Question

A planet is revolving around the sun in a circular orbit with a radius r. The time period is T. If the force between the planet and star is proportional to r^-3/2 , then the square of time period is proportional to [AIMS 2018]

Moderate

Solution

As force, $F = \frac{G M m}{r^{3 / 2}} = m ω^{2} r$
$\Rightarrow \frac{G M m}{r^{3 / 2}} = \frac{4 π^{2} m r}{T^{2}}$ $(∵ ω = \frac{2 π}{T})$
$\Rightarrow T^{2} = (\frac{4 π^{2}}{G M}) \cdot r^{5 / 2} \Rightarrow T^{2} \propto r^{5 / 2}$

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