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. T = 2π √((m/k)) . 0.6 = 2π √((0.9/k)) ⇒ (0.6/2π) = √((0.9/k)) . (0.0955)² = (0.9/k) ⇒ k = (0.9/0.00912) ≈ 98.68 N/m . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 98.68 N/m follows, reflecting SHM dependence on amplitude A, ω and system parameters.