Question

A satellite of mass m moves in a circular orbit of radius r around the Earth with orbital speed v. What is the expression for its angular momentum about the centre of the Earth?

Hint:

For circular motion the velocity is always perpendicular to the radius, so L = m v r.

Answer 1

The angular momentum of a satellite in a circular orbit is L = m v r.

Step 1: Definition of angular momentum.
For a particle moving in a circle of radius r with linear speed v, the angular momentum about the centre is \(L = m v r \sin\theta\), where theta is the angle between the position vector and the velocity vector.

Step 2: Applying it to a circular orbit.
In a circular orbit, the velocity of the satellite is always tangential, that is, perpendicular to the radius vector joining it to the centre of the Earth. So the angle theta is 90 degrees, and \(\sin 90^\circ = 1\).

Step 3: Simplified formula.
This makes the expression simplify to \(L = m v r\), where m is the mass of the satellite, v is its orbital speed, and r is the radius of the orbit.

Step 4: Answer.
So the angular momentum of a satellite moving in a circular orbit is given by L = m v r, and it stays constant throughout the orbit because both v and r remain constant for a circular path.