Question

In order to shift a body of mass $m$ from a circular orbit of radius $3\text{R}$ to a higher orbit of radius $5\text{R}$ around the earth, the work done is

• A. $\frac{3GMm}{5R}$
  
• B. $\frac{GMm}{2R}$
  
• C. $\frac{GMm}{15R}$
  
• D. $\frac{2GMm}{15R}$
  

Answer 1

The correct option is C $\frac{GMm}{15R}$


The total mechanical energy of a satellite in a circular orbit of radius $r$ is given by $$E=-\frac{GMm}{2r}$$ The work done to move a satellite from one orbit to another is the difference between the total mechanical energy in the two orbits.

Total mechanical energy of the satellite at $r=3R$,

$E_1=-\frac{GMm}{2\left(3R\right)}=-\frac{GMm}{6R}$

Total mechanical energy of the satellite at $r=5R$,

$E_2=-\frac{GMm}{2\left(5R\right)}=-\frac{GMm}{10R}$

So, workdone required,

$W=E_2-E_1=-\frac{GMm}{10R}+\frac{GMm}{6R}$

$\Rightarrow W=\frac{GMm}{15R}$

Hence, option (c) is the correct answer.

Key Concept: The workdone required to shift a satellite from one orbit to the other is equal to the difference between the total mechanical energy of the two orbits.