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Two satellites $\mathrm{P}$ and $\mathrm{Q}$ are moving in different circular orbits around the Earth (radius $R$ ). The heights of $\mathrm{P}$ and $\mathrm{Q}$ from the Earth surface are $h_{\mathrm{p}}$ and $h_{\mathrm{Q}}$, respectively, where $h_{\mathrm{p}}=\mathrm{R} / 3$. The accelerations of $\mathrm{P}$ and $\mathrm{Q}$ due to Earth's gravity are $g_{\mathrm{p}}$ and $g_{\mathrm{Q}}$, respectively. If $g_{\mathrm{p}} / g_{\mathrm{Q}}=36 / 25$, what is the value of $h_Q$ ?
$3 R / 5$
$R / 6$
$6 R / 5$
$5 R / 6$
Solution
(image)
$\frac{g_p}{g_0}=\frac{\frac{G M}{r_p^2}}{\frac{G M}{r_Q^2}}=\left(\frac{r_0}{r_p}\right)^2$
$\frac{36}{25}=\left(\frac{r_0}{r_p}\right)^2$
$\frac{r_0}{r_p}=\frac{6}{5}$
$r_Q=\frac{6}{5} r_p$
$R+h_Q=\frac{6}{5}\left(R+\frac{R}{3}\right)$
$h_Q=\frac{24}{15} R-R=\frac{9}{15} R=\frac{3}{5} R$
