Definition

Torque (τ) is the rotational equivalent of force - it causes angular acceleration:

• T = r × F = rF sin θ

where r is the position vector from the axis of rotation to where the force is applied.

Cross Product

The cross product (×) produces a vector perpendicular to both input vectors:

• A × B = |A||B| sin θ n̂

where n̂ is the unit vector perpendicular to both A and B (right-hand rule).

Moment of Inertia

Moment of inertia (I) is the rotational equivalent of mass - resistance to angular acceleration:

• I = Σmᵢrᵢ²

For common shapes:

• Point mass: I = mr²
• Solid disk/cylinder (about center): I = ½MR²
• Solid sphere (about center): I = ⅖MR²
• Rod (about center): I = 1/12 ML²

Rotational Analogs

Many linear quantities have rotational equivalents:

• Position x → Angle θ
• Velocity v → Angular velocity ω