Energy required to move a body of mass m from an orbit of radius 2R to 3R is

The correct option is GMm/6R.

To find the energy required to move a body of mass m from an orbit of radius 2R to 3R, we calculate the change in its gravitational potential energy.

The gravitational potential energy $U$ of a body of mass $m$ at a distance $r$ from the center of a planet of mass $M$ is given by the formula:
$U=-\frac{GMm}{r}$
where $G$ is the universal gravitational constant.

Initially, the body is at a distance of radius $r_1=2R$. Therefore, its initial potential energy $U_i$ is:
$U_i=-\frac{GMm}{2R}$

Finally, the body is moved to a distance of radius $r_2=3R$. Its final potential energy $U_f$ is:
$U_f=-\frac{GMm}{3R}$

The work done or the energy required ($W$) to move the body is equal to the change in its potential energy:
$W=U_f-U_i$

Substituting the expressions for $U_i$ and $U_f$ into this equation:
$W=\left(-\frac{GMm}{3R}\right)-\left(-\frac{GMm}{2R}\right)$

Simplifying the expression:
$W=\frac{GMm}{2R}-\frac{GMm}{3R}$

Factoring out the common terms:
$W=\frac{GMm}{R}\left(\frac{1}{2}-\frac{1}{3}\right)$

Calculating the difference inside the parentheses:
$W=\frac{GMm}{R}\left(\frac{3-2}{6}\right)$
$W=\frac{GMm}{6R}$

Therefore, the energy required to move the body is GMm/6R.