Question

In a planetary motion, the areal velocity of the position vector of a planet depends on angular velocity $\left(\omega \right)$ and distance $\left(r\right)$ of the planet from the sun. The correct relation for areal velocity $\left(\frac{dA}{dt}\right)$ is

• A. $\frac{dA}{dt}\propto \omega r$
• B. $\frac{dA}{dt}\propto {\omega }^{2}r$
• C. $\frac{dA}{dt}\propto \omega r^2$
• D. $\frac{dA}{dt}\propto \sqrt{\omega r}$

Answer 1

The correct option is C $\frac{dA}{dt}\propto \omega r^2$

Areal velocity, $\frac{dA}{dt}=\frac{L}{2m}$
where, $L$ is the angular momentum.

$\Rightarrow \frac{dA}{dt}=\frac{m{r}^2\omega }{2m}$ $[\because L=m{r}^2\omega ]$
$⟹\frac{dA}{dt}\propto r^2\omega$.