Torque: τ = r × F × sin(θ) is the rotational equivalent of force. r is the distance from axis (moment arm), F is force magnitude, θ is angle between r and F. Maximum when perpendicular (θ = 90°). Units: N⋅m. Clockwise negative, counter-clockwise positive.
Moment of Inertia: I is rotational mass, resistance to angular acceleration. Depends on mass distribution: I = Σ(m_i r_i²). Common: disk I = ½MR², sphere I = ⅖MR², rod about center I = 1/12 ML². Units: kg⋅m².
Angular Momentum: L = Iω (analogous to p = mv). Conserved in closed systems. Ice skaters pull arms in to spin faster (↓I means ↑ω). Relationship: τ = dL/dt.
Rotational KE: KE_rot = ½Iω² (analogous to ½mv²). Rolling objects have both: KE_total = ½mv² + ½Iω². Work: W = τθ. Power: P = τω.
Newton's 2nd Law for Rotation: τ = Iα, where α is angular acceleration. Analogous to F = ma. Equilibrium: Στ = 0 (no rotation or constant rotation).
Applications: Gears and gear ratios, flywheels (energy storage), engines, turbines, robot joints, sports (figure skating, diving), wrenches (longer = more torque), planetary rotation, Atwood machines.