Question

If a graph is plotted between $T^2$ and $r^3$ for a planet, then its slope will be (where $M_s$ is the mass of the sun)

• A. $\frac{{4\pi }^2}{{GM}_s}$
• B. $\frac{{GM}_s}{4\pi }$
• C. ${4\pi GM}_s$
• D. ${GM}_s$

Answer 1

The correct option is A $\frac{{4\pi }^2}{{GM}_s}$

Time period of revolution of planet,
$T=\frac{2\pi r}{v}=\frac{2\pi r}{\sqrt{\frac{{GM}_s}{r}}}=\frac{2\pi r3/2}{{GM}_s}$
where $M_s$ is the mass of the sun Squaring both sides, we get
$T^2=\frac{{4\pi }^2{r}^3}{{GM}_s}$
The graph between $T^2$ and $r^3$ is a straight line whose slope is $\frac{4{\pi }^2}{{GM}_s}$