Question
Suppose the gravitational force varies inversely as the nth power of distance. Then the time period of a planet in circular orbit 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 :
$${R^{\left( {\frac{{n + 1}}{2}} \right)}}$$
Solution :
$$\eqalign{
& F = K{R^{ - n}} = MR{\omega ^2} \cr
& \Rightarrow {\omega ^2} = K{R^{ - \left( {n + 1} \right)}}\,\,\,or,\,\,\omega = K{R^{\frac{{ - \left( {n + 1} \right)}}{2}}} \cr
& \frac{{2\pi }}{T} \propto {R^{\frac{{ - \left( {n + 1} \right)}}{2}}} \cr
& \therefore T \propto {R^{\frac{{ + \left( {n + 1} \right)}}{2}}} \cr} $$