Question

A chain of length $L$ and mass $M$ is held on a frictionless table with ${\left(\frac{1}{n}\right)}^{th}$ part hanging over the edge. Work done in pulling the chain is directly proportional to:

• A. $\sqrt{n}$
• B. $n$
• C. $n^{-3}$
• D. $n^{-2}$

Answer 1

The correct option is D $n^{-2}$

The length of the chain is L and mass is M. Its $\frac{1}{n}th$ part is hanging.

So, mass of $\frac{L}{n}$ length chain will be , $\frac{M}{n}$

This mass will be at mid-point of the length.

So, effective height from the point will be, $h=\frac{L}{2n}.$

Here work done will be equal to its potential energy.

Thus $P.E.=m\times g\times h$

i.e. $P.E.=\frac{M}{n}\times g\times \frac{L}{2n}$

Thus work done will be $=\frac{mgl}{2n^2}$

So, work done is directly proportional to $n^{-2}$, as other terms are constant.

Option D is correct.