Rotational Kinematics (Advanced)

Angular velocity, acceleration, and rolling motion with calculus

Rotational kinematics parallels translational kinematics: angular displacement θ replaces x, angular velocity ω replaces v, and angular acceleration α replaces a. For constant α: ω(t) = ω₀ + αt and θ(t) = θ₀ + ω₀t + ½αt². The moment of inertia I depends on mass distribution: I = ½MR² (solid disk/cylinder), I = MR² (ring/thin cylinder), I = ⅖MR² (solid sphere). Newton's second law for rotation is τ = Iα. For rolling without slipping, the contact point has zero velocity: v_cm = Rω. The total kinetic energy is K = ½Mv²_cm + ½Iω² = ½(M + I/R²)v²_cm. When rolling down an incline of angle θ, a = gsinθ/(1 + I/(MR²)), meaning objects with larger I/MR² roll more slowly.