C.1 Simple harmonic motionIB Physics HL: Revision notes

Simple harmonic motion occurs when acceleration is proportional to displacement from equilibrium and directed towards equilibrium:

a = −ω²x

This follows from a restoring force proportional to displacement (Hooke's-law spring; pendulum at small angles). From data, a constant negative a/x shows SHM.

Displacement x, amplitude x₀, period T, frequency f and angular frequency ω are linked by T = 1/f = 2π/ω.

• Mass–spring: T = 2π√(m/k)
• Simple pendulum (small angles): T = 2π√(l/g), independent of bob mass.

In SHM the period does not depend on the amplitude.

Energy swaps between kinetic and potential twice per cycle: all potential at the extremes (speed zero), all kinetic at equilibrium (speed maximum). With no damping the total is constant; with damping, energy is dissipated and amplitude falls.

The phase angle ϕ describes where in its cycle an oscillator is at t = 0. With x = x₀ sin(ωt + ϕ):

• ϕ = 0: starts at equilibrium moving in the positive direction;
• ϕ = π/2: starts at +x₀;
• ϕ = π: starts at equilibrium moving in the negative direction;
• ϕ = 3π/2 (or −π/2): starts at −x₀.

A time difference Δt between two oscillators of the same period corresponds to a phase difference Δϕ = 2πΔt/T. A phase difference of π means they are in antiphase.

• Displacement: x = x₀ sin(ωt + ϕ)
• Velocity: v = ωx₀ cos(ωt + ϕ)
• Speed at displacement x: v = ±ω√(x₀² − x²)

So the maximum speed is ωx₀ (at x = 0) and the maximum acceleration is ω²x₀ (at x = ±x₀). Velocity is a quarter of a cycle (π/2) ahead of displacement.

• Total energy: E_T = ½mω²x₀² (constant without damping)
• Potential energy: E_P = ½mω²x²
• Kinetic energy: E_K = E_T − E_P = ½mω²(x₀² − x²)

Because E_P ∝ x², kinetic and potential energy are equal at x = ±x₀/√2, not at half the amplitude. Total energy ∝ x₀², so doubling the amplitude quadruples the energy.