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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
${R^{\left( {\frac{{n + 1}}{2}} \right)}}$
${R^{\left( {\frac{{n - 1}}{2}} \right)}}$
$R^n$
${R^{\left( {\frac{{n - 2}}{2}} \right)}}$
Solution
The necessary centripetal force required for a planet to move round the sun. $=$ gravitational force exerted on it
ie, $\quad \frac{\mathrm{mv}^{2}}{\mathrm{R}}=\frac{\mathrm{GMm}}{\mathrm{R}^{\mathrm{n}}}$
or $\quad \mathrm{v}=\left(\frac{\mathrm{GM}}{\mathrm{R}^{\mathrm{n}-1}}\right)^{1 / 2}$
Now, $T=\frac{2 \pi R}{v}=2 \pi R \times\left(\frac{R^{n-1}}{G M}\right)^{1 / 2}$
$=2 \pi\left(\frac{\mathrm{R}^{2} \times \mathrm{R}^{\mathrm{n}-1}}{\mathrm{GM}}\right)^{1 / 2}=2 \pi\left(\frac{\mathrm{R}^{(\mathrm{n}+1) / 2}}{(\mathrm{GM})^{1 / 2}}\right)$
or $\quad \mathrm{T} \propto \mathrm{R}^{(n+1) / 2}$
