Parallel Axis Theorem Overview

• Used to calculate moment of inertia for systems with varying mass distributions relative to the axis of rotation.

Case 1: Sphere and Rod System

• Axis of Rotation: Defined for calculations.
• Moment of Inertia Formula:
  • $I = I_{cm} + m imes d^2$
  • $I<em>{cm} = \frac{2}{5} m</em>s r^2$ (Sphere's moment of inertia about its center of mass)
• Variables:
  • Mass of sphere ($m_s = 10$ kg)
  • Radius of sphere ($r = 0.5$ m)
  • Length of rod ($l = 2$ m)
  • Distance ($d = l + r = 2 + 0.5 = 2.5$ m)
• Calculation Result: Total moment of inertia = 70.2 kg m²

Case 2: Sphere at Edge of Axis of Rotation

• Moment of Inertia of Sphere:
  • $I = I_{cm} + m imes r^2$
  • $d = r$
  • Resulting in: $I<em>{total} = \frac{7}{5} m</em>s r^2$
• Calculation Result: Total moment of inertia = 10.17 kg m²

Case 3: Axis of Rotation Through Center of Mass of Sphere

• Moment of Inertia of Sphere:
  • $I = \frac{2}{5} m_s r^2$ (since $d=0$)
• Calculation for Rod:
  • $I = I_{cm} + m imes (\frac{l}{2} + r)^2$
• Calculation Result: Total moment of inertia = 13.92 kg m²

Summary of Results

• Case 1 Total = 70.2 kg m²
• Case 2 Total = 10.17 kg m²
• Case 3 Total = 13.92 kg m²
• Observation: Moment of inertia increases with distance from the axis of rotation due to mass distribution.