Moment of Inertia of a Disk and Added Mass

Moment of Inertia Overview

• Discusses moment of inertia for two objects: a disk and an additional mass.

Uniform Disk Characteristics

• A uniform disk with:
  • Mass ($M_D$): 60 kg
  • Radius ($R$): 3 m
  • Axis of rotation: through its center.

Calculating Moment of Inertia

• Moment of inertia ($I_D$) formula for a disk:
  • $I<em>D = \frac{1}{2} M</em>D R^2$
  • Calculation: $I_D = \frac{1}{2} \times 60 \times 3^2 = 270 \text{ kg m}^2$

Additional Mass Characteristics

• A stone placed at a distance of 2 m from the axis of rotation:
  • Mass ($M_S$): 5 kg

Stone's Moment of Inertia Calculation

• Moment of inertia ($I_S$) for the stone, treated as a point mass:
  • $I<em>S = M</em>S r^2$
  • Calculation: $I_S = 5 \times 2^2 = 20 \text{ kg m}^2$

Combined Moment of Inertia

• Total moment of inertia ($I_{total}$) of the system (disk + stone):
  • $I<em>{total} = I</em>D + I_S = 270 + 20 = 290 \text{ kg m}^2$

Important Reminders

• Moment of inertia is a scalar quantity; add values together.
• Results depend on the axis of rotation.
• Ensure calculations are performed about the same axis.