Question

If a planet of mass m is revolving around the sun in a circular orbit of radius r with time period T, then mass of the sun is

• A. $4{\pi }^2{r}^3/GT$
• B. $4{\pi }^2{r}^3/G{T}^2$
• C. $4{\pi }^{2}r/GT$
• D. $4{\pi }^2{r}^3/G^2{T}^2$

Answer 1

The correct option is A $4{\pi }^2{r}^3/GT$

According to LAW STATED BY KEPLER

$T^2\propto R^3$

It is known as Law of periods.

Let us consider a planet $P$ of mass $m$ moving with a velocity $v$ around the sun of mas $M$ in a circular orbit of radius $r$.

The gravitational force of attraction of the sun on the planet is,

$F=GMm/r^2$

The centripetal force is, $F=m{v}^2/r$,

euqating the two forces,

$m{v}^2/r=GMm/r^2$.

$v^2=GM/r----------\left(i\right)$

If $T$ be the period of revolution of the planet around the sun, then

$v=2\pi /T-------\left(ii\right)$

Substituting $\left(ii\right)$ in $\left(i\right)$

$4{\pi }^2{r}^2/T^2=GM/r$

$r^3/T^2=GM/4{\pi }^2$

$M=4{\pi }^2{r}^3/GT$