Question

The angular momentum $\left(L\right)$ of earth revolving round the sun is proportional to $r^n$, where $r$ is the orbital radius of the earth. The value of $n$ is (assume the orbit to be circular)

• A. $\frac{1}{2}$
• B. $1$
• C. $-\frac{1}{2}$
• D. $2$

Answer 1

The correct option is A $\frac{1}{2}$

Angular momentum of earth is $L=mvr$
The centripetal force given by gravitational force of attraction is
$\Rightarrow \frac{m{v}^2}{r}=\frac{GMm}{r^2}$
$\Rightarrow \frac{m{L}^2}{m^2{r}^3}=\frac{GMm}{r^2}$
$\Rightarrow L^2=\frac{GM{m}^2{r}^3}{r^2}$
$\Rightarrow L=\sqrt{GM{m}^{2}r}$
$\Rightarrow L\propto r^{1/2}$
$\therefore n=\frac{1}{2}$