# Calculate Orbital Angular Momentum

Satellite of mass m is moving with velocity v in a circular orbit of radius r about mass M.

Find the orbital angular momentum.

Know

$$v = \sqrt{\frac{GM}{r}}$$

Orbital angular momentum of a system is defined as the angular momentum of the center of mass of the system.

Let the origin be at mass M.

$$r_{cm} = \frac{m}{M + m} r$$
$$v_{cm} = \frac{v}{r} \frac{m}{M + m} r = \frac{vm}{M+m}$$

$$l = r_{cm} \times p_{cm} = (M + m) r_{cm} v_{cm} = \frac{m^2 rv}{M+m} = \frac{m^2}{m+M} \sqrt{GMr}$$

Correct answer in text is $$m \sqrt{GMr}$$