4. SHM Graphs and Equations

SHM Graphs and Equations

Simple harmonic motion (SHM) can be described using graphs and mathematical equations. These representations allow us to determine where an oscillating object is, how fast it is moving, and how it is accelerating at any point during its motion.

Three graphs are especially important:

• Displacement–time graphs
• Velocity–time graphs
• Acceleration–time graphs

All three have repeating, sinusoidal shapes, but they do not reach their maximum and minimum values at the same time.

Learning Targets

• I can interpret displacement-time graphs for SHM.
• I can interpret velocity-time and acceleration-time graphs for SHM.
• I can determine amplitude, period, and frequency from graphs.
• I can relate graphical representations to physical motion.
• I can apply mathematical models of SHM to solve problems.

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Displacement in SHM

Displacement, x, tells us the position of an oscillating object relative to its equilibrium position.

We usually define:

x = 0

as the equilibrium position.

One side of equilibrium is assigned positive displacement and the other negative displacement.

At the two extremes:

x = +A

and:

x = −A

where A is the amplitude.

The amplitude is therefore the maximum displacement from equilibrium.

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The Displacement-Time Graph

A displacement-time graph for ideal SHM is sinusoidal.

It may look like a sine or cosine curve depending on where the object begins its motion.

The vertical axis represents:

Displacement, x

The horizontal axis represents:

Time, t

The graph tells us where the object is relative to equilibrium at every instant.

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Reading Physical Motion from a Displacement-Time Graph

Suppose an oscillator begins at:

x = +A

It then moves toward equilibrium.

The sequence is:

+A → 0 → −A → 0 → +A

This represents one complete oscillation.

At:

x = +A

the object is at one extreme.

At:

x = 0

the object is passing through equilibrium.

At:

x = −A

the object is at the opposite extreme.

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Finding Amplitude from a Graph

Amplitude is the maximum displacement from equilibrium.

Suppose a graph reaches:

+0.12 m

and:

−0.12 m

Then:

Amplitude = 0.12 m

Do not measure from the minimum to the maximum.

The distance from −A to +A is:

2A

So in this example:

Peak-to-peak displacement = 0.24 m

but:

Amplitude = 0.12 m

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Finding the Period

The period, T, is the time required for one complete oscillation.

On a displacement-time graph, measure the time between two equivalent points in consecutive cycles.

For example:

maximum → next maximum

or:

minimum → next minimum

or:

same equilibrium crossing in the same direction → next identical crossing

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Worked Example — Finding Period

Suppose a displacement-time graph has maxima at:

t = 1.0 s

and:

t = 5.0 s

Therefore:

T = 5.0 − 1.0

T = 4.0 s

The period is:

4.0 s

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Finding Frequency

Frequency is the number of complete oscillations per second.

Frequency and period are related by:

f = 1/T

where:

f = frequency in hertz (Hz)

T = period in seconds (s)

For the previous example:

T = 4.0 s

Therefore:

f = 1/4.0

f = 0.25 Hz

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Angular Frequency

SHM equations commonly use angular frequency, represented by:

ω

Angular frequency is related to ordinary frequency by:

ω = 2πf

Since:

f = 1/T

we can also write:

ω = 2π/T

The unit of angular frequency is:

radians per second (rad/s)

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Worked Example — Angular Frequency

An oscillator has:

T = 2.0 s

First calculate frequency:

f = 1/T

f = 1/2.0

f = 0.50 Hz

Now calculate angular frequency:

ω = 2πf

ω = 2π(0.50)

ω = π rad/s

or approximately:

ω = 3.14 rad/s

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Mathematical Model for Displacement

One common equation for SHM is:

x = A cos(ωt)

where:

x = displacement

A = amplitude

ω = angular frequency

t = time

This equation describes an oscillator that begins at:

x = +A

when:

t = 0

because:

cos 0 = 1

Therefore:

x = A

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Sine and Cosine Models

SHM can also be written using a sine function:

x = A sin(ωt)

The difference depends on the object's starting position.

For:

x = A cos(ωt)

the object begins at maximum positive displacement.

For:

x = A sin(ωt)

the object begins at equilibrium moving in the positive direction.

Both equations describe simple harmonic motion.

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Phase

The position of an oscillator within its cycle is called its phase.

Two oscillators can have:

• The same amplitude
• The same frequency
• The same period

but still be at different points in their cycles.

