Question

A skylab of mass $m$ kg is first launched from the surface of the earth in a circular orbit of radius $2R$ (from the centre of the earth) and then it is shifted from this circular orbit to another circular orbit of radius $3R$. The minimum energy required to place the lab in the first orbit and to shift the lab from first orbit to the second orbit is

• A. $\frac{3}{4}mgR$, $\frac{mgR}{6}$
• B. $\frac{3}{4}mgR$, $\frac{mgR}{12}$
• C. $mgR$, $mgR$
• D. $2mgR$, $mgR$

Answer 1

The correct option is B $\frac{3}{4}mgR$, $\frac{mgR}{12}$

$GM=g{R}^2$

Total energy of satellite at surface of earth, $T.E_s=-\frac{GMm}{R}=-mgR$ (as K.E is zero)

Total energy of satellite at an orbit of radius $2R$ , $T.E_1=-\frac{GMm}{2\left(2R\right)}=-\frac{mgR}{4}$

Let $E_1$ be the energy required to put a satellite in an orbit of radius $2R$.

$E_1=(-\frac{mgR}{4})-(-mgR)$

$⟹E_1=\frac{3}{4}mgR$

Now as total energy of satellite at an orbit of radius $3R$ , $T.E_2=-\frac{GMm}{2\left(3R\right)}=-\frac{mgR}{6}$

Let $E_2$ be the energy required to shift a satellite from an orbit of radius $2R$ to $3R$.

$E_2=(-\frac{mgR}{6})-(-\frac{mgR}{4})$

$⟹E_1=\frac{mgR}{12}$