Question

Kepler's third law states that square of period of revolution $(T)$ 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 $F=\frac{G M m}{r^{2}}$ 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

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