Q.If the angular momentum of a planet is given by $\vec{L}(t) = -5t^2\,\hat{i} - 6t\,\hat{j} + 3\hat{k}$, what is the torque experienced by the planet? Will the torque be in the same direction as that of the 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 …