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. $\frac{GM_{S}}{4\pi ^{2}}$
• C. $4\pi GM_{S}$
• D. $GM_{S}$

Answer 1

Answer: (A)

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