First slide

Gravitational force and laws

Question

Imagine a light planet revolving around a very massive star in a circular orbit of radius r with a period of revolution T. If the gravitational force of attraction between the planet and the star is proportional to $R^{- 3 / 2}$ , then T₂ is proportional to

Moderate

A

$R^{3}$
B

$R^{5 / 2}$
C

$R^{3 / 2}$
D

$R^{7 / 2}$

Solution

Gravitational force $= (\frac{GMm}{R^{3 / 2}})$ provides the necessary
centripetal force $(i.e., {mRω}^{2})$
So $\frac{GMm}{R^{3 / 2}} = {mRω}^{2} = mR {(\frac{2 π}{T})}^{2} = \frac{4 π^{2} mR}{T^{2}}$
or $T^{2} = \frac{4 π^{2} R^{3 / 4}}{GM}$
e,i, $T^{2} \propto R^{3 / 2}$

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