They are then said to have a:

phase difference

Phase can be measured in:

• Degrees
• Radians

One complete cycle corresponds to:

360°

or:

2π radians

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Velocity in SHM

Velocity tells us how quickly displacement is changing.

Therefore:

Velocity = gradient of the displacement-time graph

When the displacement-time graph is steepest, the magnitude of velocity is greatest.

When the displacement-time graph has a horizontal tangent, velocity is zero.

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Velocity at Maximum Displacement

At:

x = +A

or:

x = −A

the object momentarily stops before reversing direction.

Therefore:

v = 0

So:

Maximum displacement → Zero velocity

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Velocity at Equilibrium

At:

x = 0

the oscillator is passing through equilibrium.

Its speed is greatest here.

Therefore:

Equilibrium → Maximum speed

The velocity may be positive or negative depending on which direction the object is moving.

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Maximum Velocity

For SHM:

Maximum speed = Aω

Therefore:

vmax = Aω

This tells us that maximum speed increases if:

• Amplitude increases.
• Angular frequency increases.

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Worked Example — Maximum Speed

An oscillator has:

A = 0.20 m

and:

f = 2.0 Hz

First calculate angular frequency:

ω = 2πf

ω = 2π(2.0)

ω = 4π rad/s

Now:

vmax = Aω

vmax = (0.20)(4π)

vmax ≈ 2.51 m/s

Therefore:

Maximum speed ≈ 2.5 m/s

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Velocity-Time Graph

Velocity also varies sinusoidally with time.

However, the velocity graph is shifted relative to the displacement graph.

When displacement is maximum:

Velocity = 0

When displacement is zero:

|Velocity| = maximum

Therefore, displacement and velocity are:

one-quarter of a cycle out of phase

This corresponds to:

90°

or:

π/2 radians

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Direction from the Velocity Graph

The sign of velocity tells us the direction of motion.

v > 0

means motion in the positive direction.

v < 0

means motion in the negative direction.

Suppose:

x > 0

and:

v < 0

The object is on the positive side of equilibrium but moving back toward equilibrium.

Graphs allow us to determine both position and direction of motion.

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Acceleration in SHM

Acceleration is the rate of change of velocity.

Therefore:

Acceleration = gradient of the velocity-time graph

For SHM, acceleration has a particularly important relationship with displacement:

a = −ω²x

This equation is one of the defining mathematical relationships of simple harmonic motion.

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Understanding a = −ω²x

The equation tells us two important things.

First:

|a| ∝ |x|

The further the object is from equilibrium, the greater its acceleration magnitude.

Second:

The negative sign means acceleration acts in the direction opposite to displacement.

Therefore, acceleration always points:

toward equilibrium

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Acceleration at Equilibrium

At equilibrium:

x = 0

Therefore:

a = −ω²(0)

so:

a = 0

At this instant:

• Displacement is zero.
• Acceleration is zero.
• Speed is maximum.

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Acceleration at Maximum Displacement

At:

x = +A

acceleration is:

a = −ω²A

At:

x = −A

acceleration is:

a = +ω²A

Therefore:

Maximum displacement → Maximum acceleration magnitude

The acceleration always points toward equilibrium.

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Maximum Acceleration

The maximum acceleration magnitude is:

amax = Aω²

This occurs at maximum displacement.

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Worked Example — Maximum Acceleration

An oscillator has:

A = 0.050 m

and:

ω = 8.0 rad/s

Calculate its maximum acceleration.

amax = Aω²

amax = (0.050)(8.0²)

amax = (0.050)(64)

amax = 3.2 m/s²

Therefore:

Maximum acceleration = 3.2 m/s²

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Acceleration-Time Graph

Acceleration varies sinusoidally.

It has the same period as displacement and velocity.

However, acceleration and displacement always have opposite signs.

When:

x = maximum positive

then:

a = maximum negative

When:

x = maximum negative

then:

a = maximum positive

Therefore, acceleration and displacement are:

180° out of phase

or:

π radians out of phase

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Comparing the Three Graphs

For a complete understanding of SHM, displacement, velocity, and acceleration should be considered together.

