Oscillations and Waves
What this domain tests
Simple harmonic motion, spring–mass and pendulum periods, energy in SHM, and wave behaviour: speed, superposition, standing waves, organ pipes, beats and the Doppler effect.
Core rules
- SHM: a = −ω²x and x = A sin(ωt + φ); v = ω√(A² − x²) is maximum at the mean position; acceleration is maximum at the extremes where v = 0.
- T = 2π√(m/k) for a spring; T = 2π√(l/g) for a simple pendulum, independent of mass for small swings.
- Energy: total E = ½kA², with kinetic and potential energy equal at x = A/√2.
- Wave: v = fλ; on a stretched string v = √(T/μ); sound in air is about 340 m/s and roughly ∝ √T.
- Standing waves: string fixed at both ends and open pipe f_n = nv/2L (all harmonics); closed pipe f_n = (2n−1)v/4L (odd harmonics only).
- Beats: f_beat = |f₁ − f₂|; Doppler f' = f(v ± v_o)/(v ∓ v_s).
Worked example
A 0.5 kg mass hangs on a spring of stiffness 200 N/m. Find the angular frequency and period. ω = √(k/m) = √(200/0.5) = √400 = 20 rad/s. Then T = 2π/ω = 6.28/20 ≈ 0.31 s. Use kg and N/m; the amplitude never enters the period.
Traps that cost marks
- Thinking a larger amplitude changes the period of a spring–mass system.
- Claiming acceleration is zero at the extreme position.
- Mixing the harmonic series of open and closed pipes.
- Taking the beat frequency as a sum instead of a difference.