A chain is placed on a frictionless table with one fourth of it hanging over the edge. If the length of the chain is 2m and its mass is 4kg, the energy need to be spent to pull it back to the table is

The correct option is 2.5 J.

To find the energy needed to pull the hanging part of the chain back onto the table, we need to calculate the work done against gravity. The work done is equal to the increase in the potential energy of the hanging part of the chain as its center of mass is raised to the level of the table.

Let us define the parameters given in the problem:
Mass of the chain, $M=4\text{kg}$
Total length of the chain, $L=2\text{m}$
Acceleration due to gravity, $g=10{\text{m/s}}^2$ (standard approximation)

First, let's find the mass and length of the hanging portion of the chain:
Fraction of the chain hanging, $f=\frac{1}{4}$
Length of the hanging part, $h=\frac{L}{4}=\frac{2\text{m}}{4}=0.5\text{m}$
Mass of the hanging part, $m=\frac{M}{4}=\frac{4\text{kg}}{4}=1\text{kg}$

For a uniform chain, the center of mass of the hanging portion lies exactly at its midpoint. Since the length of the hanging part is $h$, its center of mass is at a distance of $\frac{h}{2}$ below the table surface:
Distance of center of mass below the table, $d=\frac{h}{2}=\frac{0.5\text{m}}{2}=0.25\text{m}$

To pull the hanging part back onto the table, we must lift its center of mass by a distance of $d$. The work done $W$ required to do this is:
$W=m\cdot g\cdot d$

Substituting the values into the equation:
$W=1\text{kg}\cdot 10{\text{m/s}}^2\cdot 0.25\text{m}$
$W=2.5\text{J}$

Therefore, the energy that needs to be spent to pull the hanging part of the chain back onto the table is 2.5 J.