Question 449: Gravitation

Question

A skylab or mass $$m \mathrm{~kg}$$ 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

Accepted Answer

Energy of the skylab in the first orbit,
$$=-\frac{G M m}{2(2 R)}=-\frac{G M m}{4 R}$$
Total energy required to place the skylab into the orbit or radius $$2 R$$ from the surface of earth
$$\begin{aligned}
& =\frac{-G M m}{4 R}-\left(-\frac{G M m}{R}ight)=\frac{3 G M m}{4 R} \
& =\frac{3 g R^2 m}{4 R}=\frac{3}{4} m g R
\end{aligned}$$
Energy of the skylab in the second orbit
$$=-\frac{G M m}{6 R}$$
Energy needed to shift the skylab from the first orbit to the second orbit
$$=\frac{-G M m}{4 R}-\frac{G M m}{6 R}=\frac{G M m}{R} 	imes \frac{2}{24}=\frac{m g R}{12}$$

Answer

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

Answer

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

Answer

$$m g R, m g R$$

Answer

$$2 m g R, m g R$$

Gravitation

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