Unit 6: Energy and Momentum of Rotating Systems

Introduction

Rolling motion occurs when an object moves forward while also rotating about its axis. A key feature of rolling without slipping is the relationship between translational and rotational motion. Understanding rolling motion is essential for analyzing the dynamics of rotating objects in real-world applications.

Key Concepts

• Rolling Without Slipping: The point of contact between the rolling object and the surface is momentarily at rest. The condition for rolling without slipping is: v = rω, where v is linear velocity, r is the radius, and ω is angular velocity.
• Total Kinetic Energy: Rolling motion combines **translational kinetic energy** and **rotational kinetic energy**:
  • Translational: \( KE_{\text{trans}} = \frac{1}{2} m v^2 \)
  • Rotational: \( KE_{\text{rot}} = \frac{1}{2} I \omega^2 \)
• Moment of Inertia (I): Different shapes have different moments of inertia, affecting how they roll.
• Friction’s Role: Static friction prevents slipping and allows rolling, but it does no work since the point of contact does not slide.

Mathematical Routines

• Use v = rω to relate linear and rotational motion.
• For rolling kinetic energy: \[ KE_{\text{total}} = \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2 \]
• Determine acceleration using Newton’s Second Law for both translation (\(F = ma\)) and rotation (\(\tau = I\alpha\)).

Creating Representations

• Diagrams: Draw force diagrams showing friction at the contact point, the center of mass, and torque directions.
• Graphs: Sketch **KE vs. time** and **velocity vs. time** for rolling objects.

Scientific Questioning & Argumentation

AP Physics emphasizes conceptual reasoning. Consider:

• Why does a solid sphere roll down a ramp faster than a hoop?
• How does increasing the moment of inertia affect rolling motion?

Practice Activities

Summary & Exam Preparation Tips

• Rolling motion combines **translation and rotation**; use v = rω.
• Static friction enables rolling but does no work.
• Objects with smaller moments of inertia roll faster down inclines.