Question

Kepler’s second law regarding constancy of aerial velocity of a planet is a consequence of the law of conservation of

• A. Energy
• B. Angular momentum
• C. Linear momentum
• D. None of these

Answer 1

The correct option is B Angular momentum

From the Kepler's second law :$\frac{dA}{dt}=const.$

and for the area of CDS:
calculation of area:
$dA=\frac{1}{2}\times r\times d\theta \times rdA=\frac{1}{2}{r}^{2}d\theta$

Area changes w.r.t. time,
$\frac{dA}{dt}=\frac{1}{2}{r}^2\frac{d\theta }{dt}$
where, $\frac{d\theta }{dt}=\omega$
so, $\frac{dA}{dt}=\frac{1}{2}{r}^2\omega$= Constant

For option A:
Kinetic energy,
$KE=\frac{1}{2}m{v}^2=\frac{1}{2}m{r}^2{\omega }^2$
in the above expression $\frac{1}{2}m{r}^2\omega =constant$
where as rest of $\omega$ still changes the physical quantity.
Hence, this option is wrong.

For option B:
Angular Momentum $L=mvr=m\left(r\omega \right)r=m{r}^2\omega$
here, all the quatities are constant, hence angular momentum supports the Kepler's secound law.

For option C:
Linear Momentum $p=mv=mr\omega$
here, $mr\omega \times \frac{r}{r}\frac{m{r}^2\omega }{r}$
In the above expression $m{r}^2\omega$ are constant, but $r$ is not constant, which can change the physical limits.
Hence option C is wrong.

So, only option B is correct.