The big idea: Set a mass bouncing on a spring and film it: it dips, rises, dips again, the same trip over and over. That is simple harmonic motion (SHM).
| Quantity | What it means |
|---|---|
| displacement x | how far from the centre |
| velocity v | how fast, and which way |
| acceleration a | how its velocity is changing |
Three shifted waves: All three change as it swings. Plotted against time, all three are smooth waves (sinusoids) — but shifted relative to one another.
New words — equilibrium, phase, antiphase: Equilibrium = the centre, where the object would rest if it weren't moving (x = 0).
Phase = where you are in the cycle. Two curves are in phase if they peak together, and antiphase if one peaks exactly as the other troughs (half a cycle apart).
Phase angle φ (HL) = where in the cycle the motion is at t = 0, as an angle in radians (one cycle = 2π).
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Watch the graph build step by step in study mode.
The two phase rules to remember: Velocity leads displacement by a quarter-cycle (90°) — v is biggest at the centre, zero at the ends.
Acceleration is antiphase to displacement (180°) — a always points back toward the centre, opposite to x. That's the SHM rule a = -ω²x.
Sketching v from x
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Each full oscillation takes one period T. Because the curve is symmetric, one cycle splits into four equal quarters, and each quarter takes T/4.
The defining rule for SHM links the acceleration to the displacement, and it is given in the data booklet:
- acceleration of the oscillator (m s⁻²)
- angular frequency (rad s⁻¹)
- displacement from equilibrium (m)
- period — time for one full oscillation (s)
- frequency — oscillations per second (Hz)
- angular frequency (rad s⁻¹)
| Point in the cycle | Displacement x | Velocity v | Acceleration a |
|---|---|---|---|
| At the centre (equilibrium) | zero | maximum | zero |
| At a turning point (the ends) | maximum | zero | maximum |
Quarter-cycle timings: Centre → end takes T/4. End → centre takes another T/4. Centre → opposite end → back to centre is T/2 (half a cycle).
These fractions of T are the heart of most timing questions.
A mass on a spring completes one full oscillation every 0.50 s. Find its angular frequency ω.
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The SHM equations (HL)
The graphs have equations. Put in a time and they give the displacement and the velocity at that instant; put in a position and the third gives the speed there. All three are given in the data booklet.
- amplitude (m)
- angular frequency (rad s⁻¹)
- time (s)
- phase angle: where the cycle starts (rad)
- displacement (m)
- amplitude (m)
- angular frequency (rad s⁻¹)
- time (s)
- phase angle: where the cycle starts (rad)
- velocity (m s⁻¹)
- velocity (m s⁻¹)
- angular frequency (rad s⁻¹)
- amplitude (m)
- displacement (m)
Choosing φ, and radians: Starts at equilibrium moving the + way: φ = 0, so x = x₀ sin ωt.
Released from +x₀ (the end): φ = π/2, so x = x₀ cos ωt.
ωt + φ is an angle in radians: set the calculator to radians.
The two maximums: v max = ωx₀, at the centre (x = 0 in v = ω√(x₀² − x²)).
a max = ω²x₀, at the ends (x = x₀ in a = −ω²x).
An object oscillates with SHM of amplitude 0.12 m and frequency 2.0 Hz. Calculate its speed when it is 0.060 m from the equilibrium position.
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How this is tested — these graphs almost always come up as a quick timing or phase question:
Paper 1A
- Read a time to reach a point in the cycle — each quarter takes T/4 (e.g. equilibrium → maximum displacement).
- Or identify the phase between x, v and a.
Paper 2
- Describe how v and a change as the object moves, using a = −ω²x.
- Use x = x₀ sin(ωt + φ), v = ωx₀ cos(ωt + φ) or v = ±ω√(x₀² − x²) for a value at a time or a position (HL).
The classic trap: Thinking velocity is biggest at the ends — it is biggest at the centre, and zero at the ends.
Split the cycle into quarters: Mark the cycle: centre → end → centre → other end → centre. Each step is T/4.
So equilibrium to maximum displacement is the first quarter = T/4.
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Watch the graph build step by step in study mode.
A particle moves with simple harmonic motion of period 0.80 s. It starts at the equilibrium position. Find the time it takes to first reach maximum displacement.
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