Question

A small body is initially at a distance $r$ from the centre of earth. $r$ is greater than the radius of the earth. If it takes $W$ $\text{joules}$ of work to move the body from this position to another position at a distance $2r$ measured from the centre of earth, then how many $\text{joules}$ would be required to move it from this position to a new position at a distance of $3r$ from the centre of the earth.

• A. $\frac{W}{5}$
• B. $\frac{W}{3}$
• C. $\frac{W}{2}$
• D. $\frac{W}{6}$

Answer 1

The correct option is B $\frac{W}{3}$

Let, $M$ is the mass of earth and $R$ is its radius.

Work done to move the body in the gravitational field is equal to change in the potential energy of the body.

$\text{Case I}:$ Work done in moving the mass from the distance $r$ to $2r$ is given $W$. Hence,

$W=U_f-U_i$

Where, $U_i$ and $U_f$ are the initial and final potential energies of the particle.

$W=\frac{-GMm}{2r}-\frac{-GMm}{r}$

$\Rightarrow W=\frac{GMm}{2r}....\left(1\right)$

$\text{Case I}:$ Let, the work done in moving the mass from the distance $2r$ to $3r$ is $W^{\prime }$. Now,

$W^{\prime }=\frac{-GMm}{3r}-\frac{-GMm}{2r}$

$\Rightarrow W^{\prime }=\frac{GMm}{6r}....\left(2\right)$

Comparing equation $\left(1\right)$ and $\left(2\right)$, we get

$W^{\prime }=\frac{2W}{6}$

$\therefore W^{\prime }=\frac{W}{3}$

Hence, option (b) is the correct answer.

$\text{Note:}$ A uniform spherical mass can be considered to be a point mass concentrated at its center to calculate gravitational potential energy, if the other body is outside the spherical mass.