First slide

Motion of planets and satellites

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^5/2, then the square of the time period will be proportional to

Moderate

A

$r^{3}$
B

$r^{2}$
C

$r^{2 .5}$
D

$r^{3 .5}$

Solution

$\frac{{mv}^{2}}{r} = \frac{GMm}{r^{5 / 2}} [∵ F \propto \frac{1}{r^{5 / 2}}]$
$\begin{array}{l} or {rω}^{2} = \frac{GM}{r^{5 / 2}} \\ or r \frac{4 π^{2}}{T^{2}} = \frac{GM}{r^{5 / 2}} \\ or T^{2} \propto r^{7 / 2} or T^{2} \propto r^{3 .5} \end{array}$

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