Suppose the gravitational force varies invers | vedclass

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Question

The necessary centripetal force required for a planet to move round the sun. $=$ gravitational force exerted on it

ie, $\quad \frac{\mathrm{mv}^{2}

Vedclass · Accepted Answer

${R^{\left( {\frac{{n + 1}}{2}} \right)}}$

Vedclass · Answer

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 $\quad \mathrm{v}=\left(\frac{\mathrm{GM}}{\mathrm{R}^{\mathrm{n}-1}}ight)^{1 / 2}$

Now, $T=\frac{2 \pi R}{v}=2 \pi R 	imes\left(\frac{R^{n-1}}{G M}ight)^{1 / 2}$

$=2 \pi\left(\frac{\mathrm{R}^{2} 	imes \mathrm{R}^{\mathrm{n}-1}}{\mathrm{GM}}ight)^{1 / 2}=2 \pi\left(\frac{\mathrm{R}^{(\mathrm{n}+1) / 2}}{(\mathrm{GM})^{1 / 2}}ight)$

or $\quad \mathrm{T} \propto \mathrm{R}^{(n+1) / 2}$

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

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