Question

A satellite of mass \(m\) is revolving around earth of mass \(M\) in an orbit of radius \(r\) with constant angular velocity \(\omega\). The angular momentum of the satellite is (G = gravitational constant)

• A. \( m(GMr) \)
• B. \( m(GMr)^{1/2} \)
• C. \( (GMmr)^{1/2} \)
• D. \( \left(\dfrac{GMr}{m}\right)^2 \)

Hint:

For circular orbits, always use the balance between gravitational force and centripetal force to find orbital speed before calculating angular momentum.

Answer 1

The Correct Option is B

Step 1: Expression for orbital velocity.
For a satellite revolving in a circular orbit around the earth, the gravitational force provides the centripetal force. Hence, \[ \frac{GMm}{r^2} = \frac{mv^2}{r}. \] Solving for velocity, \[ v = \sqrt{\frac{GM}{r}}. \]
Step 2: Formula for angular momentum.
Angular momentum of a particle moving in a circular orbit is given by \[ L = mvr. \]
Step 3: Substituting the value of velocity.
\[ L = m \left(\sqrt{\frac{GM}{r}}\right) r = m\sqrt{GMr}. \]
Step 4: Conclusion.
Thus, the angular momentum of the satellite is \( m(GMr)^{1/2} \).