Oscillations and Waves

A particle is in simple harmonic motion (SHM) if its acceleration is proportional to displacement and directed towards the equilibrium position:

a = −ω² · x → the defining equation.

Solving the differential equation gives:

x(t) = A sin(ωt + φ)

where A = amplitude, ω = angular frequency, φ = phase constant.

Velocity and acceleration follow from x:

• v = dx/dt = Aω cos(ωt + φ) → v_max = Aω at x = 0
• a = d²x/dt² = −Aω² sin(ωt + φ) = −ω²x → a_max = Aω² at x = ±A

Energy in SHM:

• KE = ½mv² = ½m ω² (A² − x²)
• PE = ½kx² = ½m ω² x²
• Total E = ½ m ω² A² = constant.

KE is maximum at x = 0 (mean position); PE is maximum at x = ±A (extremes). They oscillate twice per cycle while total stays constant.

Time period of common SHM systems:

• Mass on spring: T = 2π √(m/k)
• Simple pendulum (small amplitude): T = 2π √(L/g)
• Physical pendulum: T = 2π √(I/(mgd))

Common pitfalls:

1. The pendulum formula requires small angles (<10°). For large amplitudes, motion is approximately periodic but not SHM.
2. ω in SHM is angular frequency, not rotational angular velocity. Same units (rad/s) but different physical meaning.
3. A 50% phase shift means time period halves? No — phase shift is measured in radians; one full cycle is 2π.