Question

Suppose the gravitational force varies inversely as the $nth$ power of distance. Then the time period of a planet in circular orbits of radius '$R$' around the sun will be proportional to

• A. $R^n$
• B. $R^{\left(\frac{n-1}{2}\right)}$
• C. $R^{\left(\frac{n+1}{2}\right)}$
• D. $R^{\left(\frac{n-2}{2}\right)}$

Answer 1

The correct option is C $R^{\left(\frac{n+1}{2}\right)}$

$F=K{R}^{-n}=MR{\omega }^2\Rightarrow {\omega }^2=K{R}^{-(n+1)}$
or $\omega =K^{\prime }{R}^{\frac{-(n+1)}{2}}$ [where $K^{\prime }=K^{1/2}$, a constant]
$\frac{2\pi }{T}\propto R^{\frac{-(n+1)}{2}}$
$\therefore T\propto R^{\frac{(n+1)}{2}}$