Question

In a planetary motion, the areal velocity of position vector of a planet depends on angular velocity $(\omega )$ and distance $(r)$ 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

Answer: (C)

Areal velocity, $\frac{dA}{dt}=\frac{L}{2m}$
where, $L=$ angular momentum
$\Rightarrow \frac{dA}{dt}=\frac{m{r}^2\omega }{2m}\propto \omega r^2$
$\left[\because L=m{r}^2\omega \right]$