1. Characteristics of SHM

Learning Outcomes

I can define simple harmonic motion (SHM).
I can identify systems that undergo SHM.
I can explain the role of restoring forces in SHM.
I can distinguish SHM from other forms of oscillatory motion.
I can describe the motion of oscillating systems using scientific terminology.

Key Topics:

Periodic motion vs. SHM.
Restoring force proportional to displacement: $F = - k x$ .

What Is Oscillatory Motion?

Oscillatory motion is repeated movement back and forth around a central position.

Many systems in physics can oscillate.

Examples include:

A mass attached to a spring
A swinging pendulum
A vibrating guitar string
A diving board moving up and down
A tuning fork after it is struck

The central position around which the object oscillates is called the equilibrium position.

When an oscillating object reaches one side of its motion, it reverses direction and moves back toward equilibrium.

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What Is Simple Harmonic Motion?

Simple harmonic motion, or SHM, is a special type of oscillatory motion.

In SHM, the restoring force acting on an object is:

Directed toward the equilibrium position.
Directly proportional to the object's displacement from equilibrium.

Mathematically:

$F \propto - x$

where:

F = restoring force
x = displacement from equilibrium
The negative sign indicates that the force acts in the opposite direction to the displacement

This relationship is the defining characteristic of simple harmonic motion.

The Equilibrium Position

The equilibrium position is the position where the system would naturally remain if it were not disturbed.

At equilibrium:

$x = 0$

For a horizontal mass-spring system, this is the position where the mass rests when no net force is acting on it.

If the mass is pulled away from equilibrium and released, it begins to oscillate.

The object repeatedly moves:

$left \to centre \to right \to centre$

and the pattern repeats.

Displacement in SHM

Displacement describes the object's position relative to equilibrium.

We normally represent displacement using:

At equilibrium:

$x = 0$

On one side:

$x > 0$

On the opposite side:

$x < 0$

The sign tells us which side of equilibrium the object is on.

What Is a Restoring Force?

A restoring force is a force that acts to return an object toward its equilibrium position.

Imagine pulling a mass attached to a spring to the right.

The spring pulls the mass back toward the centre.

If the mass moves to the left of equilibrium, the spring pushes or pulls it toward the centre again.

Therefore, the restoring force always acts opposite to the displacement.

Displacement right→Force left Displacement left→Force right

This constant tendency to return toward equilibrium produces the oscillation.

Restoring Force and Displacement

For SHM, the magnitude of the restoring force increases as the object moves farther from equilibrium.

If the displacement doubles, the restoring force doubles.

If the displacement triples, the restoring force triples.

Therefore:

$F \propto x$

in magnitude.

Including the direction gives:

$F \propto - x$

This means the restoring force is both:

proportional to displacement and opposite in direction.

Hooke's Law and SHM

A mass attached to an ideal spring provides an important example of SHM.

For a spring:

$F = - k x$

where:

F = restoring force
k = spring constant
x = displacement
The negative sign indicates the restoring direction

This is Hooke's law.

If the spring obeys Hooke's law, the restoring force is proportional to displacement.

Therefore, an ideal mass-spring system can undergo simple harmonic motion.

Example: A Mass on a Spring

Imagine a mass attached to a horizontal spring.

Position A: Maximum Displacement

The mass is pulled to the right.

Its displacement is maximum.

The restoring force is also maximum and acts toward the left.

The mass is momentarily stationary before reversing direction.

Position B: Equilibrium

The mass passes through the centre.

Its displacement is:

$x = 0$

The restoring force is zero.

However, the mass is moving at its maximum speed.

Position C: Opposite Maximum Displacement

The mass reaches the other side.

Its displacement is again maximum in magnitude.

Its speed is zero for an instant.

The restoring force is maximum and points back toward equilibrium.

The cycle then repeats.

Amplitude

The amplitude is the maximum displacement of an oscillating object from its equilibrium position.

It is usually represented by:

If an object moves between:

$x = - 5 cm$

and:

$x = + 5 cm$

then its amplitude is:

A = 5 cm

The total distance between the two extreme positions is 10 cm, but the amplitude is only 5 cm.

