Question

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.

• A. $t\propto R^{(n+1)/2}$
• B. $T\propto R^{n/2}$
• C. $T\propto R^{3/2}$ for any n
• D. $T\propto R^{\frac{n}{2}+1}$

Answer 1

The correct option is A $t\propto R^{(n+1)/2}$

Central force = $F_c=\frac{m{v}^2}{R}=\frac{k}{R^n}$

$m{v}^2=\frac{k}{R^{n-1}}$

$v=\frac{k^{\prime }}{R^{\frac{n-1}{2}}}$

Now period of rotation $T=\frac{2\pi R}{v}$

$T\propto R\times R^{\frac{n-1}{2}}$

$T\propto R^{\frac{n-1}{2}+1}$

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

Hence option A is the correct one.