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Reading SHM graphs and phase

Key idea: In SHM, velocity is a quarter-cycle out of phase with displacement, and acceleration is in antiphase with displacement. Graphs show these fixed phase relationships.

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  • GCE A-Level H2 Physics 2027
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Enlarged diagram

The equations of SHM say a lot, but a graph shows at a glance where a body is, how fast it is moving and which way it is being pulled at every instant. Drawn on the same time axis, the displacement, velocity and acceleration graphs are three versions of one curve, shifted against each other by fixed fractions of a cycle.

Displacement, velocity and acceleration graphs

Start with a body that passes equilibrium in the positive direction at t = 0:

x = x₀ sin ω t v = v₀ cos ω t a = -ω²x

Velocity is the gradient of the displacement–time graph, and acceleration is the gradient of the velocity–time graph. The graphs below plot each quantity as a fraction of its maximum, so that their shapes and timings can be compared on one set of axes.

One SHM cycle: displacement, velocity and acceleration

The sine-origin SHM solution has normalised displacement sin(2πt/T), velocity cos(2πt/T), and acceleration −sin(2πt/T). All three share period T.

Scroll across the figure to read all labels.

The sine-origin SHM solution has normalised displacement sin(2πt/T), velocity cos(2πt/T), and acceleration −sin(2πt/T). All three share period T.The sine-origin SHM solution has normalised displacement sin(2πt/T), velocity cos(2πt/T), and acceleration −sin(2πt/T). All three share period T.
For x = x₀ sin(ωt), normalised velocity leads displacement by T/4 and acceleration is in antiphase with displacement.
Open full-size graph
What to noticeExact model samples joined by short line segments; not measured traces. Separate normalisations preserve signs and phase, not equality of physical values or units.
View figure data
Figure data: One SHM cycle: displacement, velocity and acceleration
t/T (unitless)Displacement x/x₀Velocity v/(ωx₀)Acceleration a/(ω²x₀)
0.0000000.000e+001.000e+000.000e+00
0.0083335.234e-029.986e-01-5.234e-02
0.0166671.045e-019.945e-01-1.045e-01
0.0250001.564e-019.877e-01-1.564e-01
0.0333332.079e-019.781e-01-2.079e-01
0.0416672.588e-019.659e-01-2.588e-01
0.0500003.090e-019.511e-01-3.090e-01
0.0583333.584e-019.336e-01-3.584e-01
0.0666674.067e-019.135e-01-4.067e-01
0.0750004.540e-018.910e-01-4.540e-01
0.0833335.000e-018.660e-01-5.000e-01
0.0916675.446e-018.387e-01-5.446e-01
0.1000005.878e-018.090e-01-5.878e-01
0.1083336.293e-017.771e-01-6.293e-01
0.1166676.691e-017.431e-01-6.691e-01
0.1250007.071e-017.071e-01-7.071e-01
0.1333337.431e-016.691e-01-7.431e-01
0.1416677.771e-016.293e-01-7.771e-01
0.1500008.090e-015.878e-01-8.090e-01
0.1583338.387e-015.446e-01-8.387e-01
0.1666678.660e-015.000e-01-8.660e-01
0.1750008.910e-014.540e-01-8.910e-01
0.1833339.135e-014.067e-01-9.135e-01
0.1916679.336e-013.584e-01-9.336e-01
0.2000009.511e-013.090e-01-9.511e-01
0.2083339.659e-012.588e-01-9.659e-01
0.2166679.781e-012.079e-01-9.781e-01
0.2250009.877e-011.564e-01-9.877e-01
0.2333339.945e-011.045e-01-9.945e-01
0.2416679.986e-015.234e-02-9.986e-01
0.2500001.000e+006.123e-17-1.000e+00
0.2583339.986e-01-5.234e-02-9.986e-01
0.2666679.945e-01-1.045e-01-9.945e-01
0.2750009.877e-01-1.564e-01-9.877e-01
0.2833339.781e-01-2.079e-01-9.781e-01
0.2916679.659e-01-2.588e-01-9.659e-01
