Question

Astronomers observe two separate solar systems each consisting of a planet orbiting a sun. The two orbits are circular and have the same radius R. It is determined that the planets have angular momenta of the same magnitude L about their suns, and that the orbital periods are in the ratio of three to one; i.e., T1 = 3T2. The ratio m1/m2 of the masses of the two planets is

(A) 1
(C) (3)^1/2
(C) 2
(D) 3
(E) 9

Answer 1

Answer:

(D) 3

Explanation:

The angular momentum is given by:

$\vec{L}=\vec{r}\ X \ \vec{p}$

Thus, the magnitude of the angular momenta of both solar systems are given by:

$L_1=Rm_1v_1=Rm_1(\omega R)=R^2m_1(\frac{2\pi}{T_1})=2\pi R^2\frac{m_1}{T_1}\\\\L_2=Rm_2v_2=2\pi R^2\frac{m_2}{T_2}$

where we have taken that both systems has the same radius.

By taking into account that T1=3T2, we have

$L_1=2\pi R^2\frac{m_1}{3T_2}=\frac{1}{3}2\pi R^2\frac{1}{T_2}m_1=\frac{1}{3}\frac{L_2}{m_2}m_1$

but L1=L2=L:

$L=\frac{1}{3}L\frac{m_1}{m_2}\\\\\frac{m_1}{m_2}=3$

Hence, the answer is (D) 3

HOPE THIS HELPS!!