Question

A satellite revolves around the earth of radius $R$ in a circular orbit of radius $3R.$ The percentage increase in energy required to lift it to an orbit of radius $5R$ is

• A. 10%
• B. 20%
• C. 30%
• D. 40%

Answer 1

Answer: (D)

40%

Answer 2

Gravitational potential energy of a body at a point is defined as the amount of work done in bringing the given body from infinity to that point against gravitational force
i.e. Gravitational potential energy = Gravitational potential $\times$ mass of the body
Given, $R_{1}=3 R, R_{2}=5 R$
In the first condition,
$E_{1} =-\frac{G M m}{R_{1}}$
$E_{1} =-\frac{G M m}{3 R}$ ...(i)
In the second condition,
$E_{2}=-\frac{G M m}{R_{2}}$
$E_{2}=-\frac{G M m}{5 R}$ ...(ii)
Change in energy required to lift it,
$\Delta E=E_{2}-E_{1}$
$\Rightarrow \Delta E=-\frac{G M m}{5 R}+\frac{G M m}{3 R}$
$\Rightarrow \Delta E=\frac{2}{15} \frac{G M m}{R}$ ...(iii)
On dividing E (ii) by E (i), we get
$\frac{\Delta E}{E_{1}} =\frac{2}{15} \times \frac{3}{1}=\frac{2}{5}=0.4$
$\frac{\Delta E}{E} \times 100 \% =0.4 \times 100=40 \%$