Unit 6.4: Conservation of Angular Momentum

Introduction

Conservation of angular momentum is a core principle in physics stating that if no external torque acts on a system, its total angular momentum remains constant. This concept governs rotational motion, much like conservation of linear momentum governs linear motion. A familiar example is a figure skater who spins faster by pulling their arms in, illustrating how angular momentum conservation affects rotational speed.

Key Concepts

Angular Momentum (L):
- For a point particle: L = m v r sinθ, where m is mass, v is velocity, r is the distance from the rotation axis, and θ is the angle between the velocity and position vectors.
- For a rigid body: L = I ω, where I is the moment of inertia and ω is the angular velocity.
Conservation Law: If the net external torque is zero (Στ = 0), then L_initial = L_final.
Condition: Conservation holds only when no external torques (e.g., friction or applied forces) act on the system.

Mathematical Routines

To solve problems using conservation of angular momentum:

Define the system and confirm no external torques are present.
Compute the initial angular momentum (L_initial).
Set L_initial = L_final and solve for the unknown (e.g., final angular velocity).
For rigid bodies, use the equation I_initial ω_initial = I_final ω_final.

Tip: The moment of inertia (I) depends on mass distribution—recalculate it if the system’s shape changes!

Creating Representations

Visual tools help clarify conservation:

Diagrams: Sketch the system before and after an event, labeling ω and I.
Graphs: Plot ω vs. time; a horizontal line indicates conserved angular momentum.

Reminder: Use the right-hand rule to determine the direction of L.

Scientific Questioning & Argumentation

Apply the concept to explain real-world phenomena:

Question: Why does a spinning top stay upright?
Answer: Its angular momentum resists external torques, maintaining stability.
Question: How does a cat land on its feet?
Answer: By twisting its body to adjust I, the cat conserves L and rotates to an upright position.

Exam Tip: Always justify whether conservation applies by checking for external torques.

Practice Activities

Activity 1: Figure Skater

A skater spins at 2 rad/s with a moment of inertia of 5 kg·m². They pull their arms in, reducing I to 3 kg·m². What’s their final angular velocity if no external torques act?

Activity 2: Colliding Disks

Disk 1 (mass 2 kg, radius 0.1 m) spins at 10 rad/s, while Disk 2 (mass 3 kg, radius 0.15 m) is at rest. They collide and stick together. Find the final angular velocity and sketch the system.

Summary & Exam Preparation Tips

Key takeaways:

L is conserved when Στ = 0.
For rigid bodies: L = I ω.
Changes in I affect ω.

Exam preparation:

Check for external torques before assuming conservation.
Use diagrams to track system changes.
Ensure units are consistent (L in kg·m²/s).