Animated Solution for Physics - Gravitation: If the angular momentum of a planet of mass $m$, moving around sun in a circular orbit is $L$ about the centre of the sun, its areal velocity is

Have you ever wondered how planets maintain their majestic, rhythmic motion around the Sun? Johannes Kepler did, and his Second Law of Planetary Motion—the Law of Equal Areas—is one of the most beautiful geometric truths in physics. But what drives this law? The secret lies in a fundamental conserved quantity: angular momentum.

Let's embark on a journey to connect the geometry of a planet's orbit with the physics of its motion.

Imagine a planet of mass $m$ gliding along a circular orbit of radius $r$ around the Sun. As it moves from a point $P$ to a nearby point $P^{\prime }$ in an infinitesimally small time interval $dt$, the radius vector connecting the Sun to the planet sweeps out a tiny sector of area, $dA$.

If the angle swept during this time is $d\theta$, we can approximate this tiny sector as a triangle. The area of this sector is given by:

But we are interested in the areal velocity—the rate at which this area is swept out over time. By dividing our area equation by the time interval $dt$, we get:

The term $\frac{d\theta }{dt}$ is something very familiar to us from circular motion: it is the angular velocity, $\omega$. Substituting this into our areal velocity equation, we find:

This equation is elegant, but it doesn't yet involve the planet's mass or its angular momentum. We need to bridge the gap between kinematics and dynamics.

The angular momentum $L$ of a particle moving in a circular orbit is the product of its mass, its linear velocity $v$, and its orbital radius $r$:

Since linear velocity and angular velocity are related by $v=r\omega$, we can rewrite the angular momentum as:

Notice something magical? The term $r^2\omega$ appears in both our areal velocity equation and our angular momentum equation! Let's isolate $r^2\omega$ from the angular momentum equation:

Now, for the final stroke of logic. We substitute this expression for $r^2\omega$ back into our areal velocity equation:

And there we have it! The areal velocity of the planet is exactly $\frac{L}{2m}$. Because the gravitational force from the Sun is a central force, it exerts no torque on the planet. Therefore, the angular momentum $L$ remains perfectly constant. Since the mass $m$ is also constant, the areal velocity $\frac{dA}{dt}$ must be constant as well.

This is the mathematical heartbeat of Kepler's Second Law: a planet sweeps out equal areas in equal times, all thanks to the conservation of angular momentum.