Lecture Notes: Moment of Inertia and Rotational Motion

Introduction

• Problem Statement: Comparison of a solid sphere, a solid cylinder, and a hollow cylinder of the same mass rolled down a ramp.
  • Objective: Determine which reaches the bottom first.
  • Key Concept: Moment of Inertia (MOI)

Moment of Inertia

• Definition: A scalar value indicating the difficulty of changing an object's rotational velocity around a given axis.
  • High MOI: Difficult to accelerate angularly.
  • Low MOI: Easier to rotate.
• Factors Affecting MOI:
  • Mass of the object
  • Distribution of mass relative to the axis of rotation
  • Shape and axis of rotation

Examples of Moment of Inertia

1. Two Wheels with Same Mass, Different Radii:

   • Wheel A has larger radius than Wheel B.
   • Conclusion: Wheel A has greater MOI than Wheel B.

2. Mass Change Effect:

   • Example: Wheel with mass 20 kg, radius 0.2 m:
     • MOI = 20 * (0.2)^2 = 0.8 kg·m²
   • Example: Wheel with mass 25 kg, same radius:
     • MOI = 25 * (0.2)^2 = 1 kg·m²
   • Conclusion: Increase in mass increases MOI.

3. Radius Change Effect:

   • Example: Two wheels, same mass, different radii (0.2 m and 0.5 m):
     • MOI increases as radius increases.

4. Axis of Rotation Change:

   • Example: Spinning a rod:
     • MOI about center axis is ml²/12
     • MOI about end axis is ml²/3
   • Conclusion: MOI is greater when mass is distributed far from the axis.

Conclusion on Moment of Inertia

• MOI depends on:
  • Mass
  • Distribution of mass
  • Axis of rotation
• Different shapes have specific MOI formulas.

Application to Initial Problem

• Solid Sphere MOI: 2/5 Mr²
• Solid Cylinder MOI: 1/2 Mr²
• Hollow Cylinder MOI: Mr²
• Analysis:
  • Given the same mass, the sphere has the least MOI.
  • Conclusion: Solid sphere will reach the bottom first due to its lower moment of inertia.

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