Practice question

**Damped oscillations** occur when resistive forces dissipate energy, amplitude decays exponentially as A(t) = A₀ e^(-b t/2m), b damping coefficient (kg/s), frequency slightly reduced ω' = √(ω₀² - (b/2m)²). Damping arises from friction or viscosity, energy loss per cycle proportional to velocity squared, motion eventually stops. Total energy: E = (1/2) k A² = 0.5 × 800 × (0.05)² = 1 J . Potential energy: U = (1/2) k x² = 0.5 × 800 × (0.025)² = 0.25 J . Kinetic energy: K = E - U = 1 - 0.25 = 0.75 J . Applying x = A cos(ωt + φ), v = -ωA