Question

suppose the gravitational force varies inversely as the nth power of distance . then the time period of a planet in circular orbit of radius R around the sun will be proportional to?

Answer 1

you mean, gravitational force is inversely proportional to Rⁿ , where R is distance between planet and sun, right ?

we can write it, $F_G=\frac{k}{R^n}$ where k is proportionality constant.

In circular orbit, gravitational force between planet and sun is balanced by centripetal force.

so, $F_G=m\omega^2R$, where m is mass of planet.

or, $\frac{k}{R^n}=m\omega^2R$

or, $\omega^2=\frac{k}{mR^{(n+1)}}$

or, $\omega=\sqrt{\frac{k}{mR^{(n+1)}}}$

or, $T=2\pi\sqrt{\frac{mR^{(n+1)}}{k}}$

hence, it is clear that time period is directly proportional to $R^{\frac{(n+1)}{2}}$

Answer 2

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