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 then as per Newton's law of gravitation force of attraction between them is $F=\frac{GMm}{2}$, here $G$ is gravitational constant. The relation between $G$ and $K$ is described as

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

Answer 1

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

As we know, orbital speed, $V_{orb}=\sqrt{\frac{GM}{r}}$
Time period $T=\frac{2\pi r}{v_{orb}}=\frac{2\pi r}{\sqrt{GM}}\sqrt{r}$
Squaring both sides,
$T^2={\left(\frac{2\pi r\sqrt{r}}{\sqrt{GM}}\right)}^2=\frac{4{\pi }^2}{GM}\cdot r^3$
$\Rightarrow \frac{T^2}{r^3}=\frac{4{\pi }^2}{GM}=K$
$\Rightarrow GMK=4{\pi }^2$.