Conservation of Angular Momentum

Lesson ~10 min read 8 MCQs

In simple terms: In simple terms, conservation of angular momentum means a spinning system keeps its total "spin" constant unless an outside twist, or torque, acts on it.

Why this matters

Have you ever watched a figure skater start a spin with their arms stretched out, and then, by simply pulling their arms in, they suddenly whirl into a blur? It looks like magic. They don't push off the ice again. No one gives them a shove. Yet, their speed dramatically increases. How?

This isn't magic; it's physics. It's the same principle that lets a diver control their somersaults in mid-air or a gymnast stick a landing. The secret lies in a powerful conservation law that governs everything that spins. In this lesson, we'll break down the concept of angular momentum and see how understanding it allows us to predict the motion of rotating systems, from skaters on ice to planets in orbit.

Concept overview

flowchart TD
    A[Start: Analyze a rotating system] --> B{Is there a net external torque?};
    B -- Yes --> C[Angular momentum L is NOT conserved];
    C --> D[Change in L is ΔL = τ_net * Δt];
    B -- No --> E[Angular momentum L IS conserved];
    E --> F[L_initial = L_final];
    F --> G{Does rotational inertia I change?};
    G -- Yes --> H[Angular speed ω must change to keep L constant];
    G -- No --> I[Angular speed ω remains constant];

Core explanation

Hello everyone. Let's dive into one of the most elegant principles in physics: the conservation of angular momentum.

You've already mastered the conservation of linear momentum. Remember? If there's no net external force on a system, its total momentum (p = mv) stays constant. This is why if you're standing on a skateboard and you throw a heavy ball forward, you and the skateboard roll backward.

Angular momentum has a very similar conservation law, but for rotation.

What is Angular Momentum?

For a single particle or a rigid body spinning about an axis, we define angular momentum, L, as:

L = Iω

Let's break that down:

I is the rotational inertia. It’s the rotational equivalent of mass. It tells you how difficult it is to change an object's rotation. It depends not just on mass, but on how that mass is distributed relative to the axis of rotation. Mass farther from the axis means a larger I.
ω (omega) is the angular velocity. It’s the rotational equivalent of velocity. It tells you how fast the object is spinning (in radians per second) and in what direction (clockwise or counter-clockwise).

So, angular momentum L is a measure of an object's "quantity of rotation."

The Law of Conservation of Angular Momentum

Here is the central rule:

If the net external torque on a system is zero, then the total angular momentum of that system is conserved.

In equation form, if τ_net, ext = 0, then:

L_initial = L_final

This means:

I_initial * ω_initial = I_final * ω_final

This simple equation is incredibly powerful. It explains the figure skater. Let's model that scenario, which is just like the interactive you see. Imagine a student, Priya, standing on a frictionless rotating platform, holding two heavy weights.

1
Initial State
Priya starts spinning slowly with her arms extended. Because the weights are far from her body (the axis of rotation), the system's rotational inertia, I_initial, is large. She has an initial angular velocity, ω_initial. Her initial angular momentum is L_initial = I_initial * ω_initial.
2
The Action
Now, Priya pulls the weights inward, close to her body. The force she uses is internal to the system (the system is Priya + weights). It's her muscles acting on the weights. This action doesn't involve any push or pull from the outside world, like the floor or a friend. Therefore, there is no net external torque.
3
Final State
With the weights held close, the system's mass is now concentrated near the axis of rotation. This makes the final rotational inertia, I_final, much smaller.

L_initial = L_final

Since I_final is now much smaller than I_initial, what must happen to ω to keep the equation balanced? The final angular velocity, ω_final, must become much larger.

Let's use the numbers from our visual.

Initially: I_initial = 5.0 kg·m² and ω_initial = 2.0 rad/s.
So, L = (5.0)(2.0) = 10.0 kg·m²/s.
Priya pulls the weights in, and the new rotational inertia is I_final = 2.0 kg·m².
Since L must still be 10.0, we can find the new speed: 10.0 = (2.0) * ω_final ω_final = 10.0 / 2.0 = 5.0 rad/s

She spins two and a half times faster, just by changing her shape!

Choosing Your System Wisely

This brings us to a critical skill in physics: defining your system (LO 6.4.B). The conservation law only applies if you choose a system where the net external torque is zero.

System: Priya + weights
The force she uses to pull her arms in is an internal force. It creates an internal torque, but by Newton's third law, the weights exert an equal and opposite torque on her arms. These internal torques cancel out. Since the platform is frictionless, there are no external torques. Angular momentum is conserved.
System: Just Priya
If we only looked at Priya, the force from the weights pulling on her hands would be an external force. Her angular momentum would change.
System: Just one weight
The force from Priya's hand pulling the weight inward is an external force on the weight. Its angular momentum is not conserved.

