# Question #5073f

Dec 6, 2017

$W . {D}_{\text{when} ' n = 4} \Rightarrow \frac{M g L}{32}$

#### Explanation:

Lets take a general situation:

Let a chain of mass M and length L is held on a frictionless table in such a way that $\frac{1}{n}$th part is hanging below the edge.

The portion hanging from the table is $\frac{L}{n}$

Required work done$=$change in potential energy of chain

Now,let Potential energy (U)$=$0 at the table level.

Potential energy initial$\Rightarrow$mgh { h is the length of the
hanging portion from centre of mass]

For regularly shaped uniform bodies, P.E change can be calculated by considering their mass to be centered at the geometrical point.

$h = \frac{L}{2 n}$

Mass of $L$ length is M

Mass of $\frac{L}{n}$ length is $\frac{M}{L} \times \frac{L}{n} = \frac{M}{n}$

${U}_{i} = - m g h = - m g \left\{\frac{L}{2 n}\right\} = - \left\{\frac{M}{n}\right\} g \left\{\frac{L}{2 n}\right\} = - \frac{M g L}{2 {n}^{2}}$

${U}_{f} = 0$

Therefore required work done$= {U}_{f} - {U}_{i} = \textcolor{red}{\frac{M g L}{2 {n}^{2}}}$

$W . {D}_{\text{when} ' n = 4} \Rightarrow \frac{M g L}{32}$