0.3000009.511e-01-3.090e-01-9.511e-01
0.3083339.336e-01-3.584e-01-9.336e-01
0.3166679.135e-01-4.067e-01-9.135e-01
0.3250008.910e-01-4.540e-01-8.910e-01
0.3333338.660e-01-5.000e-01-8.660e-01
0.3416678.387e-01-5.446e-01-8.387e-01
0.3500008.090e-01-5.878e-01-8.090e-01
0.3583337.771e-01-6.293e-01-7.771e-01
0.3666677.431e-01-6.691e-01-7.431e-01
0.3750007.071e-01-7.071e-01-7.071e-01
0.3833336.691e-01-7.431e-01-6.691e-01
0.3916676.293e-01-7.771e-01-6.293e-01
0.4000005.878e-01-8.090e-01-5.878e-01
0.4083335.446e-01-8.387e-01-5.446e-01
0.4166675.000e-01-8.660e-01-5.000e-01
0.4250004.540e-01-8.910e-01-4.540e-01
0.4333334.067e-01-9.135e-01-4.067e-01
0.4416673.584e-01-9.336e-01-3.584e-01
0.4500003.090e-01-9.511e-01-3.090e-01
0.4583332.588e-01-9.659e-01-2.588e-01
0.4666672.079e-01-9.781e-01-2.079e-01
0.4750001.564e-01-9.877e-01-1.564e-01
0.4833331.045e-01-9.945e-01-1.045e-01
0.4916675.234e-02-9.986e-01-5.234e-02
0.5000001.225e-16-1.000e+00-1.225e-16
0.508333-5.234e-02-9.986e-015.234e-02
0.516667-1.045e-01-9.945e-011.045e-01
0.525000-1.564e-01-9.877e-011.564e-01
0.533333-2.079e-01-9.781e-012.079e-01
0.541667-2.588e-01-9.659e-012.588e-01
0.550000-3.090e-01-9.511e-013.090e-01
0.558333-3.584e-01-9.336e-013.584e-01
0.566667-4.067e-01-9.135e-014.067e-01
0.575000-4.540e-01-8.910e-014.540e-01
0.583333-5.000e-01-8.660e-015.000e-01
0.591667-5.446e-01-8.387e-015.446e-01
0.600000-5.878e-01-8.090e-015.878e-01
0.608333-6.293e-01-7.771e-016.293e-01
0.616667-6.691e-01-7.431e-016.691e-01
0.625000-7.071e-01-7.071e-017.071e-01
0.633333-7.431e-01-6.691e-017.431e-01
0.641667-7.771e-01-6.293e-017.771e-01
0.650000-8.090e-01-5.878e-018.090e-01
0.658333-8.387e-01-5.446e-018.387e-01
0.666667-8.660e-01-5.000e-018.660e-01
0.675000-8.910e-01-4.540e-018.910e-01
0.683333-9.135e-01-4.067e-019.135e-01
0.691667-9.336e-01-3.584e-019.336e-01
0.700000-9.511e-01-3.090e-019.511e-01
0.708333-9.659e-01-2.588e-019.659e-01
0.716667-9.781e-01-2.079e-019.781e-01
0.725000-9.877e-01-1.564e-019.877e-01
0.733333-9.945e-01-1.045e-019.945e-01
0.741667-9.986e-01-5.234e-029.986e-01
0.750000-1.000e+00-1.837e-161.000e+00
0.758333-9.986e-015.234e-029.986e-01
0.766667-9.945e-011.045e-019.945e-01
0.775000-9.877e-011.564e-019.877e-01
0.783333-9.781e-012.079e-019.781e-01
0.791667-9.659e-012.588e-019.659e-01
0.800000-9.511e-013.090e-019.511e-01
0.808333-9.336e-013.584e-019.336e-01
0.816667-9.135e-014.067e-019.135e-01
0.825000-8.910e-014.540e-018.910e-01
0.833333-8.660e-015.000e-018.660e-01
0.841667-8.387e-015.446e-018.387e-01
0.850000-8.090e-015.878e-018.090e-01
0.858333-7.771e-016.293e-017.771e-01
0.866667-7.431e-016.691e-017.431e-01
0.875000-7.071e-017.071e-017.071e-01
0.883333-6.691e-017.431e-016.691e-01
0.891667-6.293e-017.771e-016.293e-01
0.900000-5.878e-018.090e-015.878e-01
0.908333-5.446e-018.387e-015.446e-01
0.916667-5.000e-018.660e-015.000e-01
0.925000-4.540e-018.910e-014.540e-01
0.933333-4.067e-019.135e-014.067e-01
0.941667-3.584e-019.336e-013.584e-01
0.950000-3.090e-019.511e-013.090e-01
0.958333-2.588e-019.659e-012.588e-01
0.966667-2.079e-019.781e-012.079e-01
0.975000-1.564e-019.877e-011.564e-01
0.983333-1.045e-019.945e-011.045e-01
0.991667-5.234e-029.986e-015.234e-02
1.000000-2.449e-161.000e+002.449e-16

Reading across at the same instant:

  • at t = 0: x = 0, v = +v₀ (the displacement graph is steepest) and a = 0;
  • at t = T/4: x = +x₀, v = 0 (the graph is flat at the peak) and a = -ω²x₀, its greatest negative value;
  • at t = T/2: x = 0 again, now with v = -v₀.

