Question

Kepler's third law states that square of period of revolution $\left(T\right)$ of a planet around the sun, is proportional to third power of average distance $r$ between sun and planet
i.e $T^2=K{r}^3$ here $K$ is constant.
If the masses of sun and planet are $M$ and $m$ respectively than as per Newton's law of gravitation force of attraction between them is
$F=\frac{GMm}{r^2}$, here $G$ is gravitational constant The relation between $G$ and $K$ is described as

• A. $K=G$
• B. $K=\frac{1}{G}$
• C. $GK=4{\pi }^2$
• D. $GMK=4{\pi }^2$

Answer 1

The correct option is D $GMK=4{\pi }^2$

The gravitational force is providing centripetal force to the planets oribital circular motion

$\frac{GMm}{r^2}=\frac{m{v}^2}{r}$

$\Rightarrow v=\sqrt{\frac{GM}{r}}$

The time period for planet in one revolution will be $T=\frac{2\pi r}{v}=\frac{2\pi r}{\sqrt{\frac{GM}{r}}}$

squaring each side

$T^2=\frac{4{\pi }^2{r}^3}{GM}$

comparing with given equation $T^2=K{r}^3$

$K=\frac{4{\pi }^2}{GM}$

$KGM=4{\pi }^2$

hence, correct answer is option D