(455) HL Moment of inertia [IB Physics HL]

Moment of Inertia

• Defined as an object's resistance to change in rotational motion.

• Represented by the letter I.

• Key equation: I = Σ m r².

  • This indicates the sum of the products of mass and the square of the distance from the axis of rotation.

  • Important to consider how the mass distribution affects moment of inertia.

Units

• Moment of inertia has units: kilogram meter squared (kg m²).

Importance in Physics

• Moment of inertia is crucial for understanding rotational dynamics.

• Typically calculated using pre-defined equations for different shapes (e.g., spheres, cylinders).

• Complicated by shape and symmetry; thus, equations for common forms are often provided.

Examples of Moment of Inertia for Common Shapes

• Solid Sphere: I = (2/5) m R².

• Solid Cylinder: I = (1/2) m R².

Calculation Using the Equation

• If masses are not simple shapes, use I = Σ m r², adding up for each mass at distance r from the rotation axis.

• Example breakdown:

  • For multiple objects: I = m₁ r₁² + m₂ r₂² + ... + m₃ r₃²

Practical Example: Rolling Spheres

• Consider two spheres on an incline:

  • Sphere 1: Solid sphere (I₁ = (2/5) m R²)

  • Sphere 2: Hollow sphere (I₂ = (2/3) m R²)

• Question: Which sphere reaches the bottom first?

• Analysis: Compare moment of inertia values:

  • I₂ > I₁ → Sphere 1 has smaller moment of inertia; it will reach the bottom first.

Kinetic Energy Considerations

• Rotational kinetic energy formula: KE_rotation = (1/2) I ω².

• Smaller moment of inertia (I) allows more energy to be translated into speed (translational kinetic energy).

• Sphere 1 (solid sphere) will reach the bottom first due to its smaller moment of inertia, resulting in greater translational speed.

Conclusion

• Recap of moment of inertia, its equation, example comparisons, and the relationship between rotational and translational kinetic energy.