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. (a) 2 cos (ω t) : SHM. (b) cos ω t + cos 3ω t : Periodic (period (2π/ω) ), not SHM (multiple frequencies). (c) 3 sin (2ω t) : SHM. (d) e⁻ω t : Not periodic. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A²,