Angular Momentum (3D)

Conservation of angular momentum, precession, and gyroscopic effects

Angular momentum L = Iω is a vector quantity pointing along the axis of rotation (right-hand rule). In the absence of external torque, angular momentum is conserved: L_i = L_f. For a spinning ice skater, pulling arms in decreases I, so ω must increase to conserve L. A gyroscope tilted at angle θ from vertical experiences gravitational torque τ = Mgd·sinθ perpendicular to L, causing L to precess around the vertical axis at rate Ω = Mgd/(Iω). The faster the spin, the slower the precession. In collisions between rotating objects (e.g., two disks), the total angular momentum is conserved: I₁ω₁ + I₂ω₂ = (I₁ + I₂)ω_f for a perfectly inelastic collision.