Question 240: Gravitation

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

Accepted Answer

For moving a planet around the sun in the circular orbit,
The necessary centripetal force = Gravitational force exerted on it
$$	herefore$$ $${{m{v^2}} \over r} = {{GMm} \over {{R^n}}}$$
$$ \Rightarrow $$ $$v = \sqrt {{{GM} \over {{R^{n - 1}}}}} $$
We know, $$T = {{2\pi R} \over v}$$
$$ = 2\pi R 	imes \sqrt {{{{R^{n - 1}}} \over {GM}}} $$
= $$2\pi  	imes \sqrt {{{{R^2} 	imes {R^{n - 1}}} \over {GM}}} $$
= $$2\pi  	imes {{{R^{{{n + 1} \over 2}}}} \over {\sqrt {GM} }}$$
$$	herefore$$ $$T \propto {R^{{{  \left( {n + 1} ight)} \over 2}}}$$

Answer

$${R^n}$$

Answer

$${R^{\left( {{{n - 1} \over 2}} \right)}}$$

Answer

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

Answer

$${R^{\left( {{{n - 2} \over 2}} \right)}}$$

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 $R$ around the sun will be proportional to

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