Question

Two planets A and B having orbital radius R and 4R are initially at eh closest position and rotating in the same direction. If angular velocity of planet B is ${\omega }_0$, then after how much time will both the planets be again in the closest position? (Neglect the interaction between planets).

• A. $\frac{2\pi }{7{\omega }_0}$
• B. $\frac{2\pi }{9{\omega }_0}$
• C. $\frac{2\pi }{{\omega }_0}$
• D. $\frac{2\pi }{5{\omega }_0}$

Answer 1

The correct option is A $\frac{2\pi }{7{\omega }_0}$

By kepler's third law,

$T^2\alpha r^3$

$\frac{T_A^2}{T_B^2}=\frac{r^3}{{4r}^3}$

$\frac{T_A}{T_B}=(\frac{1}{4^3}{)}^1/2=\frac{1}{8}$

$T_B=8T_A$

$\frac{2\pi }{{\omega }_B}=8\frac{2\pi }{{\omega }_A}$

${\omega }_A=8{\omega }_B$ But ${\omega }_B={\omega }_0$

${\omega }_A=8{\omega }_0$

Relative angular velocity of A with respect to B would be,

${\omega }_A-{\omega }_B=8{\omega }_0-{\omega }_0=7{\omega }_0$

Hence the two planers will be closer again after a period of $t=\frac{2\pi }{7{\omega }_0}$