Gravitation 4 Question 1

2. A particle is moving with a uniform speed in a circular orbit of radius $R$ in a central force inversely proportional to the $n$th power of $R$. If the period of rotation of the particle is $T$, then

(2018 Main)

(a) $T \propto R^{n / 2}$

(b) $T \propto R^{3 / 2}$ for any value of $n$

(c) $T \propto R^{\frac{n}{2}+1}$

(d) $T \propto R^{\frac{n+1}{2}}$

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Answer:

Correct Answer: 2. (d)

Solution:

$$ \frac{d A}{d t}=\frac{L}{2 m} $$

$\quad F \propto \frac{1}{R^{n}}$

$$ \begin{array}{r} \frac{m v^{2}}{R} \propto \frac{1}{R^{n}} \quad \text { or } \quad v \propto \frac{I}{R^{\frac{n-1}{2}}} \\ T=\frac{2 \pi R}{v} \Rightarrow T \propto \frac{R}{v} \\ \Rightarrow \quad T \propto R^{1+\frac{n-1}{2}} \text { or } T \propto R^{\frac{n+1}{2}} \end{array} $$



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