Physics GRE GR 1777, Problem 018 Solution

18. Classical Mechanics (Angular Momentum)

Solution
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


Answer
(C) \sqrt{2}
Reference
https://en.wikipedia.org/wiki/Angular_momentum#Scalar_—_angular_momentum_in_two_dimensions

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