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 $\left(\frac{d A}{d t}\right)$ is

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

Answer 1

Answer: (C)

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