Practice question

**Resonance phenomenon** amplifies response when driving frequency matches natural frequency ω_d ≈ ω₀, large amplitude even with small F₀, as damping limits growth. Natural frequency determined by system parameters, resonance condition crucial for understanding vibrations and energy absorption, e.g., bridge collapse, tuning. The periodicity arises from the repeating nature of sine and cosine functions, which have a fixed period ( 2π/ω ), ensuring the motion repeats consistently. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result The periodicity of trigonometric functions follows, reflecting SHM dependence on amplitude A, ω and system parameters.