Gravitation 4 Question 4
5. Imagine a light planet revolving around a very massive star in a circular orbit of radius $R$ with a period of revolution T. If the gravitational force of attraction between the planet and the star is proportional to $R^{-5 / 2}$, then
$(1989,2 \mathrm{M})$
(a) $T^{2}$ is proportional to $R^{2}$
(b) $T^{2}$ is proportional to $R^{7 / 2}$
(c) $T^{2}$ is proportional to $R^{3 / 2}$
(d) $T^{2}$ is proportional to $R^{3.75}$
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Answer:
Correct Answer: 5. (b)
Solution:
- $\frac{m v^{2}}{R} \propto R^{-5 / 2}$
$$ \begin{aligned} & \therefore \quad v \propto R^{-3 / 4} \\ & \text { Now, } \quad T=\frac{2 \pi R}{v} \quad \text { or } T^{2} \propto \Big(\frac{R}{v}\Big)^{2} \\ & \text { or } \\ & T^{2} \propto \Big({\frac{R}{R^{-3 / 4}}}\Big)^{2} \text { or } T^{2} \propto R^{7 / 2} \end{aligned} $$
