Animated Solution for Physics - Rotational Motion: Angular momentum of the particle rotating with a central force is constant due to

Have you ever wondered why planets continue to orbit the sun for billions of years without spiraling out of control? Or why an electron stays bound to a nucleus? The answer lies in a profound principle of physics: the conservation of angular momentum under a central force.

Let's embark on a journey to decode this elegant concept and understand exactly why the angular momentum remains perfectly constant.

Imagine a particle moving through space. Now, add a special condition: the force acting on this particle is always directed towards (or away from) a single, fixed point. We call this point the center of force, and the force itself is known as a central force.

Gravity is the most famous example. The sun pulls the Earth directly towards its center. The electrostatic force is another; a proton pulls an electron directly towards itself.

In our mathematical setup, we define the position of the particle using a position vector, $\vec{r}$, which originates from the center and points to the particle. The central force, $\vec{F}$, acts along this exact same line. If it's an attractive force, it points directly opposite to $\vec{r}$.

Torque

To understand rotational motion, we need to look at torque ($\vec{\tau }$). Just as a linear force changes an object's linear momentum, torque changes an object's angular momentum.

Mathematically, torque is defined as the cross product of the position vector and the force vector:

The magnitude of this cross product depends on the angle $\theta$ between the two vectors:

Here is where the magic happens. For a central force, the force vector $\vec{F}$ lies exactly on the same line as the position vector $\vec{r}$.

If the force is attractive (like gravity), it points inwards, making the angle between $\vec{r}$ and $\vec{F}$ exactly $180^{\circ }$. If the force is repulsive, they point in the same direction, making the angle $0^{\circ }$.

What is the sine of $180^{\circ }$ or $0^{\circ }$? It is exactly zero!

Therefore, the cross product completely vanishes. The torque exerted by any central force about its center is perfectly zero:

Newton's second law for rotation states that the net torque acting on a system is equal to the rate of change of its angular momentum ($\vec{L}$):

Since we just proved that the torque is zero, it immediately follows that:

When the derivative of a quantity is zero, that quantity is a constant. Therefore, the angular momentum $\vec{L}$ does not change with time. It is conserved.

This simple mathematical truth has staggering consequences. Because the angular momentum is constant, the particle's orbit must remain in a single, fixed plane. Furthermore, it leads directly to Kepler's Second Law of planetary motion, which states that a line joining a planet and the sun sweeps out equal areas during equal intervals of time.

So, the next time you look up at the night sky, remember: the cosmic dance of the planets is perfectly choreographed by the elegant reality of zero torque!