Suppose the gravitational force varies inversely as the n th power of the distance. Thus the time period of a planet moving in a circular orbit of radius r around the sun will be proportional to

Options

A. `r^((n+1)/2)`
B. `r^n`
C. `r^((n-1)/2)`
D. `r^(-n)`

Answer

A

Explanation

Gravitational force =`k/(r_n)` [k is constant]
This gravitational force gives rise to centripetal force. If the mass of the planet is m and its velocity is v,centripetal force `=(mv^2)/r`.
So, `(mv^2)/r =k/(r^n) or, v=sqrt((k)/(mr^(n-1)))`
Time period ,
`T=(2pir)/v=2pi sqrt(m/k)sqrt(r^2*r^(n-1)) ="constant" xxr^((n+1)/2)`
`therefore T propr^((n+1)/2)`