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) $T \propto R^{\frac{n+1}{2}}$
Show Answer
Answer:
Correct Answer: 2. (d)
Solution:
$$ \frac{d A}{d t}=\frac{L}{2 m} $$
$\quad F \propto \frac{1}{R^{n}}$
$$ \begin{array}{r} \frac{m v^{2}}{R} \propto \frac{1}{R^{n}} \quad \text { or } \quad v \propto \frac{I}{R^{\frac{n-1}{2}}} \\ T=\frac{2 \pi R}{v} \Rightarrow T \propto \frac{R}{v} \\ \Rightarrow \quad T \propto R^{1+\frac{n-1}{2}} \text { or } T \propto R^{\frac{n+1}{2}} \end{array} $$
