185-187] ATWOOD S MACHINE. 177 



Hence the equation of energy can be written 



J (m + m ) x 2 = (m ra ) ## + const. 

 By differentiating this equation we obtain 



(m + m ) 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 2ram 



giving 1 = - ,g. 



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 

 off. 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 



2 (m - m ) IP 



Q= i - _ 



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 {6 v /( 1 + m/M)}. 



3. Two particles of masses J/, 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 Jf s orbit is 



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