A uniform chain of mass `m` and length `l` hanges on a thread and touches the surface of a table by its lower end. Find the force exerted by the table on the chain when half of its length has fallen on the table. The fallen part does not form heap.

Options

A. `(mg)/(2)`
B. `(3mg)/(2)`
C. `mg`
D. None of these

Answer

B

Explanation

Ans.(2)
At any instant of time let velocity of part of chain which is just coming in contact with surface `=v` and `dx=` length of chain reaches on surface in time `dt` then momentum carried by this part dp`=(dm)v`
`:.` rate of change of momentum of chain
`(dp)/(dt)=(dm)/(dt)v=(lamdadx)/(dt)v=lamdav^(2)=((m)/(l))v^(2)`
as `v=sqrt(2g((l)/(2)))=sqrt(gl)` so `(dp)/(dt)=(m)/(l)(gl)=mg`
weight of part on the surface `=(mg)/(2)`
`:.` net force by chain on surface `=mg+(mg)/(2)=(3)/(2)mg`