The velocity reaches its maximum a quarter of a cycle before the displacement does: velocity leads displacement by π/2 rad. The acceleration is always opposite in sign to the displacement: they are in antiphase, a phase difference of π rad. Starting the clock at a different moment shifts all three graphs together, but does not change these phase differences.

Check your understanding 1 Starting at the top

A body in SHM is released from rest at x = +x₀ at t = 0. What are its velocity and acceleration at t = 0, and which equation describes its displacement?

Velocity and acceleration at release
Displacement equation for release from rest at +x₀
Show answer

v = 0 and a = -ω²x₀, its greatest negative value. Its displacement is x = x₀ cos ω t.

Set the oscillator to Mass on a spring, with Damping: None and a positive starting displacement. Predict the release state above, then start the motion and pause it near equilibrium. The marker shows the same instant on all three time graphs. This release starts at the positive extreme, so its displacement follows a cosine; the sine graph above starts at an equilibrium crossing.

Follow one release

t = 0.00 s

A 0.50 kilogram mass on a spring of constant 20 newtons per metre, displaced 0.15 metres with no damping. Not released yet.

m
kg
N/m
More settings
Period, T
0.993 s
ω = 2π/T
6.32 rad/s
Displacement, x
0.150 m
Velocity, v
0.000 m/s
Acceleration, a
−6.00 m/s²
Maximum speed
— m/s
Displacement against time
Velocity against time
Acceleration against time

Try this

0 of 2 done

  1. Pause the motion when the acceleration is greatest. (not done yet)

  2. Pause the motion as the mass passes through the equilibrium position. (not done yet)

Check your understanding 2 Use the graph gradients

On the first passage through equilibrium after release from a positive displacement, what are the signs of the displacement–time gradient and acceleration? Use the simulation or the cosine model.

First equilibrium crossing
Show answer

The displacement graph slopes downwards through zero, so the velocity is negative and has its greatest magnitude. The acceleration is zero because the restoring displacement is zero. The steepest graph is where the speed is greatest, not where the displacement is greatest.

Worked example 1

Values at the start and at the first maximum

Problem

A body moves according to x = 0.0400 sin(5.00t), in SI units. Find x, v and a at t = 0 and at its first positive maximum displacement.

Show full solution

  1. Constants

    Method

    Read x₀ and ω from the equation, and find T.

    Reason

    They fix every value on the three graphs.

    Working

    Answer

    x₀ = 0.0400 m, ω = 5.00 rad s⁻¹, v₀ = ω x₀ = 0.200 m s⁻¹ and T = 2π/5.00 = 1.26 s

  2. At the start

    Reason

    A sine graph starts at equilibrium, moving in the positive direction.

    Working

    Answer

    At t = 0: x = 0, v = +0.200 m s⁻¹, a = 0

  3. At the first maximum

    Working

    Answer

    At t = T/4 = 0.314 s: x = +0.0400 m, v = 0, a = -(5.00)²(0.0400) = -1.00 m s⁻²

Check your understanding 3 Half and three-quarters of a cycle

For the same body, find x, v and a at t = T/2 and t = 3T/4.

Unit: m
Unit: m s⁻¹
Unit: m s⁻²
Unit: m
Unit: m s⁻¹
Unit: m s⁻²
Show answer

At T/2: x = 0, v = -0.200 m s⁻¹, a = 0, passing equilibrium in the negative direction. At 3T/4: x = -0.0400 m, v = 0, a = +1.00 m s⁻², at the negative extreme.

Phase

The phase of an oscillation says how far through its cycle it is, as an angle. In x = x₀ sin(ω t + φ), the angle φ sets where in the cycle the motion is at t = 0. Two oscillations of the same frequency with different values of φ have a phase difference. To measure it from graphs, compare matching points, such as two maxima, or two crossings of equilibrium in the same direction, and use Δφ = (2πΔ t)/T.

