Question

If the angular momentum of a planet of mass $m$, moving around the Sun in a circular orbit is $L$, about the centre of the Sun, its areal velocity is:

• A. $\frac{L}{m}$
• B. $\frac{4L}{m}$
• C. $\frac{L}{2m}$
• D. $\frac{2L}{m}$

Answer 1

The correct option is C $\frac{L}{2m}$

$\text{Areal velocity}=\frac{dA}{dt}$

For small angle $\left(d\theta \right)$ swept by the planet,
$\text{Area},dA=\frac{1}{2}\left(r\right)\left(rd\theta \right)$

$\Rightarrow \frac{dA}{dt}=\frac{1}{2}{r}^2\frac{d\theta }{dt}$

$\Rightarrow \frac{dA}{dt}=\frac{1}{2}{r}^2\omega$

Also, $L=mvr=m{r}^2\omega$

$\Rightarrow \frac{dA}{dt}=\frac{1}{2}\frac{L}{m}$

Hence, option $\left(C\right)$ is correct.