Question

What is the slope of the graph plotted between $T^{2}$ and $r^{3}$ for a planet, (where $M_{S}$ is the mass of the sun)

• A. $\frac{4 \pi ^{2}}{G M_{S}}$
• B. $\frac{G M_{S}}{4 \pi }$
• C. $4\pi GM_{S}$
• D. $GM_{S}$

Answer 1

Answer: (A)

Time period of a revolution of a planet,
$T=\frac{2 \pi r}{v}=\frac{2 \pi r}{\sqrt{\frac{G M_{S}}{r}}}=\frac{2 \pi r^{3 / 2}}{\sqrt{G M_{S}}}$
Where $M_{S}$ is the mass of the sun
Squaring both side, we get
$T^{2}=\frac{4 \pi ^{2} r^{3}}{G M_{S}}$
The graph between T2 and r3 is a straight line whose slope is $\frac{4 \pi ^{2}}{G M_{s}}$