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Question

Physics Question on Newtons law of gravitation

Suppose the gravitational force varies inversely as the nth power of distance. Then the time period of a planet in circular orbit of radius RR around the sun will be proportional to

A

R(n+1)/2R^{(n+1) /2}

B

R(n−1)/2R^{(n - 1) /2}

C

RnR^{n}

D

R(n−2)/2R^{(n -2) /2}

Answer

R(n+1)/2R^{(n+1) /2}

Explanation

Solution

The necessary centripetal force required for a planet to move round the sun
= gravitational force exerted on it
mv2R=GMemRn\frac{m v^{2}}{R}=\frac{G M_{e} m}{R^{n}}
or v=(GMRn−1)12v=\left(\frac{G M}{R^{n-1}}\right)^{\frac{1}{2}}
as T=2πRv=2πR×(Rn−1GM)12T=\frac{2 \pi R}{v}=2 \pi R \times\left(\frac{R^{n-1}}{G M}\right)^{\frac{1}{2}}
T=2π[R(n+1)2(GMe)12]T=2 \pi\left[\frac{R^{\frac{(n+1)}{2}}}{\left(G M_{e}\right)^{\frac{1}{2}}}\right]
∴T∝R(n+1)2\therefore T \propto R^{(n+1) 2}