The gravitational potential energy required to raise a satellite of mass ' $m$ ' to height ' $h$ ' above the…

The gravitational potential energy required to raise a satellite of mass ' $m$ ' to height ' $h$ ' above the earth's surface is ' $E_1$ '. 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
  1. $\mathrm{h}: \mathrm{R}$
  2. $h: 2 R$
  3. $\mathrm{R}: \mathrm{h}$
  4. $2 \mathrm{~h}: \mathrm{R}$

Solution

Energy required to raise a satellite upto a height $h$, $\mathrm{E}_1=\Delta \mathrm{U}=\frac{\mathrm{mgh}}{1+\frac{\mathrm{h}}{\mathrm{R}}}$...(i)
Energy required to put satellite into orbit, $\mathrm{E}_2=\frac{1}{2} \mathrm{mv}_0^2=\frac{1}{2} \mathrm{~m}\left(\frac{\mathrm{GM}}{\mathrm{r}}\right) \quad$ as $\mathrm{v}_0$ is orbital speed $=\frac{1}{2} m\left(\frac{\mathrm{GM}}{\mathrm{R}+\mathrm{h}}\right)$
Dividing numerator and denominator by $\mathrm{R}^2$, $\begin{aligned} E_2 & =\frac{1}{2} m\left(\frac{\frac{G M}{R^2}}{\frac{R+h}{R} \times \frac{1}{R}}\right) \\ & =\frac{1}{2} m\left(\frac{g}{1+\frac{h}{R}}\right) R \\ E_2 & =\frac{m g R}{2\left(1+\frac{h}{R}\right)} \quad \ldots\left(\because g=\frac{G M}{R^2}\right) \\ \frac{E_1}{E_2} & =\frac{2 h}{R} \end{aligned}$ $\ldots[$ From(i) and (ii) $]$

Asked in: MHT CET 2024 (02 May Shift 2)

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