In the graph below, B reaches each maximum one-sixth of a cycle after A, so B lags A by 2π/6 = π/3 rad.

Phase difference between equal-frequency oscillations

Normalised sinusoidal displacements against shared phase angle. B has the same amplitude and period as A but its positive maximum occurs π/3 rad later.

Scroll across the figure to read all labels.

Normalised sinusoidal displacements against shared phase angle. B has the same amplitude and period as A but its positive maximum occurs π/3 rad later.Normalised sinusoidal displacements against shared phase angle. B has the same amplitude and period as A but its positive maximum occurs π/3 rad later.
B lags A by π/3 rad, equivalent to a time delay of T/6. Compare matching directional crossings or matching maxima.
Open full-size graph
What to noticeConstructed sine curves, not measurements. The horizontal axis is shared phase angle, not displacement. One full cycle advances phase by 2π rad.
View figure data
Figure data: Phase difference between equal-frequency oscillations
Phase angle, ωt (rad)A: sin(ωt)B: sin(ωt − π/3)
0.000000.000e+00-8.660e-01
0.052365.234e-02-8.387e-01
0.104721.045e-01-8.090e-01
0.157081.564e-01-7.771e-01
0.209442.079e-01-7.431e-01
0.261802.588e-01-7.071e-01
0.314163.090e-01-6.691e-01
0.366523.584e-01-6.293e-01
0.418884.067e-01-5.878e-01
0.471244.540e-01-5.446e-01
0.523605.000e-01-5.000e-01
0.575965.446e-01-4.540e-01
0.628325.878e-01-4.067e-01
0.680686.293e-01-3.584e-01
0.733046.691e-01-3.090e-01
0.785407.071e-01-2.588e-01
0.837767.431e-01-2.079e-01
0.890127.771e-01-1.564e-01
0.942488.090e-01-1.045e-01
0.994848.387e-01-5.234e-02
1.047208.660e-010.000e+00
1.099568.910e-015.234e-02
1.151929.135e-011.045e-01
1.204289.336e-011.564e-01
1.256649.511e-012.079e-01
1.309009.659e-012.588e-01
1.361369.781e-013.090e-01
1.413729.877e-013.584e-01
1.466089.945e-014.067e-01
1.518449.986e-014.540e-01
1.570801.000e+005.000e-01
1.623169.986e-015.446e-01
1.675529.945e-015.878e-01
1.727889.877e-016.293e-01
1.780249.781e-016.691e-01
1.832609.659e-017.071e-01
1.884969.511e-017.431e-01
1.937329.336e-017.771e-01
1.989689.135e-018.090e-01
2.042048.910e-018.387e-01
2.094408.660e-018.660e-01
2.146758.387e-018.910e-01
2.199118.090e-019.135e-01
2.251477.771e-019.336e-01
2.303837.431e-019.511e-01
2.356197.071e-019.659e-01
2.408556.691e-019.781e-01
2.460916.293e-019.877e-01
2.513275.878e-019.945e-01
2.565635.446e-019.986e-01
2.617995.000e-011.000e+00
2.670354.540e-019.986e-01
2.722714.067e-019.945e-01
2.775073.584e-019.877e-01
2.827433.090e-019.781e-01
2.879792.588e-019.659e-01
2.932152.079e-019.511e-01
2.984511.564e-019.336e-01
3.036871.045e-019.135e-01
3.089235.234e-028.910e-01
3.141595.666e-168.660e-01
3.19395-5.234e-028.387e-01
3.24631-1.045e-018.090e-01
3.29867-1.564e-017.771e-01
3.35103-2.079e-017.431e-01
3.40339-2.588e-017.071e-01
3.45575-3.090e-016.691e-01
3.50811-3.584e-016.293e-01
3.56047-4.067e-015.878e-01
3.61283-4.540e-015.446e-01
3.66519-5.000e-015.000e-01
3.71755-5.446e-014.540e-01
3.76991-5.878e-014.067e-01
3.82227-6.293e-013.584e-01
3.87463-6.691e-013.090e-01
3.92699-7.071e-012.588e-01
3.97935-7.431e-012.079e-01
4.03171-7.771e-011.564e-01
4.08407-8.090e-011.045e-01
4.13643-8.387e-015.234e-02
4.18879-8.660e-011.225e-16
4.24115-8.910e-01-5.234e-02
4.29351-9.135e-01-1.045e-01
4.34587-9.336e-01-1.564e-01
4.39823-9.511e-01-2.079e-01
4.45059-9.659e-01-2.588e-01
4.50295-9.781e-01-3.090e-01
4.55531-9.877e-01-3.584e-01
4.60767-9.945e-01-4.067e-01
4.66003-9.986e-01-4.540e-01
4.71239-1.000e+00-5.000e-01
4.76475-9.986e-01-5.446e-01
4.81711-9.945e-01-5.878e-01
4.86947-9.877e-01-6.293e-01
4.92183-9.781e-01-6.691e-01
4.97419-9.659e-01-7.071e-01
5.02655-9.511e-01-7.431e-01
5.07891-9.336e-01-7.771e-01
5.13127-9.135e-01-8.090e-01
5.18363-8.910e-01-8.387e-01
5.23599-8.660e-01-8.660e-01
5.28835-8.387e-01-8.910e-01
5.34071-8.090e-01-9.135e-01
5.39307-7.771e-01-9.336e-01
5.44543-7.431e-01-9.511e-01
5.49779-7.071e-01-9.659e-01
5.55015-6.691e-01-9.781e-01
5.60251-6.293e-01-9.877e-01
5.65487-5.878e-01-9.945e-01
5.70723-5.446e-01-9.986e-01
5.75959-5.000e-01-1.000e+00
5.81195-4.540e-01-9.986e-01
5.86431-4.067e-01-9.945e-01
5.91667-3.584e-01-9.877e-01
5.96903-3.090e-01-9.781e-01
6.02139-2.588e-01-9.659e-01
6.07375-2.079e-01-9.511e-01
6.12611-1.564e-01-9.336e-01
6.17847-1.045e-01-9.135e-01
6.23083-5.234e-02-8.910e-01
6.28319-1.133e-15-8.660e-01