At maximum positive displacement:

x = +A

v = 0

a = maximum negative

At equilibrium while moving in the negative direction:

x = 0

v = maximum negative

a = 0

At maximum negative displacement:

x = −A

v = 0

a = maximum positive

At equilibrium while moving in the positive direction:

x = 0

v = maximum positive

a = 0

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A Quarter-Cycle Pattern

A useful way to understand SHM is to divide one oscillation into four equal parts.

Starting at maximum positive displacement:

0T

x = +A
v = 0
a = maximum negative

¼T

x = 0
v = maximum negative
a = 0

½T

x = −A
v = 0
a = maximum positive

¾T

x = 0
v = maximum positive
a = 0

1T

x = +A
v = 0
a = maximum negative

Then the cycle repeats.

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Relating Graphs to a Spring

Imagine a mass attached to a horizontal spring.

At maximum extension:

• Displacement is maximum.
• Velocity is zero.
• Restoring force is maximum toward equilibrium.
• Acceleration magnitude is maximum.

As the mass approaches equilibrium:

• Displacement decreases.
• Speed increases.
• Acceleration magnitude decreases.

At equilibrium:

• Displacement is zero.
• Speed is maximum.
• Acceleration is zero.

The process then reverses as the mass moves toward maximum compression.

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Relating Graphs to a Pendulum

A small-angle pendulum follows the same general pattern.

At maximum displacement:

• Speed = 0
• Restoring acceleration magnitude = maximum

At equilibrium:

• Speed = maximum
• Tangential restoring acceleration = 0

The same mathematical model can therefore describe both systems approximately.

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Velocity as a Function of Displacement

Velocity can also be related directly to displacement:

v² = ω²(A² − x²)

or:

v = ±ω√(A² − x²)

This equation allows us to calculate the velocity at a particular displacement without knowing the time.

The positive or negative sign depends on the direction of motion.

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Worked Example — Velocity at a Position

An oscillator has:

A = 0.10 m

ω = 5.0 rad/s

At one instant:

x = 0.060 m

Calculate the speed.

v = ω√(A² − x²)

v = 5.0√(0.10² − 0.060²)

v = 5.0√(0.0100 − 0.0036)

v = 5.0√0.0064

v = 5.0(0.080)

v = 0.40 m/s

Therefore:

Speed = 0.40 m/s

The velocity could be +0.40 m/s or −0.40 m/s depending on the direction of motion.

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Worked Example — Using the SHM Equation

An oscillator follows:

x = 0.15 cos(4t)

where x is measured in metres and t in seconds.

Compare this with:

x = A cos(ωt)

Therefore:

A = 0.15 m

and:

ω = 4 rad/s

The period is:

T = 2π/ω

T = 2π/4

T ≈ 1.57 s

The frequency is:

f = 1/T

f ≈ 0.637 Hz

So:

Amplitude = 0.15 m

Period ≈ 1.57 s

Frequency ≈ 0.64 Hz

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Worked Example — Finding Acceleration

For the same oscillator:

x = 0.15 cos(4t)

suppose the displacement at a particular instant is:

x = 0.10 m

Use:

a = −ω²x

a = −(4²)(0.10)

a = −1.6 m/s²

The negative sign means the acceleration points in the negative direction.

Because the object currently has positive displacement, this acceleration points:

toward equilibrium

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Graph of Acceleration Against Displacement

Instead of plotting acceleration against time, we can plot:

acceleration against displacement

Since:

a = −ω²x

the graph is a straight line through the origin with a negative gradient.

The gradient is:

−ω²

This provides another way to identify SHM experimentally.

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Using the Gradient to Find Angular Frequency

Suppose an acceleration-displacement graph has gradient:

−25 s⁻²

Since:

Gradient = −ω²

then:

ω² = 25

Therefore:

ω = 5 rad/s

The period is:

T = 2π/ω

T = 2π/5

T ≈ 1.26 s

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Energy and the SHM Graphs

The graphical relationships also help us understand energy.

At maximum displacement:

v = 0

Therefore:

Kinetic energy = minimum

and potential energy is maximum.

At equilibrium:

|v| = maximum

Therefore:

Kinetic energy = maximum

and potential energy is minimum.

In an ideal oscillator:

Total mechanical energy remains constant.

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Recognizing SHM from Graphs

A sinusoidal displacement-time graph is strong evidence of oscillatory motion.

However, the defining condition for SHM is:

a ∝ −x

Therefore, an acceleration-displacement graph that forms a straight line through the origin with negative gradient is particularly useful evidence of SHM.

