Question

A planet of mass $m$ moves around the Sun along an elliptical path with a period of revolution $T$. During the motion, the planet's maximum and minimum distance from Sun is $R$ and $\frac{R}{3}$ respectively. If $T^{2}=\alpha R^{3}$, then the magnitude of constant $\alpha$ will be

• A. $\frac{10}{9} \cdot \frac{\pi}{ Gm }$
• B. $\frac{20}{27} \cdot \frac{\pi^{2}}{ Gm }$
• C. $\frac{32}{27} \cdot \frac{\pi^{2}}{ Gm }$
• D. $\frac{1}{18} \cdot \frac{\pi^{2}}{ Gm }$

Answer 1

Answer: (C)

Given, mass of planet $=m$
Maximum and minimum distance of the planet from Sun is $R$ and $\frac{R}{3}$, respectively.
$\therefore $ Semi-major axis of elliptical path of planet around the Sun,
$a=\frac{R+\frac{R}{3}}{2}=\frac{2 R}{3}$
$\therefore $ Time period of planet is given by
$T=2 \pi \sqrt{\frac{a^{3}}{G m}}=2 \pi \sqrt{\frac{\left(\frac{2 R}{3}\right)^{3}}{G m}}$
$T=2 \pi \sqrt{\frac{8 R^{3}}{27 \,G m}}$
Square on the both sides, we get
$\therefore T^{2} =4 \pi^{2} \cdot \frac{8 R^{3}}{27\, G m} $
$T^{2} =\frac{32 \pi^{2}}{27 \,G m} \cdot R^{3}=\alpha \,R^{3} $
$ \alpha =\frac{32 \pi^{2}}{27\, G m}$
Hence, the magnitude of constant $\alpha$ will be $\frac{32}{27} \cdot \frac{\pi^{2}}{G m}$