Question

Planet X and Planet Y both have circular orbits around a star with twice the mass of our sun. Planet X is the same distance from the star as the earth, while Planet Y is four times further away.
If both Planets have equal angular momentum, how do their masses compare?

• A. Planet Y has 4 times the mass of Planet X
• B. Planet X has 4 times the mass of Planet Y
• C. Planet X has 2 times the mass of Planet Y
• D. Planet Y has 8 times the mass of Planet X
• E. Planet X has 8 times the mass of Planet Y

Answer 1

Answer: (C)

Planet X has 2 times the mass of Planet Y

Let the mass of sun be $M_s$.

Thus mass of the star $M=2M_s$

Given : $R_Y=4R_X$

$\therefore$ Orbital velocity of planet X $V_X=\sqrt{\frac{G\left(2M_s\right)}{R_X}}$

Also orbital velocity of planet Y $V_Y=\sqrt{\frac{G\left(2M_s\right)}{R_Y}}=\sqrt{\frac{G\left(2M_s\right)}{4R_X}}$

$⟹V_Y=\frac{V_X}{2}$

Given : $L_X=L_Y$ i.e $M_X{R}_X{V}_X=M_Y{R}_Y{V}_Y$

$\therefore$ $M_X{R}_X{V}_X=M_Y\times \left(4R_X\right)\times \frac{V_X}{2}$ $⟹M_X=2M_Y$