Question 314: Gravitation

Question

The gravitational potential energy required to raise a satellite of mass ' $m$ ' to height ' $h$ ' above the earth's surface is ' $\mathrm{E}_1$ '. Let the energy required to put this satellite into the orbit at the same height be ' $E_2$ '. If $M$ and $R$ are the mass and radius of the earth respectively then $E_1: E_2$ is

Accepted Answer

Energy required to raise a satellite upto a height h,
$$\mathrm{E}_1=\Delta \mathrm{U}=\frac{\mathrm{mgh}}{1+\frac{\mathrm{h}}{\mathrm{R}}}\quad	ext{.... (i)}$$
Energy required to put satellite into orbit,
$E_2=\frac{1}{2} m v_0^2=\frac{1}{2} m\left(\frac{G M}{r}ight) \quad$ as $v_0$ is orbital speed
$=\frac{1}{2} \mathrm{~m}\left(\frac{\mathrm{GM}}{\mathrm{R}+\mathrm{h}}ight)$
Dividing numerator and denominator by $\mathrm{R}^2$,
$$\begin{aligned}
E_2 & =\frac{1}{2} m\left(\frac{\frac{G M}{R^2}}{\frac{R+h}{R} 	imes \frac{1}{R}}ight) \
& =\frac{1}{2} m\left(\frac{g}{1+\frac{h}{R}}ight) R \quad \ldots\left(\because g=\frac{G M}{R^2}ight)\
E_2 & =\frac{m g R}{2\left(1+\frac{h}{R}ight)} \quad	ext{... (ii)}\
\frac{E_1}{E_2} & =\frac{2 h}{R}\quad	ext{.. [From (i) and (ii)]}
\end{aligned}$$

Answer

$\mathrm{h}: \mathrm{R}$

Answer

$\mathrm{h}: 2 \mathrm{R}$

Answer

$\mathrm{R}: \mathrm{h}$

Answer

$2 \mathrm{~h}: \mathrm{R}$

Gravitation

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