Every SHM equation so far assumed the object started tidily — right at the centre or right at the edge. But what if two pendulums are swinging and one is a little ahead of the other? To describe an oscillator that starts partway through its cycle, we add a phase angle. It’s the final tool in the SHM kit: a single number that says “how far into the cycle did this one begin?”
📘 What you need to know
The phase angle (φ) describes how far into its cycle an oscillator is at t = 0
It’s measured relative to an oscillator that starts at equilibrium (x = 0 when t = 0)
With a phase angle, the equations become x = x0 sin(ωt + φ), and similarly for v and a
Two oscillators are “in phase” when φ = 0 and “out of phase” otherwise
A phase difference of π radians (180°) means perfectly opposite; π/2 (90°) is a quarter-cycle apart
Sine and cosine are simply π/2 out of phase: cos ωt = sin(ωt + π/2)
What is a phase angle?
Imagine two identical pendulums with the same frequency and amplitude. You release one, then release the other a moment later. They swing with the same rhythm, but one is always a step behind. That constant “step” between them is the phase difference, and the angle that measures it is the phase angle, φ (phi).
Formally, the phase angle is the difference in angular position compared to an oscillator that starts at equilibrium (x = 0 at t = 0). Since one complete cycle is 2π radians, the phase angle can be anywhere from 0 to 2π.
Think of one oscillation as a trip round a circle. The phase angle φ is simply where on that circle the object sits at t = 0.
The equations with a phase angle
Adding the phase angle to the SHM equations just shifts the starting point of the curve. The displacement, velocity and acceleration become:
When φ = 0, these collapse back to the familiar “starts at equilibrium” equations — the object begins at x = 0. A non-zero φ simply means it began somewhere else in the cycle.
WE 1
An oscillator is described by x = x₀ sin(ωt + φ) with amplitude x₀ = 0.10 m and phase angle φ = π/6. What is its displacement at t = 0?
Step 1 — at t = 0 the equation gives x = x₀ sin(φ)Step 2 — substitute φ = π/6 (= 30°)x = 0.10 × sin(π/6) = 0.10 × 0.5x = 0.05 m (5.0 cm)A phase angle of π/6 means the object starts already halfway out — not at the centre.
In phase and out of phase
When two oscillators have a phase difference of zero, they move in perfect step — they’re in phase. Any other phase difference means they’re out of phase. Two special cases come up all the time:
🎯 The phase differences worth memorising
φ = 0 (in phase): the two move together, peaking and crossing zero at the same instants.
φ = π/2 (90°, quarter cycle): one is a quarter of a cycle ahead — like displacement and velocity in SHM.
φ = π (180°, half cycle): perfectly opposite — when one is at its positive peak, the other is at its negative peak.
Same amplitude and period, but the red curve leads the blue by a quarter cycle. The gap between matching features is the phase difference.
Sine and cosine: π/2 apart
This is where phase angles connect back to something you already know. A cosine curve is just a sine curve shifted by a quarter cycle. In other words, sine and cosine are π/2 out of phase:
The sine–cosine phase linkx0 cos ωt = x0 sin(ωt + π/2)
That’s exactly why the “starts at amplitude” equations used cosine: starting at the peak is the same as starting at equilibrium but a quarter-cycle ahead. The phase angle is the bridge between the two forms.
Lead and lag: mind the sign
A phase angle can be added or subtracted, and the sign tells you whether the curve shifts left or right — but watch out, because the sign is the opposite of what most people expect:
Equation
Shift
Meaning
x0 sin(ωt − φ)
Right (later)
The oscillation lags — it happens φ behind
x0 sin(ωt + φ)
Left (earlier)
The oscillation leads — it happens φ ahead
The sign trap catches almost everyone: a +φ shifts the curve left (the motion happens earlier — it leads), while a −φ shifts it right (later — it lags). It feels backwards, but think of it this way: with +φ, the bracket reaches any given value sooner, so that stage of the motion arrives earlier. Sketch it once and it sticks.
Turning a phase angle into a time delay
A phase difference can be converted into an actual time delay between two oscillators. Since the whole cycle (2π) takes one period T, a phase angle φ corresponds to a time φ/ω:
Phase angle to time delayΔt = φ / ω
WE 2
Two identical pendulums swing with the same period of 1.2 s, but one lags the other by a phase angle of π/2. What is the time delay between them?
Step 1 — find ωω = 2π/T = 2π/1.2 = 5.24 rad s⁻¹Step 2 — convert phase to time, Δt = φ/ωΔt = (π/2) / 5.24 = 1.571 / 5.24Δt = 0.30 sSense check: π/2 is a quarter of a cycle, and T/4 = 1.2/4 = 0.30 s. Matches.
Phase differences summarised
Phase difference
In degrees
Fraction of cycle
Relationship
0
0°
None
In phase (move together)
π/2
90°
Quarter cycle
Quarter-cycle apart
π
180°
Half cycle
Completely opposite (antiphase)
2π
360°
Full cycle
Back in phase
💡 Top tips
Phase angle φ = starting position in the cycle, measured from an oscillator that starts at x = 0.
+φ leads (shifts left), −φ lags (shifts right) — the opposite of the intuitive guess.
cos ωt = sin(ωt + π/2) — sine and cosine are always π/2 apart.
Convert phase to time with Δt = φ/ω; a phase of π/2 is a quarter period.
Keep the calculator in radians whenever a phase angle is inside sin or cos.
⚠ Common mistakes
Getting the sign backwards — +φ means lead (earlier), not lag
Thinking π radians is a quarter cycle — it’s half a cycle (90° is the quarter)
Forgetting that sine and cosine differ by π/2, not π
Working in degrees when the phase angle is inside sin or cos — use radians
Confusing a phase angle (radians) with a time delay (seconds) — link them with Δt = φ/ω
Quick recap: A phase angle φ says how far into its cycle an oscillator starts, giving x = x0 sin(ωt + φ). Two oscillators are in phase when φ = 0, a quarter-cycle apart at π/2, and opposite at π. Sine and cosine differ by π/2; a +φ leads while −φ lags; and a phase converts to a time delay via Δt = φ/ω.
That completes the whole simple harmonic motion story — from describing an oscillation, through its conditions and equations, to its energy and now phase. Next in this topic, these ideas step up a level: when many oscillators are linked together, the disturbance travels, and we get waves. Head to the next sub-section to see how a single oscillation becomes a wave that carries energy across space.
Finished SHM? Let’s lock it in.
Book a free meeting and we’ll run through phase angles, the lead/lag sign rule and past-paper SHM questions together.
