Conservation of Angular | AP Physics 1 Unit 6 Study Guide

Getting Started

Consider an ice skater spinning in place. They begin with their arms outstretched, rotating at a certain speed, but when they pull their arms in close to their body, they suddenly spin much faster. This phenomenon, seen in everything from planetary orbits to children on a playground merry-go-round, is governed by a fundamental conservation law. The core question we will explore is: How can a system change its speed of rotation without any external push or pull, and how can we predict the outcome of such a change?

What You Should Be Able to Do

After working through this section, you should be able to:

Define a system and identify the conditions required for its angular momentum to be conserved.
Predict the final angular velocity of a system if its mass distribution (and thus its rotational inertia) changes.
Calculate the total angular momentum of a system composed of multiple parts.
Explain how the choice of a system boundary can determine whether a torque is considered internal or external, and therefore whether the system's angular momentum is constant.

Key Concepts & Mechanisms

This topic is best understood through the lens of Interactions & Conservation, which focuses on how a defined system interacts with its environment and which physical quantities remain constant during those interactions.

System & Preconditions

To apply conservation of angular momentum, we must first clearly define our system and the ideal conditions.

System: The object or collection of objects we are analyzing. For the ice skater, the system could be just the skater. For two colliding asteroids, the system could be both asteroids together.
Environment: Everything outside the defined system.
Torque ( $τ$ ): The rotational equivalent of a force, which can cause an object to change its rotational motion. It is measured in Newton-meters (N·m).
External Torque ( $τ_{e x t}$ ): A torque exerted on the system by an object in the environment. For the skater, the friction from the ice is an external torque.
Internal Torque: A torque exerted by one part of the system on another part. When the skater pulls her arms in, her torso exerts a torque on her arms, and her arms exert an equal and opposite torque on her torso. These are internal torques.

The single most important precondition for the conservation of angular momentum is that the net external torque on the system must be zero ( $Σ τ_{e x t} = 0$ ). When this condition is met, the system is called an isolated system in terms of rotation. In many problems, we make the idealization that dissipative forces like friction and air resistance are negligible, allowing us to treat the system as isolated.

Key Steps / Relations

Define Angular Momentum.
Angular Momentum ( $L$ ) is the measure of an object's rotational motion about an axis. It is the rotational analog of linear momentum and is a vector quantity, though in this course, we often deal with its magnitude. For a rigid body rotating about a fixed axis, its magnitude is given by:
$L = I ω$
- $L$ is the angular momentum (SI units: kg·m²/s).
- Rotational Inertia ( $I$ ) is a scalar that measures an object's resistance to a change in its rotational motion. It depends on the object's mass and how that mass is distributed relative to the axis of rotation (SI units: kg·m²).
- Angular Velocity ( $ω$ ) is the rate of rotation (SI units: rad/s).
Sum the Parts.
The total angular momentum of a system is the sum of the angular momenta of its constituent parts. For a system of two objects rotating about the same axis:
$L_{t o t a l} = L_{1} + L_{2} = I_{1} ω_{1} + I_{2} ω_{2}$
Care must be taken with direction; counter-clockwise is typically positive and clockwise is negative.
Apply the Conservation Law.
The Law of Conservation of Angular Momentum states: If the net external torque on a system is zero, the total angular momentum of that system remains constant.
This means the system's initial angular momentum is equal to its final angular momentum:
$L_{ini t ia l} = L_{f ina l}$
For a single object that changes its shape (like our skater), this becomes:
$I_{ini t ia l} ω_{ini t ia l} = I_{f ina l} ω_{f ina l}$

Outputs & Effects

What Changes: In an isolated system, internal forces and torques can cause the system's configuration to change. This often results in a change to the system's total rotational inertia ( $I$ ). As a consequence, the system's angular velocity ( $ω$ ) must also change to keep the product $I ω$ constant.
What Remains Constant: The total angular momentum ( $L$ ) of the isolated system.

This leads to a critical inverse relationship: if a spinning, isolated system rearranges its mass to decrease its rotational inertia, its angular velocity must increase, and vice versa.

Regulation & Limits

Domain of Validity: The law is only valid for systems where the net external torque is zero or negligibly small. If a significant external torque (like a brake being applied) acts on the system, angular momentum is not conserved.
System Choice is Key: Consider a student standing on a turntable who is initially at rest. If the student begins to run along the edge of the turntable, the turntable will begin to rotate in the opposite direction.
- If the system is just the student, the friction from the turntable is an external torque, and the student's angular momentum changes.
- If the system is the student and the turntable together, the friction force between them is internal. With no external torques (assuming a frictionless axle), the total angular momentum of the student-turntable system remains constant at its initial value: zero. The student gains angular momentum in one direction, and the turntable gains an equal amount in the opposite direction.

Key Models & Diagrams

The process of solving a conservation of angular momentum problem can be visualized with the following workflow.

Situation & Representation	System Analysis	Governing Equation	Predicted Observable
Initial State: A figure skater spins with arms outstretched. Sketch: Wide profile, low $ω_{i}$ .	System: Skater. Analysis: Assume friction is negligible, so $Σ τ_{e x t} = 0$ . Angular momentum is conserved.	$L_{ini t ia l} = L_{f ina l}$ $I_{i} ω_{i} = I_{f} ω_{f}$	The final angular velocity, $ω_{f}$ , can be predicted.
Final State: The skater pulls their arms in. Sketch: Narrow profile, high $ω_{f}$ .	System: Skater. Analysis: Pulling arms in is an internal process. Mass is now closer to the axis, so $I_{f} < I_{i}$ .	$ω_{f} = ω_{i} (\frac{I _{i}}{I _{f}})$	Since $I_{i} > I_{f}$ , the ratio $(I_{i} / I_{f})$ is greater than 1. Therefore, $ω_{f} > ω_{i}$ . The skater speeds up.

