Simple harmonic motion

Learning objectives

Define simple harmonic motion by the condition that the acceleration is proportional to the displacement from equilibrium and is always directed towards it, a = −ω²x

Relate period, frequency and angular frequency using T = 1/f and ω = 2πf, and explain why the period of a simple harmonic oscillator is independent of its amplitude

Apply T = 2π√(m/k) to a mass–spring system and T = 2π√(L/g) to a simple pendulum oscillating through a small angle, and use each relationship to determine k or g experimentally

(HL) Apply x = x₀ sin(ωt) or x = x₀ cos(ωt) together with the corresponding velocity and acceleration equations, and use v = ±ω√(x₀² − x²) to find the speed at a given displacement

(HL) Use the phase angle φ in x = x₀ sin(ωt + φ) and v = ωx₀ cos(ωt + φ) to describe the state of an oscillator at t = 0 and the phase difference between two oscillators

(HL) Calculate the total energy of an oscillator from ET = ½mω²x₀², and describe how the kinetic and potential energy vary with time and with displacement

Sketch and interpret graphs of displacement, velocity, acceleration and energy against time and against displacement, identifying the phase relationships between them