EXAMPLES. 323 



80. There is a system of n moveable pulleys m 15 m 2 , ... m n and n corre 

 sponding counterpoises of masses fi lt ju 2 , ... /i n . Each pulley and its counter 

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

 highest cord (connecting m^ and p.^) 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^ T%, ... T n are the tensions in the cords, 



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

 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 t 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 ~ qjOB)/(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 



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 /xw 2 /{(/x + 1)1}, and that the initial radius of curvature of 

 B s path will be (/*+ 1) I. 



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