Oscillations and Waves
Oscillations and waves questions test simple harmonic motion, pendulum and spring periods, energy exchange, wave speed, standing waves, sound and resonance.
Core rules
- SHM: a = −ω²x, x = A cos(ωt + φ), ω = 2π/T = 2πf. Acceleration is opposite to displacement.
- Spring–mass T = 2π√(m/k); simple pendulum T = 2π√(L/g) for small angles.
- Energy: E = ½kA² = ½mω²A². Kinetic energy is maximum at equilibrium, potential energy maximum at the extremes.
- Wave relation v = fλ. The source fixes frequency; the medium fixes speed, so on refraction frequency stays constant while wavelength changes.
- String wave speed v = √(T/μ). Standing waves on a string fixed at both ends have f_n = nv/(2L).
- Resonance occurs when driving frequency equals a natural frequency.
Worked example
A pendulum of length 1.0 m swings with small amplitude where g = 10 m/s². T = 2π√(L/g) = 2π√(1.0/10) = 2π × 0.316 ≈ 2.0 s. The mass does not appear because the restoring force and inertia both scale with mass.
Traps
Using degrees in ωt; applying the pendulum formula at large angles; confusing wave speed with particle speed; claiming frequency changes when a wave enters a new medium; assuming doubling amplitude doubles period.