Question

If the angular momentum of a planet of mass m moving around the Sun in a circular orbit is L about the centre of the Sun, then its areal velocity is:
A. $\dfrac{{4L}}{m}$
B. $\dfrac{L}{m}$
C. $\dfrac{L}{{2m}}$
D. $\dfrac{{2L}}{m}$

Answer 1

Hint
The angular momentum of a body depends directly on its velocity and mass. The areal velocity is the rate at which the area is swept by the planet. Consequently, it also depends on the mass and velocity of the planet.
$L = mvr$, where L is the angular momentum of the planet, m is the mass, v is the velocity and r is the radius from the centre that is the Sun.

Complete step by step answer
The areal velocity is the area covered per unit time, hence it is given as:
$A = \dfrac{{{\text{area}}}}{{{\text{time}}}}$
The time period is calculated as the total distance covered divided by the velocity at which the planet moves. The total distance is equal to the circumference of the circular orbit, hence:
Time period $ = \dfrac{{2\pi r}}{v}$
The area covered in this time period will be the total area of the circular orbit. Thus:
Area $ = \pi {r^2}$
The areal velocity will now become:
$A = \dfrac{{\pi {{\text{r}}^2}v}}{{2\pi r}} = \dfrac{{{\text{r}}v}}{2}$ [Eq. 1]
We know that the angular momentum is given as:
$L = mvr$
Finding this relation in terms of the velocity v, we get:
$v = \dfrac{L}{{mr}}$
Substituting this value in the Eq. 1 gives us:
$A = \dfrac{{\text{r}}}{2} \times \dfrac{L}{{mr}} = \dfrac{L}{{2m}}$
Hence, the correct answer is option (C).

Note
Kepler's Law states that the angular momentum of a planet remains constant as it moves in an orbit around the Sun. As the areal velocity is directly dependent on the angular momentum and inversely proportional to the mass, none of these quantities are variable. Hence, the areal velocity also remains constant during planetary motion.