 

 at all in power : a force of 1 pound will only support a weight of 

 the same amount. What, then, is the use of it ? Simply that it 

 enables us to change the direction in which any force acts often 

 as great an advantage as an increase of power would be. 



Suppose, for instance, a man wants to raise a heavy bale to 

 the top of a warehouse. He might go to the upper story, and 

 lift it by pulling in a rope tied round it ; but this would be a 

 very bad way of applying his force, and would soon tire him, 

 and, at the same time, there would be a great danger of his 

 overbalancing himself and falling. Instead of this, the rope is 

 passed over a pulley fixed to an arm which projects from the 

 warehouse, and he stands within, or 

 on the ground, and by pulling down- 

 wards raises the bale. 



In the single movable pulley there 

 is, as we saw, an actual gain, a power 

 of one pound balancing a weight of 

 two pounds. The single fixed pulley, 

 or runner, is often used with this, not 

 that it increases the advantage gained, 

 but merely to change the direction, it 

 being usually more convenient for the 

 power to act downwards than upwards. 

 No vy, if we turn our attention to the 

 figure, we shall learn from it a new 

 and very important principle, which, 

 though it strictly belongs to dynamics, 



time being taken into account, we had better inquire into now, 

 as it will help us more clearly to understand our subject. 



We have supposed that there is equilibrium in the system, 

 that is, that the power exactly balances the weight, being, in bhis 

 case, just one-half of it. Now, let the power be slightlv in- 

 creased so that P falls ; the weight, w, will then be raised ; and let 

 this continue till it has been raised through a space of, say, two 

 inches. Now, since it is supported by the cords, or rather the 

 two parts A B, A c of the same cord, it is plain that each must 

 be shortened by the same quantity namely, two inches and to 

 effect this, p must move through four inches, or double the space 

 passed over by w. Thus while, on the one hand, a power of 

 | one pound will balance a weight of two, the power must, on the 

 other hand, be moved through four inches in order to move the 

 weight through two, or, in other words, the power is only one-hal/ 

 the weight ; but in order to raise the weight any given distance, 

 it must move over twice that distance. This rule is frequently 

 expressed as follows : 



Whatever is gained in power is lost in time ; that is, if by any 

 of the mechanical powers we are enabled to overcome a larger 

 resistance than the power could if directly applied, in just that 

 proportion will the space through which the power moves be 

 larger than that through which the weight moves, and therefore 

 the time occupied will be proportionally longer. 



This, simple principle applies to all the mechanical powers, and, 

 when fully understood, clears up many difficulties, and removes 

 apparent paradoxes. It is, therefore, important for us to be 

 clear about it. A machine cannot create force; it merely 

 modifies the force or motion of the power, so as to cause it best 

 to produce the resistance or motion required. The power must 

 be supplied by some " prime mover," as it is called, such as 



Pig. 71. 



the force of the wind or the tide, the strength of men or 

 animals, or the force of heat, which converts water into steam. 

 But when we have the force wo can store it up, as is the case 

 in a clock or watch, where the real driving power, which is the 

 force o r the hand exerted in winding it up, is laid up in the 

 1 spring and used gradually ; the spring not being, as it is 

 | commonly called, the moving power, but merely a kind of 

 reservoir in which it is stored up for future use. Or we can 

 modify and change the mode of action of our force, as in a 

 water-mill, where the onward force of the stream is changed 

 into the circular motion of the millstone, or applied in any other 

 way that may be desired. 



If by the movable pulley a man can lift a weight of two 

 hundred-weight to any given height say, for instance, four feet 

 when without it he can only lift one hundred- weight, it will take 

 him twice as long to do it, and the practical result will be the 

 same as if he divided the weight into two of one hundred- weight 

 each, and lifted each separately ; the only difference being one 

 of convenience, for by it he can lift a weight which otherwise 

 would be too heavy for him to move. 



If we for a moment retrace our steps, and see the application 

 of this principle to the mechanical powers we have already 

 considered, we shall perceive more distinctly its full meaning 

 and importance. 



Let us take again the case of a lever of the first kind. 

 Suppose the power to act 

 at a distance from the f ul- JJ 1 

 crum, F, five times as great 

 as the weight does ; let p F 

 be, for instance, 10 feet, 

 and F w 2 feet. Then a 

 power of 3 pounds will 

 balance a weight of 15 

 pounds. If P be now 

 slightly increased, w will be raised that is, a weight of 15 

 pounds will be lifted by a force only slightly greater than 3 

 pounds. This at first sight seems very strange, and a contra- 

 diction of our principle that a machine cannot create force. 

 But let us suppose that p has moved to P', so that the lever is 

 now in the position of the dotted line p* w 7 , w having likewise 

 moved to w' ; p will have passed over the arc F p', and w over 

 w w 7 ; and since F p is five times as great as F w, p p' is also, by 

 the laws of geometry, five times as great as \vw'. You can 

 easily satisfy yourself of this by actual measurement. And so, 

 though the power has lifted a weight five times as great as 

 itself, in order to do so, it has had to pass over a space five 

 times as large as that passed over by th weight. 



We thus get another way of expressing our principle, which 

 should be caref ally remembered : The power multiplied by the 

 space through which it moves is equal to the weight multiplied 

 by the space through which it moves. Of course, as before 

 explained, when we speak of the power multiplied by the 

 distance, we mean the product of the two numbers which 

 represent the number of units of weight in the power, what- 

 ever they be, and the number of units of length in the distance. 



In the case above let P P' bo 5 inches, w w' will be 1 inch, 

 and 3 (the number of pounds in the power), multiplied by 5 (the 

 number of inches through which it has passed), is equal to 15 

 multiplied by 1. This equation, as it is called, is plainly true, 

 each product being 15. 



This principle, which is called the LAW OF VIRTUAL VELO- 

 CITIES, is the fundamental principle in mechanics, and holds 

 good in all the mechanical powers. On account of its import- 

 ance, it has been called the Golden Eule of Mechanics. 



We will now trace its application to the wheel and axle, 

 which is, in reality, only a modification of the lever, being an 

 arrangement whereby an endless succession of levers may be 

 brought into play, for any two radii of the wheel and axle in 

 the same straight line may be considered as a simple lever. 



If the power be slightly increased it descends, and, when tho 

 wheel has turned just once round, will have fallen through ;; 

 space equal to the circumference of the wheel. In the same 

 time, the weight will have been raised through a space equal tc 

 the circumference of the axle. But tho circumference of circles 

 always bear the same ratio to one another that their radii bear. 

 If, then, the radius of the wheel be 12 inches, and that of th" 

 axle 3 inches, the power will pass through four times as great u 

 space as the weight, but will only be one-fourth of it. 



