Q.State the law of conservation of angular momentum.

Conservation of Angular Momentum

The Intuition First

Imagine you're sitting on a spinning office chair with your arms stretched out. Someone gives you a gentle push, and you start rotating slowly. Now pull your arms in tight against your chest. What happens? You spin faster. Push your arms back out — you slow down again.

Nothing external pushed you to go faster or slower. The change came from inside — from how you arranged your mass relative to the axis of rotation.

That's the core idea: Angular momentum is a quantity that stays constant for a rotating system unless an external torque acts on it. When you pulled your arms in, you didn't change your angular momentum — you changed your distribution of mass, and your rotation speed had to adjust to keep the total constant.

The Precise Statement

Where:

• $\vec{L}$ is the angular momentum of the system
• $\vec{\tau}_{\text{ext}}$ is the net external torque acting on the system

In words: The total angular momentum of an isolated system (no external torque) remains constant in both magnitude and direction.

What Is Angular Momentum?

For a point mass $m$ moving with velocity $\vec{v}$ at position $\vec{r}$ from a reference point:

For a rigid body rotating about a fixed axis:

Where:

• $I$ = moment of inertia (how mass is distributed relative to the axis)
• $\omega$ = angular velocity (how fast it spins)

Why It Works: The Physics

Newton's second law for rotation says:

If $\tau_{\text{ext}} = 0$, then $\frac{dL}{dt} = 0$, so $L$ is constant.

Since $L = I\omega$, if $I$ changes (you pull arms in), $\omega$ must change in the opposite way to keep $L$ the same:

Smaller $I$ → larger $\omega$ (spin faster). Larger $I$ → smaller $\omega$ (spin slower).

Real-World Examples

Common Mistake to Avoid …