Lecture on Moment of Inertia of a Disk and Mass

Introduction

• The lecture covers the concept of moment of inertia for two objects:
  • A uniform disk
  • A mass placed on or around the spinning disk
• Objective: Calculate how the moment of inertia changes when combining these two objects.

System Description

• Uniform Disk:
  • Mass (M) = 60 kg
  • Radius (R) = 3 meters
  • Rotating about an axis passing through its center.
• Stone/Mass:
  • Considered as a point mass
  • Placed at a distance of 2 meters from the axis of rotation.
  • Mass of stone (m) = 5 kg

Calculation of Moment of Inertia

• Moment of Inertia of the Disk (I_d):

  • Formula: (I_d = \frac{1}{2} M R^2)
  • Substituting values: (I_d = \frac{1}{2} \times 60 \times 3^2 = 270) kg·m²

• Moment of Inertia of the Stone (I_s):

  • Treated as a point mass
  • Formula: (I_s = m r^2)
  • Substituting values: (I_s = 5 \times 2^2 = 20) kg·m²

• Total Moment of Inertia (I_total):

  • Formula: (I_{total} = I_d + I_s)
  • Calculation: (I_{total} = 270 + 20 = 290) kg·m²

Important Notes

• Moment of inertia is a scalar quantity, meaning it can be directly added.
• Ensure calculations are done about the same axis of rotation to maintain consistency.
• Results differ if the axis of rotation changes.

Conclusion

• The combined moment of inertia for the disk and mass system is 290 kg·m².
• The moment of inertia increases when the mass is added at a distance from the axis of rotation.

Additional Information

• For any questions, comments are encouraged.
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