1. Introduction

The Conservation of Angular Momentum is one of the most fundamental principles in rotational mechanics.

It states that if the net external torque acting on a system is zero, the total angular momentum of the system remains constant.

This principle explains phenomena ranging from spinning figure skaters and rotating platforms to planetary motion and galaxies.

2. Review of Angular Momentum

Before studying conservation, recall that angular momentum for a rigid body rotating about a fixed axis is:

$$L=I\omega$$

where:

• $L$ = angular momentum
• $I$ = moment of inertia
• $\omega$ = angular velocity

For a particle:

L⃗=r⃗×p⃗\vec{L} = \vec{r} \times \vec{p}L=r×p​

where:

p⃗=mv⃗\vec{p}=m\vec{v}p​=mv

Angular momentum is a vector quantity.

3. Relationship Between Torque and Angular Momentum

The rotational form of Newton’s Second Law is:

∑τ=dLdt\sum \tau = \frac{dL}{dt}∑τ=dtdL​

This equation states that torque changes angular momentum.

It is analogous to:

∑F=dpdt\sum F = \frac{dp}{dt}∑F=dtdp​

in linear motion.

4. Condition for Conservation

Angular momentum is conserved whenever the net external torque is zero.

∑τexternal=0\sum \tau_{\text{external}} = 0∑τexternal​=0

If:

dLdt=0\frac{dL}{dt}=0dtdL​=0

then:

$$L=\text{constant}$$

This is the mathematical statement of angular momentum conservation.

5. Conservation Equation

For any isolated rotational system:

Li=LfL_i=L_fLi​=Lf​

For a rigid body:

Iiωi=IfωfI_i\omega_i = I_f\omega_fIi​ωi​=If​ωf​

where:

• IiI_iIi​ = initial moment of inertia
• IfI_fIf​ = final moment of inertia
• ωi\omega_iωi​ = initial angular velocity
• ωf\omega_fωf​ = final angular velocity

6. Physical Meaning

Angular momentum conservation means that a system cannot spontaneously gain or lose angular momentum.

If the moment of inertia changes, the angular velocity must adjust to keep angular momentum constant.

If:

$$I\downarrow$$

then:

$$\omega \uparrow$$

If:

$$I\uparrow$$

then:

$$\omega \downarrow$$

7. Figure Skater Example

A spinning skater pulls their arms inward.

This decreases the moment of inertia.

If<IiI_f<I_iIf​<Ii​

Since angular momentum is conserved:

Iiωi=IfωfI_i\omega_i = I_f\omega_fIi​ωi​=If​ωf​

the angular velocity increases.

The skater spins faster without any external torque.

8. Rotating Platform Example

A person stands on a frictionless rotating platform holding dumbbells.

When the person pulls the dumbbells inward:

• moment of inertia decreases
• angular velocity increases

When the dumbbells are extended outward:

• moment of inertia increases
• angular velocity decreases

The total angular momentum remains unchanged.

9. Planetary Motion

Planets orbiting the Sun approximately conserve angular momentum.

For orbital motion:

$$L=mvr$$

When a planet moves closer to the Sun:

$$r\downarrow$$

its speed increases.

When a planet moves farther away:

$$r\uparrow$$

its speed decreases.

This behavior is a direct consequence of angular momentum conservation.

10. Angular Momentum in Collisions

Angular momentum is often conserved during rotational collisions.

If external torque is negligible:

Li=LfL_i=L_fLi​=Lf​

This principle is especially useful when kinetic energy is not conserved.

Examples include:

• objects sticking together
• rotating-platform collisions
• impact problems

11. Example Problem

A student stands on a frictionless turntable.

Initially:

Ii=8 kg⋅m2I_i = 8\,kg\cdot m^2Ii​=8kg⋅m2 ωi=2 rad/s\omega_i = 2\,rad/sωi​=2rad/s

The student pulls in their arms, reducing the moment of inertia to:

If=4 kg⋅m2I_f = 4\,kg\cdot m^2If​=4kg⋅m2

Find the final angular velocity.

Solution

Using conservation of angular momentum:

Iiωi=IfωfI_i\omega_i = I_f\omega_fIi​ωi​=If​ωf​

Substitute values:

(8)(2)=(4)ωf(8)(2) = (4)\omega_f(8)(2)=(4)ωf​ 16=4ωf16 = 4\omega_f16=4ωf​ ωf=4 rad/s\omega_f = 4\,rad/sωf​=4rad/s

The angular velocity doubles.

12. Conservation Does Not Always Mean Energy Conservation

A common misconception is that conserving angular momentum automatically means conserving kinetic energy.

This is not true.

Angular momentum may remain constant while rotational kinetic energy changes.

For a rotating body:

Krot=12Iω2K_{\text{rot}} = \frac12 I\omega^2Krot​=21​Iω2

When a skater pulls in their arms:

• angular momentum stays constant
• rotational kinetic energy increases

The extra energy comes from the work done by the skater’s muscles.

13. Systems of Multiple Objects

For systems containing several interacting objects:

Ltotal,i=Ltotal,fL_{\text{total},i} = L_{\text{total},f}Ltotal,i​=Ltotal,f​

provided that:

∑τexternal=0\sum \tau_{\text{external}} = 0∑τexternal​=0

Internal torques cancel in pairs and do not affect total angular momentum.

14. Common AP Physics C Applications

Rotating Platforms

Objects moving toward or away from the axis.

Figure Skaters

Changing body configuration changes moment of inertia.

Planetary Orbits

Orbital speed changes with distance from the central body.

Rotational Collisions

Angular momentum often remains conserved even when kinetic energy does not.

15. Common AP Physics C Mistakes

Mistake 1

Using conservation of angular momentum when external torque is present.

Always verify:

∑τexternal=0\sum \tau_{\text{external}}=0∑τexternal​=0

Mistake 2

Confusing angular momentum with rotational kinetic energy.

They are different physical quantities.

Mistake 3

Ignoring the change in moment of inertia.

Angular velocity changes whenever the moment of inertia changes.

16. Problem-Solving Strategy

Step 1

Identify the system.

Step 2

Determine whether external torque is negligible.

Step 3

Write:

Li=LfL_i=L_fLi​=Lf​

Step 4

Substitute:

Iiωi=IfωfI_i\omega_i = I_f\omega_fIi​ωi​=If​ωf​

for rigid-body problems.

Step 5

Solve for the unknown variable.

Summary

Angular momentum of a rotating rigid body:

$$L=I\omega$$

Rotational Newton’s Second Law:

∑τ=dLdt\sum\tau = \frac{dL}{dt}∑τ=dtdL​

Conservation condition:

∑τexternal=0\sum\tau_{\text{external}}=0∑τexternal​=0

Conservation equation:

Li=LfL_i=L_fLi​=Lf​

or

Iiωi=IfωfI_i\omega_i = I_f\omega_fIi​ωi​=If​ωf​

Key ideas:

• angular momentum is conserved when net external torque is zero
• changes in moment of inertia cause inverse changes in angular velocity
• conservation applies to rotating bodies, collisions, and orbital motion
• angular momentum conservation is one of the most powerful tools in rotational mechanics

The Conservation of Angular Momentum provides a unifying principle that explains rotational behavior across systems ranging from simple laboratory experiments to planetary and astronomical motion.