Angular momentum is always conserved in any interaction, but it might be transferred between objects or between a system and its environment. To use the simple L_initial = L_final equation, you must define your system so that all the messy pushes and pulls are internal.

What if There IS an External Torque?

If there is a net external torque, the angular momentum of the system will change. The relationship is:

ΔL = τ_net, ext * Δt

This is the definition of angular impulse. A net external torque applied over a time interval causes a change in angular momentum. Think about pushing a merry-go-round to get it started. You apply a torque over a few seconds, which gives it angular momentum.

Worked examples

Let's walk through a couple of problems to make this concrete.

Example 1

The Spinning Student

A student with a rotational inertia of 3.0 kg·m² is rotating at 2.0 rad/s on a frictionless stool. They are holding two 1.0 kg masses, each 1.0 m from the axis of rotation. The student then pulls the masses inward so they are only 0.2 m from the axis. What is their new angular speed? (Treat the masses as point masses).

1. Define the System and Check for Conservation

Our system is the student + the two masses.
The force used to pull the masses in is internal.
Friction is negligible, so there is no net external torque.
Therefore, angular momentum is conserved: L_initial = L_final.

2. Calculate Initial Rotational Inertia (I_initial)

The total I is the sum of the parts: I_student + I_masses.
The rotational inertia of a point mass is mr². Since there are two, I_masses = 2 * m * r_initial².
I_initial = I_student + 2 * m * r_initial²
I_initial = 3.0 kg·m² + 2 * (1.0 kg) * (1.0 m)² = 3.0 + 2.0 = 5.0 kg·m²
- (This matches the setup from our core explanation!)

3. Calculate Final Rotational Inertia (I_final)

The student's I doesn't change, but the radius of the masses does.
I_final = I_student + 2 * m * r_final²
I_final = 3.0 kg·m² + 2 * (1.0 kg) * (0.2 m)² = 3.0 + 2 * (1.0) * (0.04) = 3.0 + 0.08 = 3.08 kg·m²

4. Apply Conservation of Angular Momentum

I_initial * ω_initial = I_final * ω_final
(5.0 kg·m²) * (2.0 rad/s) = (3.08 kg·m²) * ω_final
10.0 = 3.08 * ω_final
ω_final = 10.0 / 3.08 ≈ 3.25 rad/s

As expected, because the rotational inertia decreased, the angular speed increased.

Example 2

The Clay and the Turntable

A 2.0 kg turntable with a radius of 0.5 m is at rest. It is free to rotate about its center. A 0.2 kg lump of clay is thrown with a speed of 10 m/s and hits the edge of the turntable, sticking to it. The clay's path is tangent to the turntable's edge. What is the angular speed of the turntable and clay after the collision? (Model the turntable as a solid disk, I = ½MR²).

1. Define the System

The system is the turntable + the lump of clay.
The collision force between the clay and the turntable is internal. There are no other horizontal forces or torques.
Therefore, angular momentum is conserved.

2. Calculate Initial Angular Momentum (L_initial)

Initially, the turntable is at rest (ω_turntable = 0), so it has no angular momentum.
The clay is moving in a straight line, but it has angular momentum relative to the turntable's axis. For a point mass, L = r * p_perp = r * (mv). Here, the velocity is perpendicular to the radius at the point of impact.
L_initial = L_clay = r * m_clay * v = (0.5 m) * (0.2 kg) * (10 m/s) = 1.0 kg·m²/s

3. Calculate Final Rotational Inertia (I_final)

After the collision, the clay and turntable rotate together. Their rotational inertias add up.
I_turntable = ½ * M * R² = ½ * (2.0 kg) * (0.5 m)² = 0.25 kg·m²
I_clay = m_clay * R² = (0.2 kg) * (0.5 m)² = 0.05 kg·m²
I_final = I_turntable + I_clay = 0.25 + 0.05 = 0.30 kg·m²
- Common mistake alert: Students often forget to include the clay's rotational inertia in the final state. The clay is now part of the spinning object!

4. Apply Conservation

L_initial = L_final
1.0 kg·m²/s = I_final * ω_final
1.0 = (0.30 kg·m²) * ω_final
ω_final = 1.0 / 0.30 ≈ 3.33 rad/s

The system, initially with L=1.0, ends up spinning together with the same total angular momentum.

Try it yourself

Here's a chance to apply what you've learned.

