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,
$a=\frac{R+\frac{R}{3}}{2}=\frac{2R}{3}$
$\therefore$ Time period of planet is given by
$T=2\pi \sqrt{\frac{a^3}{Gm}}=2\pi \sqrt{\frac{{\left(\frac{2R}{3}\right)}^3}{Gm}}$
$T=2\pi \sqrt{\frac{8R^3}{27Gm}}$
Square on the both sides, we get
$\therefore T^2=4{\pi }^2\cdot \frac{8R^3}{27Gm}$
$T^2=\frac{32{\pi }^2}{27Gm}\cdot R^3=\alpha R^3$
$\alpha =\frac{32{\pi }^2}{27Gm}$
Hence, the magnitude of constant $\alpha$ will be $\frac{32}{27}\cdot \frac{{\pi }^2}{Gm}$