Question

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

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

Answer 1

Answer: (B)

As, $\frac{GMm}{r^2}=\frac{m{v}^2}{r}=$ centripetal force
$\Rightarrow v^2=\frac{GM}{r}\Rightarrow T=\frac{2\pi r}{v}$
$\Rightarrow T^2=\frac{4{\pi }^2{r}^2}{v^2}$
Out putting the value of $v^2$ , we get
$T^2=\frac{4{\left(\pi \right)}^2{r}^2}{\left(\frac{GM}{r}\right)}\Rightarrow T^2=\frac{4{\left(\pi \right)}^2{r}^3}{GM}$
$\Rightarrow T^2=K{r}^3$
Here, $K$ is given by,
$\frac{4{\pi }^2}{GM}=K\Rightarrow GMK=4{\pi }^2$