MECHANICS 717 character as that of a free mass falling in a vertical line, hut the acceleration is less, in the ratio of the difference of the two masses to their sum. 174. This is the essence of the arrangement called Atwood s Machine, which used to be employed for the demon stration (in a rough way) of the first and second laws of motion, in certain simple cases. The main feature of the method is the artificial reduction of the acceleration, so that the motion of the falling body is rendered slow enough to be followed by the eye with some degree of accuracy. To prove the first lav, a bar of metal was laid across one of two equal masses suspended as in the example ; and the system was allowed to move under acceleration until the preponderat ing mass passed through a ring which arrested the bar. The .subsequent motion, with no acceleration, was then observed by noting the passage of the falling mass in front of a vertical scale, while the observer also listened to the ticking of a pendulum escapement. For the verification of the second law, so far as uniform force is concerned, the apparatus was adj listed by trial so that the extra load was detached from the preponderating mass after 1, 2, 3, &c., beats of the pendulum ; and the subsequent uniform speed was found to be nearly in proportion to these numbers. And, again, to prove that momentum acquired is, c&teris paribus, proportional to the force, the effects of bars of different masses were compared by the same process. If x and I- x be the portions of the string on opposite sides of the pulley at time t, we have m ~(l - x) = m cj - T = - m x . Hence by elimination of T we have and by elimination of x m + m 2 mm - t (j , as before. in + m When one of the masses is vibrating pendulum-wise, the problem assumes a very much more difficult aspect. We will take it later as an example of the application of Lagrange s general method, nn- 175. Let us now suppose these masses, so connected, it to be thrown like a chain-shot. We see by 166 that their centre of inertia moves as if the masses were concentrated there. Also that the moment of momentum is unaffected. Hence we have only to find the initial position and motion of the centre of inertia, and the plane and amount of the initial moment of momentum ; and the complete determina tion of the motion follows. This case is precisely the same as that of a well-thrown quoit, the rotation of which is about its axis of symmetry. It is, so far as 166 goes, the case of an ill-thrown quoit, which appears to wabble about in an irregular manner. But these are matters properly to be treated under Kinetics of a Rigid System. tsses 176. Suppose, next, two masses m^ and m. 2 to be con- a- nected together by an elastic string, the extension of the ted string being proportional to the tension. Let m l be held 5tic in the hand, while m. 2 hangs at rest. Then let the system ing. be allowed to fall. What is the nature of the motion 1 ? Without mathematical investigation it is easy to see that, the moment the masses are left free to fall, the tension of the stretched string will gradually draw them together. When it has thus contracted to its normal length, I, the relative speed of the two masses will have a definite value. This will continue to be the relative speed until they have passed one another and again arrived at a mutual distance /. At that instant the tension of the string comes into play again ; the relative speed becomes less and less, finally vanishing when the distance between the masses is what it was at starting. Then the relative speed becomes again one of approach, increasing steadily till the dis tance between the masses is I. This maximum speed of approach continues till, after again passing one another, the particles once more reach the relative distance I. And so on. All this time, however, their common centre of inertia has been steadily falling with uniformly accelerated speed, as if the masses had been concentrated at it into one. Since I is the unstretched length of the string, if we call E its modulus of elasticity, its tension at any other length, X, is by Hooke s law. Hence, if initially m^ were at the origin, and the axis of x be taken vertically downwards, we have for the initial coordinate of m.-, (s)o I. When the masses are moving, the third law informs us that the tension of the string acts equally and in opposite directions on them. Thus the equations of motion are m.^ = m. 2 g - T . By eliminating T we have at once mix + m. 2 x. 2 = (7% + m. 2 }g . But m 1 x 1 + m. 2 x. i = (m 1 + m 2 )^, if { be the coordinate of the centre of inertia of the two masses. Hence f-0, the ordinary equation for the fall of a stone. Thus Since = 0, *! = 0, x, 2 = ( 7 -^9 + 1 l, x 2 = 0, when < = 0, we have
1 y + : h(tn 1 + m 2 )gt 2 . and tnus f t"ti^t!c-^~~ f iTtnX. 2 So long as x a -x 1 >l we have also Hence, multiplying the first of the equations of motion by ??i 2 > and the second by ?H I} and taking the difference, we have ... .. . , ,- K (x. 2 -x l mjin^s^ x^) (mi+mtfrjij
*
The integral is a 1 -x l = l+ Pcos(nt + Q) , where ?i 2 =- 7 1 2 E. Im-jin.f Also, by the data at starting, we have Q = 0, F = ^. Hence, finally, E ^ (1 - cosnt) whence the value of x. 2 can easily be found. As soon as we have nt>%w these values cease to represent the coordinates of the two masses, because they are deduced from equations involving constraint which, in the case supposed, has ceased for a time. At the instant nt = l z ir the relative speed of the masses is m.,ffl /(&! + ?>i. 2 )K "~E~ V mi m a l and the ; r distance I. This distance diminishes thenceforward with the above speed until the uppermost stone, having passed the lower one, falls below it to a distance I, We must, in order to trace the next part of the motion, reapply the differential
equations above, integrating them, and determining the
