204 MOTION UNDER CONSTRAINTS AND RESISTANCES. [CHAP. X. 



required to be added to overcome the friction of the axle when equal weights 

 P are hung at the ends of the chain. Prove that an additional weight R will 

 produce acceleration Rg/(2P+2Q + R+ W\ where W is the weight of the 

 pulley. 



21. Two equal masses P, P are connected by a cord passing over a 

 smooth pulley, and to them are attached equal masses Q, Q by cords. 

 Initially Q lies on a horizontal plane, and P, P , Q are in motion ; Q is raised 

 from the plane and Q caught by it almost simultaneously. If V is the initial 

 subsequent velocity of P, Q, P when Q is raised just first, and V the initial 

 subsequent velocity when Q is dropped just first, prove that 



V : F = (2P+) 2 : 4P(P+Q). 



22. Two pulleys each of mass 8m hang at the ends of a chain of negligible 

 mass which passes over a fixed pulley ; a similar chain passes over each 

 pulley and carries at its ends bodies of mass 2?7i. A mass m is now removed 

 from one of the bodies and attached to one of those which hang over the other 

 pulley ; prove that the acceleration of each pulley is ^g. Prove also that the 

 two descending bodies move with the same velocity, and that the velocity of 

 one of the ascending bodies is five times that of the other. 



23. A chain of negligible mass passes over two fixed pulleys and under a 

 moveable pulley and bodies are attached to its ends. Prove that, if all the 

 parts of the chain are vertical, the moveable pulley will remain at rest if its 

 mass is twice the harmonic mean of the other two masses. 



24. A chain of negligible mass passes over a fixed pulley B and supports 

 a body of mass m at one end and a pulley C of mass p at the other. A similar 

 chain is fastened to a point A below B, passes over (7, and supports a body of 

 mass m . Prove that the acceleration of the pulley is 



g (2m - m +jo)/(4m + m +-p). 



25. Two pulleys of masses M and M are connected by a cord passing 

 over a fixed pulley. Bodies of masses m 1 and m 2 are hung ov,er M by a cord, 

 and bodies of masses MI/, ra 2 are hung over M . Prove that either pulley 

 moves with acceleration 



g (J/+ 2,* - M - 2/0/(#+ Jf + 2/i + 2/0, 



where p is the harmonic mean of m l and ra 2 , and p. is the harmonic mean of 

 m^ and m 2 . 



26. Two bodies are supported in equilibrium on a wheel and axle, and a 

 body whose mass is equal to that of the greater body is suddenly attached to 

 that body. Prove that the acceleration with which it moves is a#/(2a + 6), 

 a and b being the radii of the wheel and the axle respectively, and the inertia 

 of the machine being neglected. 



27. A body of weight P balances a body of weight W in that system of 

 pulleys in which each pulley hangs by a separate cord. Prove that if bodies 



