What makes the harmonic oscillator model applicable to such a wide range of phenomena?
Simple harmonic motion (SHM) is the physics of restoring forces. Whenever a displaced body experiences a force directed back towards its equilibrium position, and that force is proportional to the displacement, the body will oscillate sinusoidally. The defining equation a = −ω²x encodes both conditions: the negative sign means restoring (back towards equilibrium), and the proportionality means the oscillation is mathematically simple enough to solve exactly.
The model is powerful because it appears everywhere: a mass on a spring, a pendulum for small angles, a charged particle in a parabolic potential, an atom vibrating in a crystal lattice, a current oscillating in an LC circuit. Once you know SHM, you know the shape of all these motions.
A mass of 0.25 kg is attached to a spring of spring constant 16 N m−1. It is displaced 0.08 m from equilibrium and released. Write expressions for displacement, velocity, and acceleration as functions of time, and calculate the total energy of the oscillation.
Key equations
Defining equation: a = −ω²x
T = 1/f = 2π/ω
Mass-spring: T = 2π√(m/k)
Simple pendulum: T = 2π√(l/g)
HL x = x₀ sin(ωt + φ); v = ωx₀ cos(ωt + φ); v = ±ω√(x₀² − x²)
HL ET = ½mω²x₀²; Ep = ½mω²x²
What students must understand
Conditions for SHM: restoring force proportional to displacement, directed towards equilibrium
Defining equation: a = −ω²x (minus sign is critical — restoring)
Descriptors: period T, frequency f, angular frequency ω, amplitude x₀, equilibrium, displacement x
T = 2π√(m/k) for mass-spring; T = 2π√(l/g) for pendulum
Qualitative energy description: KE maximum at equilibrium, PE maximum at amplitude, total mechanical energy constant
HL Phase angle φ; full sinusoidal equations for x, v, and a
HL Speed at any displacement: v = ±ω√(x₀² − x²)
HL Quantitative energy analysis: ET = ½mω²x₀²
Linking questions
How can greenhouse gases be modelled as simple harmonic oscillators? → B.2 Greenhouse Effect
How can circular motion visualise SHM? → A.2 Forces and Momentum
How does damping affect periodic motion? → C.4 Standing Waves and Resonance
Period of a Pendulum | Simple harmonic motion and rotational motionDefinition of Amplitude and PeriodPeriod dependence for mass on springEquation for simple harmonic oscillatorsEnergy graphs for simple harmonic motion | Simple harmonic motion
Flipping Physics
Simple Harmonic Motion Introduction (SHM) via a Horizontal Mass-Spring SystemAP Physics 1 - Unit 7 Review - Oscillations - Exam Prep
The Organic Chemistry Tutor
How To Solve Simple Harmonic Motion Problems In PhysicsSimple Harmonic Motion, Mass Spring System - Amplitude, Frequency, Velocity - Physics Problems
Michel van Biezen
Physics 16.2 Simple Harmonic Motion Basics (1 of 5) Introduction 1Physics 16 Simple Harmonic Motion (1 of 19) Why is There a (-) in F=-kx?
WNY Tutor — worked problems
A 3.0 kg particle in simple harmonic motion - equation of motionA simple harmonic oscillator consists of a block of mass
Physics with Professor Matt Anderson — full course modules
Module 14 | Simple Harmonic Motion | Physics with Professor Matt Anderson