§11.3 — Angular Momentum

In physics, things are conserved — energy is conserved, linear momentum is conserved, and now we learn that angular momentum is also conserved. This is one of the deepest principles in physics.

Definition of Angular Momentum

Angular momentum is the rotational version of momentum.

For a Point Mass

If a particle of mass \( m \) moves with linear velocity \( v \) at distance \( r \) from a rotation axis:

\[ L = mvr = pr \]

where \( p = mv \) is linear momentum.

For a Rigid Body Rotating

For a rigid body rotating with angular velocity \( \omega \) about an axis:

\[ L = I\omega \]

where:

• \( I \) = moment of inertia
• \( \omega \) = angular velocity

Parallel: Linear momentum \( p = mv \); angular momentum \( L = I\omega \). Just like the force-torque relationship!

Relationship Between Torque and Momentum

Torque changes angular momentum:

\[ \tau_{\text{net}} = \frac{dL}{dt} \]

This is exactly like \( F_{\text{net}} = \frac{dp}{dt} \) for linear motion!

Consequence: Conservation Law

If no external torque acts (\( \tau_{\text{net}} = 0 \)):

\[ \frac{dL}{dt} = 0 \implies L = \text{constant} \]

Angular momentum is conserved!

Practical Examples

Example 1: A Figure Skater Spins

A skater with arms extended rotates — large moment of inertia: I₁ = 2 kg·m², low angular velocity: ω₁ = 2 rad/s.

\[ L_1 = 2 \times 2 = 4 \text{ kg·m}^2\text{/s} \]

Now the skater pulls arms to body — moment of inertia decreases: I₂ = 0.5 kg·m².

Since no external torque (only internal friction):

\[ L_2 = L_1 \implies I_2 \omega_2 = 4 \]

\[ \omega_2 = \frac{4}{0.5} = 8 \text{ rad/s} \]

Result: Angular velocity increases fourfold! (while angular momentum stays constant.)

Example 2: Collapsing Stars

A giant star rotates slowly with large radius. When a neutron star forms, radius becomes tiny:

• Before: \( R_1 = 10^{11} \) m, \( \omega_1 = 10^{-6} \) rad/s
• After: \( R_2 = 10^4 \) m, \( \omega_2 = ? \)

Moment of inertia: \( I \propto MR^2 \)

\[ \frac{I_2}{I_1} = \frac{R_2^2}{R_1^2} = \frac{(10^4)^2}{(10^{11})^2} = 10^{-14} \]

\[ \omega_2 = \omega_1 \times \frac{I_1}{I_2} = 10^{-6} \times 10^{14} = 10^8 \text{ rad/s} \]

A neutron star spins at insane speeds!

Numerical Example: Collision and Rotation

Two disks:

• Disk A: \( I_A = 0.4 \) kg·m², rotating at \( \omega_A = 5 \) rad/s
• Disk B: \( I_B = 0.6 \) kg·m², stationary (\( \omega_B = 0 \))

They collide and stick together. Final angular velocity?

Solution: No external torque (only internal friction).

\[ L_{\text{initial}} = I_A \omega_A + I_B \omega_B = 0.4 \times 5 + 0 = 2 \text{ kg·m}^2\text{/s} \]

After collision: \[ L_{\text{final}} = (I_A + I_B)\omega_f = (0.4 + 0.6)\omega_f = \omega_f \]

By conservation: \[ L_{\text{initial}} = L_{\text{final}} \]

\[ 2 = \omega_f \implies \omega_f = 2 \text{ rad/s} \]

Both disks now spin at 2 rad/s!

Angular Momentum as a Vector

Like torque, angular momentum is vectorial:

\[ \vec{L} = \vec{r} \times \vec{p} \]

Or for a rigid body:

\[ \vec{L} = I\vec{\omega} \]

The direction of \( \vec{L} \) is given by the right-hand rule.

Relationship with Energy

Rotational kinetic energy can be rewritten in terms of angular momentum:

\[ K = \frac{1}{2}I\omega^2 = \frac{1}{2}I\left(\frac{L}{I}\right)^2 = \frac{L^2}{2I} \]

If \( I \) changes while \( L \) stays constant:

• Smaller \( I \) → larger \( K \)
• Skater: pull arms in → spins faster and gains kinetic energy!

(Where does energy come from? From work done when pulling in arms.)

What You Should Know

• Definition: \( L = I\omega \) or \( L = mvr \)
• Torque relation: \( \tau_{\text{net}} = \frac{dL}{dt} \)
• Conservation: If \( \tau_{\text{net}} = 0 \), then \( L = \text{constant} \)
• Skater: Pull arms in → angular velocity increases
• Collapsing stars: Smaller radius → faster spin

Preview of §11.4

Now we understand angular momentum. But what if an object spins about two different axes? Like a gyroscope, where a wheel spins and the axle also rotates? Answer: a vector that wants to change direction — creating precession.

📚 See also: Halliday Vol 1, Ch 11, §11.3 — Angular momentum and its conservation. 🔗 Reference: §7.8 (Conservation of linear momentum) — the linear analogue.

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