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Question

A satellite is rotating around a planet in the orbi of radius r with time period T. If gravitational force changes according to $r^{5 / 2}$ the $T^{2}$ will be

A

$\propto r^{3}$

B

$\propto r^{7 / 2}$

C

$\propto r^{9 / 2}$

D

$\propto r^{3 / 2}$

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Solution

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Correct option is B)

$\frac{G M m}{r ^{5 / 2}} = \frac{m v ^{2}}{r}$
$v^{2} = \frac{G M}{r ^{3 / 2}}$
$T^{2} = (\frac{2 π r}{v})^{2}$
$= \frac{4 π ^{2} r ^{2}}{G M / r ^{3 / 2}} = \frac{4 π ^{2} r ^{7 / 2}}{G m}$
$T^{2} \propto r^{7 / 2}$

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