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. $\frac{4{\pi }^2}{G{M}_S}$
• B. $\frac{G{M}_S}{4{\pi }^2}$
• C. $4\pi G{M}_S$
• D. $G{M}_S$

Answer 1

Answer: (A)

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