content_copy

Unit 6: Energy and Momentum of Rotating Systems

6.3: Angular Momentum and Angular Impulse

Introduction

Angular momentum and angular impulse are the rotational counterparts to linear momentum and impulse. Angular momentum quantifies an object’s rotational motion, while angular impulse describes how torque alters that motion over time. These concepts are vital for analyzing spinning objects—like a figure skater speeding up by pulling in their arms or a gyroscope maintaining its orientation. Understanding them is key to mastering rotational dynamics for the AP Physics 1 Exam.

Key Concepts

  • Angular Momentum (L): Measures rotational motion, calculated as L = I ω, where:
    • I: Rotational inertia (kg·m²).
    • ω: Angular velocity (rad/s).
    Units: kg·m²/s.
  • Angular Impulse (ΔL): The change in angular momentum due to torque over time, given by ΔL = τ Δt, where:
    • τ: Torque (N·m).
    • Δt: Time interval (s).
    Units: kg·m²/s (same as angular momentum).
  • Conservation of Angular Momentum: When no external torque acts (Στ = 0), angular momentum stays constant: Linitial = Lfinal.

Mathematical Routines

Follow these steps for calculations:

  1. Angular Momentum: Use L = I ω. Identify I based on the rotation axis.
  2. Angular Impulse: Compute ΔL = τ Δt with average torque τ.
  3. Change in Angular Momentum: ΔL = Lfinal - Linitial.
  4. Conservation: If Στ = 0, set Iinitial ωinitial = Ifinal ωfinal.
Tip: Use radians per second for ω and double-check units (kg·m²/s for L and ΔL).

Creating Representations

Visual tools help on the AP Exam:

  • Diagrams: Sketch the object, axis, and arrows for ω and τ.
  • Vector Diagrams: Show L perpendicular to the rotation plane (use the right-hand rule).
  • Graphs: Plot torque vs. time; angular impulse is the area under the curve.
Reminder: Right-hand rule: Curl fingers in the rotation direction; thumb points along L.

Scientific Questioning & Argumentation

Practice AP-style reasoning:

  • “Why does a figure skater spin faster when pulling their arms in?”
    Answer: Reducing I increases ω to conserve L since Στ = 0.
  • “How does torque over time affect angular momentum?”
    Answer: Torque delivers angular impulse (ΔL = τ Δt), changing L.
Exam Tip: For conservation, state “no external torque” and use Linitial = Lfinal.

Practice Activities

Activity 1: Angular Momentum

A disk (mass 3 kg, radius 0.2 m) spins at 15 rad/s about its center. Find its angular momentum. Draw a diagram.

Activity 2: Angular Impulse

A 4 N·m torque acts on a wheel (I = 0.5 kg·m²) for 2 s. Calculate ΔL and final ω if starting from rest. Include a diagram.

Summary & Exam Preparation Tips

Master these:

  • L = I ω and ΔL = τ Δt.
  • Conservation when Στ = 0.
  • Visualize with diagrams.

Exam strategies:

  • Practice calculations and reasoning.
  • Verify units (kg·m²/s).
  • Explain using formulas and conservation.