Question

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

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

Answer 1

Answer: (B)

Energy of the skylab in the first orbit is
$-\frac{G M m}{2(2 R)}=-\frac{G M m}{4 R}$
Total energy required to place the skylab into the orbit of radius $2 R$ from the surface of earth is
$-\frac{G M m}{4 R}-\left(-\frac{G M m}{R}\right)$
$=\frac{3 G M m}{4 R}$
$=\frac{3 g R^{2} m}{4 R}=\frac{3}{4} m g R$
Energy of the skylab in the second orbit $=-(G M m) / 6 R$.
Energy needed to shift the skylab from the first orbit to the second orbit is
$-\frac{G M m}{4 R}-\frac{G M m}{6 R}$
$=\frac{G M m}{R} \times \frac{2}{24}=\frac{m g R}{12}$