Question

A planet of mass m is in an elliptical orbit about the sun $(m<<M_{\mathrm{sun}})$ with an orbital period. If A be the area of orbit, then its angular momentum would be:

• A. $\frac{2mA}{T}$
• B. $mAT$
• C. $\frac{mA}{2T}$
• D. $2mAT$

Answer 1

The correct option is A $\frac{2mA}{T}$

Given,

Swept area, $\frac{dA}{dt}=\frac{1}{2}\left(r\right)\left(r\frac{d\theta }{dt}\right)$$=\frac{1}{2}r{v}_{\theta }$

Angular momentum, $d\overrightarrow{L}=m(\overrightarrow{r}\times \overrightarrow{v})$

$d\overrightarrow{L}=m\frac{2dA}{dt}$

$L=\frac{2mA}{T}$

Hence, angular momentum is $\frac{2mA}{T}$