185-187] ATWOOD'S MACHINE. 177 



Hence the equation of energy can be written 



(m + mf) x* = (m- m') gx + const. 

 By differentiating this equation we obtain 

 (m + mf) x = (m m') g, 

 showing that the acceleration of either particle is in magnitude 



m + m' ' 



The tension T of the thread can be. obtained from the equation 

 of motion of either particle. Thus we have for m 



mx = mg T, 



. . m 2mm' 



giving L = - , a. 



m + m y 



An instrument of which the above is the principle is known as 

 Atwood's machine. It is manifest that the masses can be so 

 adjusted as to make the acceleration much smaller, and therefore 

 much more accurately measurable, than the acceleration g of a free 

 falling body. When the ratio of the masses of the two bodies is 

 known, experiments with this instrument yield a determination of 

 the value of g. 



187. Examples. 



1. In Atwood's machine the mass m' is rigid, the mass m consists of a 

 rigid portion of mass m! and a small additional piece lightly resting upon it. 

 As m descends it passes through a ring, by which the additional piece is lifted 

 Qff. Prove that, if m starts from a height h above the ring, and if the time 

 it takes to fall a distance k after passing through the ring is t, then 



~ m + m' ht 2 ' 



2. Two particles of masses M, m are connected by an inextensible thread 

 of negligible mass which passes through a small smooth ring on a smooth 

 fixed horizontal table. When the thread is just stretched, so that M is at a 

 distance c from the ring, and the particles are at rest, M is projected on the 

 table at right angles to the thread. Prove that until m reaches the ring M 

 describes a curve whose polar equation is of the form 



r = c cos {0 J(l+m/M)}. 



3. Two particles of masses Jf, m are connected by an inextensible thread 

 of negligible mass ; M describes on a smooth table a curve which is nearly a 

 circle with centre at a point 0, and the thread passes through a small smooth 

 hole at and supports m. Prove that the apsidal angle of M's orbit is 



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