Moment of Inertia and Torque Summary

Moment of Inertia

• Characterizes rotational inertia, depends on rotation axis.
• Kinetic energy for rotating body: KE = Iω², where I = moment of inertia.

Conservation of Energy

• In a system: GPEinitial + KEinitial = GPEfinal + KEfinal.
• Example shows how to relate gravitational potential energy to kinetic energy in rotating systems.

Moments of Inertia for Various Bodies

• Formulas for different geometries:
• Slender rod (axis through center): I = 1/12 ML²
• Slender rod (axis through one end): I = 1/3 ML²
• Rectangular plate (center): I = 1/12 M(a² + b²)
• Solid cylinder: I = 1/2 MR²
• Hollow cylinder: I = 1/2 M(R₁² + R₂²)

Parallel-Axis Theorem

• I = ICM + Md², where I is about any parallel axis, ICM is about the center of mass.
• Efficient way to calculate moments of inertia without recalculating for every possible axis.

Torque

• Quantifies the ability to cause rotation:
• Depends on force magnitude and distance from the axis of rotation.
• Torque τ = r × F, maximizing when F is perpendicular.
• Magnitude: τ = rFsin(φ) where φ is the angle between r and F.

Key Points on Torque and Forces

• Larger torque when force is applied further from rotation axis.
• Same force at different lever arms alters torque effectiveness.
• Essential to consider direction and application of force for torque magnitude.