GRE Physics GR 1777 Problem Solution

018. Classical Mechanics (Angular Momentum)

Solution

Angular momentum in a circular motion is

$latex L = mvr$
where $latex L$ is angular momentum, $latex m$ is the mass of the satellite, $latex v$ is the velocity, and $latex r$ 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

$latex r_A = 2r_B$

Since satellites have circular motion, the centripetal force is equal to the gravitational force

$latex \\frac{GMm}{r^2} = \\frac{mv^2}{r}&s=2$
where $latex M$ is the mass of the Earth. The orbital velocity of the satellite can be obtained as

$latex v = \\sqrt{\\frac{GM}{r}}&s=1$

Then we can wirte angular momentum as

$latex L = mvr = m \\sqrt{\\frac{GM}{r}} r = m\\sqrt{GMr}$
$latex \\therefore L \\propto \\sqrt{r}$

Therefore, the ratio of the angular momentum is

$latex L_A : L_B = \\sqrt{r_A} : \\sqrt{r_B} = \\sqrt{2} : 1$

Answer

(C) $latex \\sqrt{2}$

Reference

https://en.wikipedia.org/wiki/Angular_momentum#Scalar_—_angular_momentum_in_two_dimensions