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. $40$%
• E. $67$%

Answer 1

Answer: (D)

$40$%

Key Concept 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, $R_1=3R,R_2=5R$
In the first condition,
$E_1=-\frac{GMm}{R_1}$
$E_1=-\frac{GMm}{3R}....\left(i\right)$
In the second condition,
$E_2=-\frac{GMm}{R_2}$
$E_2=-\frac{GMm}{5R}....\left(ii\right)$
Change in energy required to lift it,
$△E=E_2-E_1$
$\Rightarrow △E=-\frac{GMn}{5R}+\frac{GMm}{3R}$
$\Rightarrow △E=\frac{2}{15}\frac{GMm}{R}....\left(iii\right)$
On dividing Eq. (ii) by Eq. (i), we get
$\frac{△E}{E_1}=\frac{2}{15}\times \frac{3}{1}=\frac{2}{5}=0.4$
$\frac{△E}{E}\times 100$% $=0.4\times 100=40$%