Key Components & Evidence

Angular Momentum ( $L$ ): The conserved quantity of rotational motion for an isolated system. Its conservation dictates the system's behavior. (Units: kg·m²/s).
Rotational Inertia ( $I$ ): The property of a system that resists changes in rotational motion. It is the variable that is often changed by internal forces. (Units: kg·m²).
Angular Velocity ( $ω$ ): The rate of spin. It changes in response to changes in rotational inertia to keep $L$ constant. (Units: rad/s).
Net External Torque ( $Σ τ_{e x t}$ ): The agent of change for a system's angular momentum. If it is zero, angular momentum is conserved. (Units: N·m).
System: The defined collection of objects. The choice of system determines which torques are external.
Conservation Law: The principle stating $L_{ini t ia l} = L_{f ina l}$ when $Σ τ_{e x t} = 0$ . This is the core predictive tool.
Inverse Relationship: The observable evidence that as $I$ decreases, $ω$ increases (and vice versa) for an isolated system.
Skater Example: The classic real-world evidence. A skater pulling their arms in reduces their rotational inertia, causing their angular velocity to increase dramatically.
Planetary Orbits: A planet's angular momentum is conserved (mostly) as it orbits the Sun. As its distance from the sun changes in an elliptical orbit, its speed changes to keep $L$ constant.

Skill Snapshots

Causation

A net external torque applied to a system causes a change in that system's total angular momentum.
A redistribution of mass within an isolated system causes a change in the system's rotational inertia.
In an isolated system, a change in rotational inertia causes an inverse change in angular velocity to maintain constant angular momentum.

Comparison

Feature	Open System	Isolated System
External Torques	Net external torque is non-zero.	Net external torque is zero.
Angular Momentum	Angular momentum is not conserved.	Angular momentum is conserved.
Example	A spinning wheel slowing down due to friction.	A diver tucking in mid-air (neglecting air resistance).

Internal vs. External Torques: Internal torques act between parts of a system and cannot change the system's total angular momentum; external torques act from the environment and are the only way to change the system's total angular momentum.
Angular vs. Linear Momentum: Angular momentum ( $L = I ω$ ) depends on mass distribution, while linear momentum ( $p = m v$ ) does not. Both are fundamental conserved quantities of isolated systems.

Change Over Time

Baseline: A spinning turntable has an initial rotational inertia $I_{i}$ and angular velocity $ω_{i}$ .
Change 1: A piece of clay is dropped vertically onto the edge of the turntable, sticking to it. This interaction increases the total mass and changes its distribution, resulting in a new, larger rotational inertia for the system ( $I_{f} > I_{i}$ ).
Change 2: Because the clay-turntable system is isolated from external horizontal torques, its angular momentum is conserved. The angular velocity must decrease to compensate for the increased rotational inertia ( $ω_{f} < ω_{i}$ ).
Continuity: The total angular momentum of the clay-turntable system just before the collision is equal to the total angular momentum just after the collision.

Common Misconceptions & Clarifications

Misconception: Any change in rotational speed requires an external torque.
- Clarification: A system can change its own rotational speed without any external influence simply by changing its shape or configuration. This internal change alters the rotational inertia ( $I$ ), which in turn alters the angular velocity ( $ω$ ) to keep the product $L = I ω$ constant.
Misconception: If angular momentum is conserved, then rotational kinetic energy must also be conserved.
- Clarification: This is false. Rotational kinetic energy is $K_{ro t} = \frac{1}{2} I ω^{2}$ . When a skater pulls her arms in, she does work. This work adds energy to the system. While $L = I ω$ is constant, $K_{ro t}$ increases. You can show algebraically that $K_{ro t} = L^{2} / (2 I)$ , so if $L$ is constant and $I$ decreases, $K_{ro t}$ must increase.
Misconception: An object must be rotating to have angular momentum.
- Clarification: A point mass moving in a straight line can have angular momentum about a point not on its path. Its angular momentum is $L = m v r_{⊥}$ , where $r_{⊥}$ is the perpendicular distance from the point to the object's line of motion. This is crucial for understanding how a non-rotating object (like a piece of clay) can transfer angular momentum to a system upon collision.

One-Paragraph Summary

Conservation of angular momentum is a fundamental principle stating that the total angular momentum of a system remains constant, provided that no net external torque acts upon it. This principle is expressed by the equation $L_{ini t ia l} = L_{f ina l}$ , which for a system changing its shape becomes $I_{i} ω_{i} = I_{f} ω_{f}$ . This relationship reveals a powerful inverse dynamic: if a system's rotational inertia ( $I$ ) decreases, its angular velocity ( $ω$ ) must increase to keep the total angular momentum constant. The strategic selection of the system boundary is critical, as it determines which torques are internal (and do not affect total $L$ ) and which are external (and do). This law provides a powerful tool for predicting the final state of a rotating system after an interaction without analyzing the complex internal torques involved.

Conservation of Angular Momentum - AP Physics 1: Algebra-Based Study Guide