Question

Suppose gravitational pull varies inversely as nth power of the distance. Show that the time period of a planet in circular orbit of radius R around the sun will be proportinal to `R^((n+1)//2)`

Answer 1

As gravitational pull of sun on the planet provides necessary centripetal force to the planet, therefore
`(GMm)/R^(n)=mR(2pi/T)^(2)=(4pi^(2)mR)/T^(2)` or `T^(2)=(4pi^(2)mR^(n+1))/(GMm)`
`T=(2pi)/sqrt(GM)R^((n+1)//2)`
Hence `TpropR^((n+1)//2)`.