Question

A uniform chain of linear mass density λ has a length L and mass M. A part (1/n) of its length hangs down from the edge of the table (assumed frictionless). The chain is gradually pulled till the entire chain is on the table. The work done, in the process, is proportional to

(1) n

(2) n/2

(3) 1/n

(4) 1/n²

Answer 1

(4) 1/n²

The linear mass density of the chain is λ = M/L

The chain is pulled without acceleration. The length hanging will continously change. Let y be the length hanging at an instant of time.

The force required to pull the chain up the table = F = (Mass)g = (λy)g = λgy

Work done to pull a small length dy up the table = dW = λgy (-dy)

∴ Total work done to pull the entire