Question

A uniform chain has a mass m and length l. It is held on a frictionless table with two third of its length hanging over the edge. Find the work done in just pulling the hanging part back on the table.

Answer 1

The work done is required to be done in the lifting of the chain up will be equal to the change in the Potential Energy of the hanging part.
So, the expression is:
Potential\;Energy = Mass \times acceleration\;due\;to\;gravity \times height \\\
$P.E. = m \times g \times h \\\$

Complete answer:
Here, the given condition is
Length of the chain is l
Mass of the chain is m
Mass of the hanging part chain is $\left( {\dfrac{{2m}}{3}} \right)$
So, according to the given condition
The change in the potential energy will be the same as bringing the centre of the mass of the chain up on the table.
So, the centre of the mass of the chain lies at the $\left( {\dfrac{{2l}}{6}} \right) = \left( {\dfrac{l}{3}} \right)$ below the table
So, the change in the potential energy
E = m \times g \times h \\\
E = \dfrac{{2m}}{3} \times g \times \dfrac{l}{3} \\\
$E = \dfrac{{2mgl}}{9} \\\$
So, the work done in just pulling the hanging part back on the table is $E = \left( {\dfrac{{2mgl}}{9}} \right)$

Note:
The Potential energy is that energy which is stored or conserved in an object or a substance. The stored energy is based on the position, and the arrangement or state of the object or that substance.
The Potential energy is that energy of an object, because of its position relative to the other objects. It is known as potential, because it has the potential to be converted into other forms of energy, like kinetic energy.
It follows the law of conservation of energy. According to the law of conservation of the energy, the energy can neither be created nor be destroyed but it can convert from one form to another form.