Problem 1: A large wooden disk (a merry-go-round) with a rotational inertia of 150 kg·m² is rotating at 0.5 rad/s. A 50 kg person, Carlos, is standing at the center. Carlos then walks out to the edge of the disk, a distance of 2.0 m from the center. What is the new angular speed of the merry-go-round?

Hint 1
What is your system? Is angular momentum conserved?
Hint 2
Calculate the initial rotational inertia. When Carlos is at the center (r=0), how much does he contribute to I?
Hint 3
Now calculate the final rotational inertia with Carlos at the edge. Treat him as a point mass.
Hint 4
Set L_initial equal to L_final and solve for the new ω. Do you expect the speed to increase or decrease?

TL;DR

In simple terms, conservation of angular momentum means a spinning system keeps its total "spin" constant unless an outside twist, or torque, acts on it.

Key terms

Angular momentum Conservation of angular momentum Rotational inertia Angular velocity Torque AP Physics 1 Rotational motion Angular impulse Physics lesson Shrutam US

You can now…

6.4.A: Describe the behavior of a system using conservation of angular momentum.
6.4.B: Describe how the selection of a system determines whether the angular momentum of that system changes.

Essential knowledge (exam-tested)

6.4.A.1: The total angular momentum of a system about a rotational axis is the sum of the angular momenta of the system's constituent parts about that axis.
6.4.A.2: Any change to a system's angular momentum must be due to an interaction between the system and its surroundings.
6.4.A.2.i: The angular impulse exerted by one object or system on a second object or system is equal and opposite to the angular impulse exerted by the second object or system on the first. This is a direct result of Newton's third law.
6.4.A.2.ii: A system may be selected so that the total angular momentum of that system is constant.
6.4.A.2.iii: The angular speed of a nonrigid system may change without the angular momentum of the system changing if the system changes shape by moving mass closer to or further from the rotational axis.
6.4.A.2.iv: If the total angular momentum of a system changes, that change will be equivalent to the angular impulse exerted on the system.
6.4.B.1: Angular momentum is conserved in all interactions.
6.4.B.2: If the net external torque exerted on a selected object or rigid system is zero, the total angular momentum of that system is constant.
6.4.B.3: If the net external torque exerted on a selected object or rigid system is nonzero, angular momentum is transferred between the system and the environment.

Concept map

flowchart TD
    A[Start: Analyze a rotating system] --> B{Is there a net external torque?};
    B -- Yes --> C[Angular momentum L is NOT conserved];
    C --> D[Change in L is ΔL = τ_net * Δt];
    B -- No --> E[Angular momentum L IS conserved];
    E --> F[L_initial = L_final];
    F --> G{Does rotational inertia I change?};
    G -- Yes --> H[Angular speed ω must change to keep L constant];
    G -- No --> I[Angular speed ω remains constant];

Read what Saavi narrates

Have you ever watched a figure skater start a spin with their arms stretched out, and then, by simply pulling their arms in, they suddenly whirl into a blur? It looks like magic, but it's pure physics. The secret is the conservation of angular momentum.

The main idea is this: a spinning object keeps its total "spin" constant, unless an outside twist, or torque, acts on it. We measure this "spin" with a quantity called angular momentum, which we label with a capital L. It's calculated as L equals I times omega, where I is rotational inertia, and omega is the angular speed.

If there's no net external torque, then the initial angular momentum equals the final angular momentum.

Let's walk through an example. Imagine a student on a spinning stool. Their initial rotational inertia, I initial, is 5 kilogram meters squared, and they're spinning at an initial angular speed, omega initial, of 2 radians per second. Their angular momentum, L, is I times omega, so 5 times 2, which equals 10.

Now, they pull their arms in, changing their shape. This makes them easier to spin, so their rotational inertia decreases to a final value, I final, of 2 kilogram meters squared.

Because no one pushed them from the outside, their angular momentum must still be 10. So, we set up the equation: L equals I final times omega final. That's 10 equals 2 times the new angular speed. Solving for that new speed, we get 10 divided by 2, which is 5 radians per second. By pulling their arms in, they went from spinning at 2 radians per second to 5 radians per second!

A common mistake I see every year is students forgetting that the rotational inertia, I, changes when the shape of the system changes. They see the skater pull their arms in and just assume the speed stays the same. But the whole reason the speed changes is because the inertia changes. Remember, it's the product, I times omega, that stays constant, not the individual parts.

So, keep this powerful idea in mind. If you see a rotating system with no external torques, you know its angular momentum is conserved. It's a fundamental rule of our universe, and a key tool for your AP exam. You've got this.

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