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

The Correct Option is A

The correct option is(A): \(\frac{4\pi^{2}}{GM_{S}}\).

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 \(\frac{4\pi^{2}}{GM_{S}}\)