Question

If the angular momentum of a planet of mass \(M\), moving around the sun in a circular orbit, is \(L\). The areal velocity of the planet about the center of the sun is

• A. \( \frac{L}{M} \)
• B. \( \frac{4L}{M} \)
• C. \( \frac{L}{2M} \)
• D. \( \frac{2L}{M} \)

Hint:

To calculate the areal velocity of a planet in a circular orbit, use the relationship with angular momentum: \( \text{Areal velocity} = \frac{L}{M} \), where \(L\) is the angular momentum and \(M\) is the mass of the planet.

Answer 1

The Correct Option is A

The angular momentum \(L\) of a planet moving around the sun in a circular orbit is given by: \[ L = M \dot r \dot v \] where \(M\) is the mass of the planet, \(r\) is the radius of the orbit, and \(v\) is the tangential velocity of the planet.
The areal velocity of the planet is defined as the rate at which the area swept out by the radius vector changes. It is given by: \[ \text{Areal velocity} = \frac{dA}{dt} \] For a circular orbit, the areal velocity is constant and is related to the angular momentum by: \[ \frac{dA}{dt} = \frac{L}{2M} \] Therefore, the areal velocity of the planet about the center of the sun is: \[ \text{Areal velocity} = \frac{L}{M} \] Thus, the areal velocity of the planet is \( \frac{L}{M} \).