A chain of length `l` and mass `m` lies in a pile on the floor. If its end A is raised vertically at a constant speed `v_0`, express in terms of the length y of chain which is off the floor at any given instant.
(a) The magnitude of the force P applied to end A.
(b) Energy lost during the lifting of the chain.

Answer

A, B

Explanation

(a) Chain has a constant speed. Therefore, net force on it should be zero. Thus,
P=Weight of length y of chain +thrust force
`=m/lyg+rhov_0^2` (here, `rho=m/l`)
`=m/l(gy+v_0^2)`
(b) Energy lost during the lifting=work done by apllied force-increase in mechanical energy of chain
`=int_0^yP*dy-(m/ly)g(y/2)-1/2(m/l*y)v_0^2`
`=(myv_0^2)/(2l)`