Question

In a planetary motion, the areal velocity of position vector of a planet depends on angular velocity ($ω$) and distance ($r$) of the planet from the sun. The correct relation for areal velocity $\fracdAdt$ is

• A. $\fracdAdt \propto ω r$
• B. $\fracdAdt \propto ω r²$
• C. $\fracdAdt \propto \frac\omegar²$
• D. $\fracdAdt \propto \frac\omegar$

Hint:

Kepler's second law: Areal velocity is constant for a planet, $\fracdAdt = \frac12 r² ω$.

Answer 1

The Correct Option is B

The question involves understanding the relation of areal velocity in planetary motion. The areal velocity of a planet refers to the rate at which area is swept by the position vector of the planet as it orbits around the sun. According to Kepler's second law (also known as the Law of Areas), the line joining a planet to the sun sweeps out equal areas in equal times. Mathematically, this can be expressed as the areal velocity \(\left(\frac{dA}{dt}\right)\).

To find the correct relation, let's analyze the dependencies:

1. Areal Velocity Formula: For a planet revolving around the sun in an elliptical orbit, the areal velocity is given by: \(\frac{dA}{dt} = \frac{1}{2} r^2 \omega\), where
   • \(r\) is the distance from the sun,
   • \(\omega\) is the angular velocity.
2. Derivation of Relation: From the formula given above, it is evident that the relationship between areal velocity, angular velocity, and distance is:
   • \(\frac{dA}{dt} \propto r^2 \omega\)

Therefore, the correct option is:

\(\frac{dA}{dt} \propto \omega r^2\)

Let's analyze other options:

• \(\frac{dA}{dt} \propto \omega r\): This suggests a linear relation with distance and angular velocity, which is incorrect as explained above.
• \(\frac{dA}{dt} \propto \frac{\omega}{r^2}\): This option suggests an inverse relation with the square of the distance, which is also incorrect.
• \(\frac{dA}{dt} \propto \frac{\omega}{r}\): This option implies an inverse relation with distance, which doesn't align with the correct formula.

In conclusion, by applying the formula for areal velocity in planetary motion, we confirm that the areal velocity is directly proportional to the square of the distance and the angular velocity. This is consistent with Kepler's laws and planetary motion dynamics.