Question

Suppose the gravitational force varies inversely as the $n^{th}$ power of distance. Then the time period of a planet in circular orbit of radius $R$ around the sun will be proportional to

• A. $T\propto R^{\frac{n+1}{2}}$
• B. $T\propto R^{\frac{n-1}{2}}$
  
• C. $T\propto R^{\frac{n+1}{3}}$
  
• D. $T\propto R^{\frac{n-1}{3}}$
  

Answer 1

The correct option is A $T\propto R^{\frac{n+1}{2}}$

From Newton's second law,

$m{\omega }^{2}R=\frac{G}{R^n}$ where G is not necessarily gravitational constant but some constant.

$\omega =\sqrt{\frac{K}{R^{n+1}}}=\frac{2\pi }{T}$

$T=2\pi \sqrt{\frac{R^{n+2}}{K}}$

Thus, $T\alpha R^{\frac{n+1}{2}}$