GRE GR 1777, Problem 018 Solution

18. Classical Mechanics (Angular Momentum)

Angular momentum in circular motion is 

L = mvr
where L is angular momentum, m is the mass of the satellite, v is the velocity, and r is the orbital radius. In this problem, two satellites are identical. So the mass of the satellite is equal. The orbital radius of A is the ratio of the angular momentum of A to the twice that of B, so
r_A = 2r_B
Since satellites have circular motion, the centripetal force is equal to the gravitational force
\frac{GMm}{r^2} = \frac{mv^2}{r}
where M is the mass of the Earth. The orbital velocity of the satellite can be obtained as
v = \sqrt{\frac{GM}{r}}
Then we can wirte angular momentum as
L = mvr = m \sqrt{\frac{GM}{r}} r = m\sqrt{GMr}
\therefore L \propto \sqrt{r}
Therefore, the ratio of the angular momentum is
L_A : L_B = \sqrt{r_A} : \sqrt{r_B} = \sqrt{2} : 1
(C) \sqrt{2}

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