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A small planet is revolving around a very massive star in a circular orbit of radius with a period of revolution . If the gravitational force between the planet and the star is proportional to , then will be proportional to
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According to the question, the gravitational force between the planet and the star is \(\mathrm{F} \propto \frac{1}{\mathrm{R}^{5 / 2}}\)
\(\therefore \mathrm{F}=\frac{\mathrm{GMm}}{\mathrm{R}^{5 / 2}}\)
Where M and M be masses of star and planet respectively.
For motion of planet in circular orbit,
\(\mathrm{mR} \omega^{2}=\frac{\mathrm{GMm}}{\mathrm{R}^{5 / 2}}\)
\(\mathrm{mR}\left(\frac{2 \pi}{\mathrm{T}}\right)^{2}=\frac{\mathrm{GMm}}{\mathrm{R}^{5 / 2}}\) \(\left(\therefore \omega=\frac{2 \pi}{\mathrm{T}}\right)\)
\(\frac{4 \pi^{2}}{\mathrm{~T}^{2}}=\frac{\mathrm{GM}}{\mathrm{R}^{7 / 2}} \Rightarrow \mathrm{T}^{2}=\frac{4 \pi^{2}}{\mathrm{GM}} \mathrm{R}^{7 / 2}\)
\(\mathrm{T}^{2} \propto \mathrm{R}^{7 / 2}\) or \(\mathrm{T}^{2} \propto \mathrm{R}^{7 / 4}\)
\(\therefore \mathrm{F}=\frac{\mathrm{GMm}}{\mathrm{R}^{5 / 2}}\)
Where M and M be masses of star and planet respectively.
For motion of planet in circular orbit,
\(\mathrm{mR} \omega^{2}=\frac{\mathrm{GMm}}{\mathrm{R}^{5 / 2}}\)
\(\mathrm{mR}\left(\frac{2 \pi}{\mathrm{T}}\right)^{2}=\frac{\mathrm{GMm}}{\mathrm{R}^{5 / 2}}\) \(\left(\therefore \omega=\frac{2 \pi}{\mathrm{T}}\right)\)
\(\frac{4 \pi^{2}}{\mathrm{~T}^{2}}=\frac{\mathrm{GM}}{\mathrm{R}^{7 / 2}} \Rightarrow \mathrm{T}^{2}=\frac{4 \pi^{2}}{\mathrm{GM}} \mathrm{R}^{7 / 2}\)
\(\mathrm{T}^{2} \propto \mathrm{R}^{7 / 2}\) or \(\mathrm{T}^{2} \propto \mathrm{R}^{7 / 4}\)
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