Question

The distance between the Sun and the Earth is $r$ and the Earth takes time $T$ to make one complete revolution around the Sun. Assuming the orbit of the Earth around the Sun to be circular, the mass of the Sun will be proportional to

• A. $\frac{r^2}{T}$
• B. $\frac{r^2}{T^2}$
• C. $\frac{r^3}{T^2}$
• D. $\frac{r^3}{T^3}$

Answer 1

The correct option is C $\frac{r^3}{T^2}$

Let $m$ and $M$ be the masses of the earth and the sun respectively and v the speed of the earth in circular orbit. To keep the earth in circular orbit, the gravitational force $\frac{GMm}{r^2}$ must balance the centripetal force $\frac{m{v}^2}{r}$, i.e.
$\frac{GMm}{r^2}=\frac{m{v}^2}{r}$
or $M=\frac{v^{2}r}{G}$
Also, $v=\frac{2\pi r}{T}$. Using this, we get $M=\frac{4{\pi }^2{r}^3}{T^{2}G}$ or $M\propto \frac{r^3}{T^2}$.
Hence the correct choice is (c).