Chapter 45 · Physics module 23
Oscillations
Understand the motion. Follow the energy.
Explore six focused lessons on SHM, energy, damping, resonance and standing waves. Change the models, work through examples and check your understanding.
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Change an input. Discover a relationship.
Chapter Overview
SHM, energy exchange, damping, resonance, and standing waves.
Each lesson brings the key relationships, a model you can control, assumptions, worked examples and a five-question quiz together. Lesson quiz answers remain in the open page only.
Learning Objectives
- Define and recognise simple harmonic motion
- Track energy and phase
- Compare damping regimes and driven response
- Apply standing-wave boundary conditions
- Use calculus to test an oscillation model
Explore the Six Lessons
Follow the sequence or select the idea you need.
Simple Harmonic Motion
The defining condition a = −ω²x, restoring forces, and the periods of springs and pendulums.
Learn · Explore · Practise →02E = ½mω²A²Energy in SHM
Kinetic and potential energy trade places under one constant total, ½mω²A².
Learn · Explore · Practise →03x″ + 2γx′ + ω₀²x = 0Damped Oscillations
Light, critical, and heavy damping; exponential decay and the Q factor.
Learn · Explore · Practise →04fᵣ = √(ω₀² − 2γ²)/(2π)Forced Oscillations & Resonance
Drive at the natural frequency and small pushes build huge amplitudes — for better or worse.
Learn · Explore · Practise →05fₙ = nv/(2L)Standing Waves
Counter-propagating waves superpose into nodes, antinodes, and the harmonics of strings and pipes.
Learn · Explore · Practise →06x = A sin(ωt + φ)Oscillations Calculus
Solve d²x/dt² = −ω²x outright, differentiate for v and a, and read the damped solution.
Extend with calculus →Interactive SHM Simulation
Explore springs, pendulums, graphs and energy. Change mass or stiffness, then compare the period and motion.
Angular frequency ω4.47 rad/s
Period T1.40 s
Frequency f0.71 Hz
Displacement x0.1 m
Velocity v0 m/s
Acceleration a-2 m/s²
Total oscillation energy0.1 J
Ideal massless spring: displacement is measured from the gravitational equilibrium. The restoring force shown is the net force. Apparatus size is schematic. The graph uses x/A, so physical values are in the readouts. Motion starts paused.
Ideal undamped, linear oscillations. Springs must remain within their proportional range; the pendulum uses the small-angle approximation. The apparatus views show two periods; the separate Graphs view covers four seconds. Normalised curves and physical readouts are labelled separately.
Open the full SHM lesson →Key Relationships
- SHM acceleration
- a = −ω²x
- Angular frequency
- ω = 2πf
- Total energy
- E = ½mω²A²
- Spring period
- T = 2π√(m/k)
- Decay envelope
- A(t) ∝ e^(−γt)
- String harmonics
- fₙ = nv/(2L)
Check the assumptions and units in each lesson before applying a formula.