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

(D) 3

Step-by-step 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((2\pi)/(T_1))=2\pi R^2(m_1)/(T_1)\\\\L_2=Rm_2v_2=2\pi R^2(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(m_1)/(3T_2)=(1)/(3)2\pi R^2(1)/(T_2)m_1=(1)/(3)(L_2)/(m_2)m_1`

but L1=L2=L:

`L=(1)/(3)L(m_1)/(m_2)\\\\(m_1)/(m_2)=3`

Hence, the answer is (D) 3

HOPE THIS HELPS!!