EXAMPLES. 345 



table before any of it reaches the cylinder, prove that the amount of energy 

 dissipated by the time the chain leaves the table is 



212. A quantity of uniform chain is coiled on a horizontal plane, and a 

 body of mass equal to that of a length I of the chain is fastened to one end 

 and projected vertically upwards with the velocity due to falling through 

 a height h ; prove that it will rise to a height 



213. A length k of a uniform chain of length l + k and mass /z(Z + ) is 

 coiled at the edge of a table and the length I hangs over the edge. Prove 

 that the amount of energy dissipated by the time the chain leaves the table 

 is a/tf**^ + *)/(* + *). 



214. A great length of uniform chain is coiled at the edge of a horizontal 

 platform, and one end is allowed to hang over until it just reaches another 

 platform distant h below the first. The chain then runs down under gravity. 

 Prove that it ultimately acquires a finite terminal velocity V, that its velocity 

 at time t is V tanh ( Vt/h), and that the length of chain which has then run 

 down is h log cosh ( Vt/h). 



215. A thread of length Zh-l passing over a smooth peg at a height h 

 above a table has attached to its ends two uniform chains, and the system is 

 released from rest when a length I of one chain is vertical and the rest of 

 that chain and the other chain are coiled on the table. Prove that the 

 chains will be momentarily at rest when the length of the vertical portion 

 I is reduced to I - x, where 



and that the maximum velocity is acquired when ^x\l = log 2. 



216. A chain whose density varies uniformly from p at one end to 2p at 

 the other end is placed symmetrically on a small smooth pulley and is then 

 let go. Prove that it leaves the pulley with velocity ^*J(lllg\ where 21 is its 

 length. 



217. Two buckets each of mass M are connected by a chain of negligible 

 mass which passes over a fixed smooth pulley. On the bottom of one of 

 them lies a length I of uniform chain, whose mass is /zZ, one end of which 

 is attached to a fixed point just above the bottom of the bucket. Prove 

 that, if the system starts to move from rest, the velocity of the bucket when 

 there remains upon it a length y of chain is V, where 



218. Two scale-pans each of mass M are supported by a cord of negligible 

 mass passing over a smooth pulley, and a uniform chain of mass m and 

 length I is held by its upper end above one of the scale-pans so that it just 

 reaches the pan. Find the acceleration of the pan when a length x of chain 

 has fallen upon it, and prove that the whole chain will have fallen upon it 

 after an interval 