The negative gradient demonstrates that:

• Acceleration is proportional to displacement.
• Acceleration acts opposite to displacement.
• The acceleration therefore acts toward equilibrium.

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Common Mistakes

Mistake: Measuring Peak-to-Peak Distance as Amplitude

Amplitude is measured:

equilibrium → maximum displacement

not:

minimum → maximum

Peak-to-peak displacement is 2A.

Mistake: Measuring Half a Cycle as the Period

The period is the time required to return to the same position moving in the same direction.

Maximum positive displacement to maximum negative displacement is only:

T/2

Mistake: Thinking Velocity Is Greatest at Maximum Displacement

At maximum displacement:

v = 0

Velocity magnitude is greatest at equilibrium.

Mistake: Thinking Acceleration Is Greatest at Equilibrium

At equilibrium:

a = 0

Acceleration magnitude is greatest at maximum displacement.

Mistake: Thinking Velocity and Acceleration Always Point Together

They do not.

An oscillator can be moving toward equilibrium while accelerating in the same direction, or moving away from equilibrium while acceleration opposes its motion.

Mistake: Forgetting the Negative Sign

In:

a = −ω²x

the negative sign is essential.

It shows that acceleration acts opposite to displacement.

Mistake: Confusing Frequency and Angular Frequency

Frequency:

f, measured in Hz.

Angular frequency:

ω, measured in rad/s.

They are related by:

ω = 2πf

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Key Terms

Simple harmonic motion (SHM) – oscillation in which acceleration is proportional and opposite to displacement.

Displacement – position relative to equilibrium.

Amplitude – maximum displacement from equilibrium.

Period – time for one complete oscillation.

Frequency – number of complete oscillations per second.

Angular frequency – rate of change of phase, measured in rad/s.

Phase – position within an oscillation cycle.

Phase difference – difference between the positions of two oscillations within their cycles.

Equilibrium position – central position where displacement and restoring acceleration are zero.

Sinusoidal – having the mathematical form of a sine or cosine curve.

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Check Your Understanding

1. What does the amplitude of a displacement-time graph represent?

2. A graph varies between +8 cm and −8 cm. What is its amplitude?

3. Successive maxima occur at 2 s and 7 s. Calculate the period.

4. Calculate the frequency for the graph in Question 3.

5. Where in an SHM cycle is speed greatest?

6. Where is acceleration magnitude greatest?

7. What is the velocity when x = +A?

8. What is the acceleration when x = 0?

9. Explain why displacement and acceleration are 180° out of phase.

10. An oscillator has a period of 0.50 s. Calculate its frequency.

11. Calculate its angular frequency.

12. An oscillator has A = 0.080 m and ω = 6.0 rad/s. Calculate its maximum speed.

13. Calculate its maximum acceleration.

14. An oscillator follows x = 0.20 cos(5t). Identify its amplitude and angular frequency.

15. Explain how an acceleration-displacement graph can demonstrate that an object is undergoing SHM.

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Key Takeaways

• SHM can be represented using displacement, velocity, and acceleration graphs.
• Displacement varies sinusoidally with time.
• Amplitude is the maximum displacement from equilibrium.
• Period is the time for one complete oscillation.
• Frequency is the reciprocal of period.
• Angular frequency is related to frequency by ω = 2πf.
• Velocity is the gradient of the displacement-time graph.
• Acceleration is the gradient of the velocity-time graph.
• Speed is maximum at equilibrium.
• Velocity is zero at maximum displacement.
• Acceleration is zero at equilibrium.
• Acceleration magnitude is maximum at maximum displacement.
• Displacement and velocity differ in phase by one-quarter of a cycle.
• Displacement and acceleration are 180° out of phase.
• SHM satisfies the relationship a = −ω²x.
• Maximum speed is Aω.
• Maximum acceleration is Aω².
• An acceleration-displacement graph for SHM is a straight line with negative gradient.
• Mathematical models allow the position, velocity, acceleration, period, and frequency of an oscillator to be calculated.

Did You Know?

SHM graphs are closely related to circular motion. Imagine watching a point moving at constant speed around a circle, but looking only at its horizontal position. The point's horizontal displacement moves back and forth in exactly the same sinusoidal pattern as ideal simple harmonic motion. This connection explains why sine, cosine, angular frequency, and phase appear so naturally in the mathematics of oscillations.