Q. A planet of mass $m$ moves around the sun of mass $M$ in an elliptical orbit. The maximum and minimum distance of the planet from the sun are $r^{1}$ and $r^{2}$, respectively. The time period of the planet is proportional to
A) $(r_{1} + r_{2})$
B) ${(r_{1} + r_{2})}^{1/2}$
C) $(r_{1} - r_{2})^{3/2}$
D) ${(r}_{1} + r_{2})^{3/2}$
Explanation
(d) According to Kepler’s third law, $T^{2}$ is proportional to cube of semi-major axis of the elliptical orbit.
$Semi–major\ axis = \frac{{r_{1} + r}_{2}}{2}$
$\therefore\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ T^{2} \propto \left\lbrack \frac{r_{1} + r_{2}}{2} \right\rbrack^{3}$
$or\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ T \propto (r_{1} + r_{2})^{3/2\ }$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.