Question

The period of revolution t of a planet moving round the sun in a circular orbit depends upon the radius r of the orbit mass m of the sun and the gravitation constant g then t is proportion to

Answer 1

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Answer 2

$\large \bf \clubs \: To \: Prove :-$

• The square of Time Period of a Planet revolving around the Sun is directly proportional to cube of Radius of orbit .

$\large \bf \clubs \: Proof:-$

Here,

The Gravitational Force between Planet and the Sun provide enough Centripetal force for the Revolution of the Planet .

Hence Equating Gravitational Force and centripetal Force :

$\begin{gathered} \sf \dfrac{GMm}{ {r}^{2} } = mr { \omega}^{2} \\ \\ \sf\frac{GM}{ {r}^{3} } = { \bigg(\frac{2 \pi}{t} \bigg)}^{2} \: \: \: ( \because \: \omega = { \frac{2\pi}{t} )}\\ \\ \purple{ \Large :\longmapsto \underline {\boxed{{\bf {t}^{2} = \frac{4 {\pi}^{2} \: {r}^{3} }{GM} \propto {r}^{3} } }}}\end{gathered}$

Hence,

$\pink{ \Large :\longmapsto \underline {\boxed{{\bf {t}^{2} \propto {r}^{3} } }}}$