Question

Earth revolves round the sun in a circular orbit of radius \(R\). The angular momentum of the revolving earth is directly proportional to

• A. \(R^2\)
• B. \(R^3\)
• C. \(R\)
• D. \(\sqrt{R}\)

Hint:

For planets in circular orbits, angular momentum depends on the square root of orbital radius.

Answer 1

The Correct Option is D

Step 1: Expression for angular momentum.
Angular momentum of a body moving in a circular orbit is \[ L = mvr \]

Step 2: Use gravitational force as centripetal force.
For Earth revolving around the Sun, \[ \frac{mv^2}{R} = \frac{GMm}{R^2} \Rightarrow v = \sqrt{\frac{GM}{R}} \]

Step 3: Substitute velocity in angular momentum.
\[ L = mR\sqrt{\frac{GM}{R}} = m\sqrt{GMR} \]

Step 4: Conclusion.
Thus, angular momentum \(L \propto \sqrt{R}\).