# Physics GRE GR 1777, Problem 018 Solution

## 18. Classical Mechanics (Angular Momentum)

##### Solution
Angular momentum in circular motion is $L = mvr$

where $L$ $L$ is angular momentum, $m$ $m$ is the mass of the satellite, $v$ $v$ is the velocity, and $r$ $r$ 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 $M$ $M$ 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) $\sqrt{2}$ $\sqrt{2}$
https://en.wikipedia.org/wiki/Angular_momentum#Scalar_—_angular_momentum_in_two_dimensions