Will the angular momentum of a planet be conserved? Justify your answer.

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Q. Will the angular momentum of a planet be conserved? Justify your answer.

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Concept understanding — 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

\vec{L} = \text{constant} \quad \text{if} \quad \vec{\tau}_{\text{ext}} = 0

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:

\vec{L} = \vec{r} \times m\vec{v}

For a rigid body rotating about a fixed axis:

L = I\omega

Where:

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

Note

Moment of inertia $I$ depends on where the mass is, not just how much. Mass far from the axis gives larger $I$ ; mass close to the axis gives smaller $I$ .

Why It Works: The Physics

Newton's second law for rotation says:

\tau_{\text{ext}} = \frac{dL}{dt}

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:

I_1 \omega_1 = I_2 \omega_2

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

Real-World Examples

Situation	What happens	Why
Ice skater pulling arms in	Spins faster	$I$ decreases, $\omega$ increases to keep $L$ constant
Diver tucking into a ball	Rotates faster in midair	Same principle — no external torque during flight
Cat falling upside-down	Twists body to land on feet	Changes $I$ of different body parts to rotate without external torque
Planet orbiting the Sun	Speeds up when closer, slows down when farther	Gravitational force is central (torque = 0), so $L$ is constant

Watch out

Angular momentum is a vector. Its direction matters too. If no external torque acts, the axis of rotation stays fixed in space. This is why a spinning gyroscope or a bicycle wheel resists being tilted.

Common Mistake to Avoid

Students often think "angular momentum is conserved" means "angular velocity is constant." That's false. Angular velocity can change if the moment of inertia changes. What stays constant is the product $I\omega$ .

Also: conservation applies only when net external torque is zero. If you apply a brake to a spinning wheel, torque is present — angular momentum is not conserved for the wheel alone (though it is conserved for wheel + brake + Earth as a system).

The Big Picture

Conservation of angular momentum is one of the three great conservation laws in physics (along with energy and linear momentum). It's a direct consequence of the rotational symmetry of space — the laws of physics don't care which direction you're facing. That deep symmetry gives us this powerful tool for solving problems, from planetary orbits to quantum spins.

When you next see an ice skater spin faster by pulling arms in, you're watching one of the most elegant principles of physics in action: nature conserves rotation.

This topic frequently turns up in searches like "Conservation of Angular Momentum: definition, formula and real-world examples" — Conservation of Angular Momentum sits squarely within the System of Particles and Rotational Motion coverage of NCERT Class 11 Physics, so it is fair game for both CBSE board questions and competitive-exam numericals. Cross-checking this explanation against the relevant NCERT Physics chapter and solving a few past-year questions will round out your preparation.

Step 1. Torque about the Sun is $\vec\tau=\vec r\times\vec F$ .

Step 2. The gravitational force $\vec F=-\dfrac{GMm}{r^2}\hat r$ always points exactly along the line joining the Sun and the planet, i.e. along (or against) $\vec r$ itself.

Step 3. The cross product of two parallel (or antiparallel) vectors is zero, so $\vec\tau=\vec r\times\vec F=0$ at every instant.

Step 4. Since $\vec\tau=d\vec L/dt$ , zero torque means $\vec L$ never changes -- angular momentum is conserved throughout the orbit.

✓Final answer

Yes, it is conserved -- because gravity is a central force, it produces zero torque about the Sun.

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Q.Will the angular momentum of a planet be conserved? Justify your answer.

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