# GRE Physics GR 1777 Problem Solution

## 018. Classical Mechanics (Angular Momentum)

### Solution

Angular momentum in a 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 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) $\sqrt{2}$