EXAMPLES. 323 



80. There is a system of n moveable pulleys m lt m 2 , ... m n and n corre- 

 sponding counterpoises of masses p lt /x 2 , ... /z n . Each pulley and its counter- 

 poise are suspended by a cord passing over the preceding pulley. The 

 highest cord (connecting m-^ and ^ passes over a fixed pulley, and no cord 

 passes over the lowest pulley m n . The suffixes indicate the order in which 

 the pulleys are slung. The pulleys are simultaneously set free. Prove that, 

 if T lt T 2J ... T n are the tensions in the cords, 



} = T p (!/ 



further, if the mass of each pulley (m) is to the mass of each counterpoise (/*) 

 as 5 : 3, prove that the downward acceleration of the p th moveable pulley is 



81. Two equal particles connected by an inextensible thread lie on a 

 smooth table with the thread straight; prove that, if one of them is pro- 

 jected on the table at right angles to the thread, the initial radius of curvature 

 of its path is twice the length of the thread. 



82. A small ring of mass m rests on a smooth straight wire, and another 

 particle of mass m' is connected with it by a thread of length a. Prove that, 

 if m' is projected in a direction at right angles to the wire from a point on it 

 at a distance a from m, the initial radius of curvature of the path is 



83. An inextensible thread passes through two smooth rings A, B on a 

 smooth table ; particles of masses p and q are attached to the ends, and a 

 particle of mass m is attached to a point between A and B. Prove that, if 

 m is projected horizontally at right angles to the thread, the initial curvature 

 of its path is (p\OA ~ qlOB)/(p + q+m). 



84. A particle of mass m on a smooth table is joined to a particle of 

 mass m' hanging just over the edge by a thread of length a at right angles 

 to the edge. Prove that, if the system starts from rest, the radius of curva- 

 ture of the path of m immediately after it leaves the table is 



2m' a 



85. Two particles A, B are connected by a fine string; A rests on a rough 

 horizontal table (coefficient of friction =/i) and B hangs vertically at a 

 distance I below the edge of the table. If A is on the point of motion, and 

 B is projected horizontally with velocity U, show that A will begin to move 

 with an acceleration p.i(?/{(fjL + !)}, and that the initial radius of curvature of 

 #s path will be GA + I) 



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