Guided practice 2

Phase from the starting conditions

About 4 min

Problem

A body moves according to x = x₀ sin(ω t + φ). At t = 0 it is at equilibrium, moving in the positive direction. Find φ in the range 0 ≤ φ < 2π.

Find the phase

Unit: rad

Hints

Hint 1

both conditions

First solve x₀ sin φ = 0. Then check which answer gives a positive velocity, v = ω x₀ cos φ.

Show solution step by step

  1. Displacement

    Reason

    At equilibrium, x = x₀ sin φ = 0.

    Working

    Answer

    φ = 0 or π

  2. Direction

    Working

    Answer

    v = ω x₀ cos φ must be positive. cos 0 = 1 but cos π = -1, so φ = 0.

Check your understanding 4 Graphs from the top

A body in SHM starts at x = +x₀. Sketch the three graphs for one cycle on paper and give the values of x, v and a at t = 0, T/4, T/2 and 3T/4. Use optional notes to record your phase decisions.

Show answer

Check that your graphs share one time axis, show the correct extrema and crossings, and have velocity as the gradient of displacement. With x = x₀ cos ω t, v = -v₀ sin ω t and a = -ω²x: x is + x₀, 0, -x₀, 0; v is 0, -v₀, 0, + v₀; a is -ω²x₀, 0, + ω²x₀, 0.

Check your understanding 5 Phase difference from two traces

Two oscillators both have a period of 1.20 s. B reaches each positive maximum 0.300 s after A. Find the phase difference. State which oscillator lags in the explanation below.

Show the steps Optional working
Unit: rad
Show answer

Δφ = 2π(0.300)/1.20 = π/2 = 1.57 rad, with B behind. Equal displacements occur twice in each cycle, once moving each way, so they need not be the same stage of the cycle. A maximum happens only once per cycle.

Check your understanding 6 Choose matching points

For the two traces above, state which oscillator lags. Explain why two equal displacements need not be matching phase points, and give a valid alternative to comparing maxima.

Show answer

B lags A because it reaches each maximum later. A displacement between the extremes occurs twice per cycle, with opposite velocity signs. Compare maxima, or equilibrium crossings in the same direction, to match both position and direction. Check that your comparison uses one common period and identifies the later trace correctly.

Common mistakes

  • Thinking two equilibrium crossings half a period apart are at the same phase; they are in opposite directions.
  • Putting the maximum velocity at the maximum displacement; the velocity is zero there.
  • Drawing the acceleration graph in phase with the displacement instead of in antiphase.
  • Using x = x₀ sin ω t for a body released from rest at + x₀.

Spotted an error? Report a correction

Syllabus and review details

Last reviewed:

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