An object of mass $m$ lying at rest at depth of ' $R / 2$ ' from the Earth's surface, is to be made

Question

An object of mass $m$ lying at rest at depth of ' $R / 2$ ' from the Earth's surface, is to be made an Earth's satellite at a height of 'R/2' from the Earth's surface. The work required to do this is... (M-mass of the Earth, R-radius of the Earth and neglect the rotation of earth.)

Professor PC Thomas Classes · Answer

The potential energy at depth is given by
$$
U_1=V_1 m=-\frac{G M}{2 R^3}\left(3 R^2-\frac{R^2}{4}\right) m=-\frac{11 G M m}{8 R}
$$
The energy of satellite at a height $\mathrm{R} / 2$ is given by $E_2=-\frac{G M m}{3 R}$ ∴ The work required $=E_2-U_1=\frac{25 G M m}{24 R}$