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

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) (3/4) m g R, (m g R/6) (B) (3/4) m g R, (m g R/12) (C) m g R, m g R (D) 2 mgR, mgR

Tardigrade Admin · Accepted Answer

Energy of the skylab in the first orbit is

- \frac{GM m}{2 ( 2 R )} = - \frac{GM m}{4 R}

Total energy required to place the skylab into the orbit of radius

2 R

from the surface of earth is

- \frac{GM m}{4 R} - (- \frac{GM m}{R})

= \frac{3 GM 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

= - (GM m) /6 R

.
Energy needed to shift the skylab from the first orbit to the second orbit is

- \frac{GM m}{4 R} - \frac{GM m}{6 R}

= \frac{GM m}{R} \times \frac{2}{24} = \frac{m g R}{12}

$\frac{3}{4} m g R, \frac{m g R}{6}$

$\frac{3}{4} m g R, \frac{m g R}{12}$

$m g R, m g R$

$2 m g R, m g R$

43​mgR,6mgR​

43​mgR,12mgR​

mgR,mgR

2mgR,mgR

$\frac{3}{4} m g R, \frac{m g R}{6}$

$\frac{3}{4} m g R, \frac{m g R}{12}$

$m g R, m g R$

$2 m g R, m g R$