Amplitude is always measured from equilibrium to an extreme position.

One Complete Oscillation

One complete oscillation occurs when an object returns to the same position while moving in the same direction.

For example:

$+ A \to 0 \to - A \to 0 \to + A$

is one complete oscillation.

The object has moved from one extreme to the other and back again.

Period

The period is the time required for one complete oscillation.

It is represented by:

and measured in:

seconds (s)

For example, if a pendulum completes one full oscillation in 2.0 seconds:

T = 2.0 s

Frequency

The frequency is the number of complete oscillations occurring each second.

It is represented by:

and measured in:

hertz (Hz)

One hertz means:

1 oscillation per second

For example:

$f = 5 Hz$

means that the system completes:

5 oscillations every second

Period and frequency are closely related:

$f = $ \frac{1}{T} $$

and:

$T = $ \frac{1}{f} $$

SHM as a Graph

If displacement is plotted against time, ideal SHM produces a repeating wave-shaped pattern.

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The graph allows us to identify:

Maximum positive displacement
Maximum negative displacement
Equilibrium position
Amplitude
Period
Repeated oscillations

The repeating shape shows that the motion is periodic.

Speed During SHM

The speed of an object undergoing SHM constantly changes.

At the extreme positions:

$x = \pm A$

the speed is:

v = 0

At equilibrium:

$x = 0$

the speed is:

maximum

This happens because the object accelerates toward equilibrium and slows down as it moves away from equilibrium.

Acceleration During SHM

Since:

$F = ma$

and the restoring force in SHM satisfies:

$F \propto - x$

the acceleration must also satisfy:

$a \propto - x$

In fact:

$a = - ω 2 x$

where ω is the angular frequency.

This is another mathematical definition of SHM.

The acceleration always points toward equilibrium.

At equilibrium:

$a = 0$

At maximum displacement:

$∣ a ∣ = maximum$

Position, Speed, and Acceleration

The behaviour of an object during SHM can be summarized as follows:

Position	Displacement	Speed	Restoring Force	Acceleration
Positive extreme	+A	Zero	Maximum	Maximum
Equilibrium	0	Maximum	Zero	Zero
Negative extreme	−A	Zero	Maximum	Maximum

At both extreme positions, the force and acceleration point toward equilibrium.

Pendulums and SHM

A swinging pendulum can also approximately undergo SHM.

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When the pendulum is displaced from its equilibrium position, gravity produces a restoring effect that pulls it back toward the centre.

However, a pendulum behaves like ideal SHM only when the oscillations are relatively small.

For larger angles, the relationship between restoring force and displacement is no longer sufficiently proportional.

Therefore:

A simple pendulum approximately undergoes SHM for small oscillations.

Examples of Systems That Can Undergo SHM

Common examples include:

Mass-Spring System

A mass attached to an ideal spring oscillates around equilibrium.

Simple Pendulum

For small angles, a pendulum approximately undergoes SHM.

Vibrating Tuning Fork

The prongs move rapidly back and forth around equilibrium.

Vibrating String

Small sections of a stretched string can oscillate approximately harmonically.

Suspended Mass

A mass hanging vertically from a spring can oscillate vertically around its equilibrium position.

Many more complicated vibrating systems can also be approximated using SHM.

Is Every Oscillation SHM?

No.

All SHM is oscillatory motion, but not all oscillatory motion is simple harmonic motion.

For motion to be SHM, the restoring force must satisfy:

$F \propto - x$

An object may move repeatedly back and forth without satisfying this condition.

For example:

A pendulum swinging through very large angles
A bouncing ball
A strongly driven mechanical system
Oscillations involving complicated or non-linear forces

These may be oscillatory, but they are not necessarily simple harmonic.

SHM vs. General Oscillatory Motion

Simple Harmonic Motion	General Oscillatory Motion
Repeats around equilibrium	Repeats around a position
Has a restoring force	May have a restoring force
Restoring force is proportional to displacement	Force may have a more complicated relationship
Force points toward equilibrium	Motion may not satisfy this condition
Produces a sinusoidal displacement-time pattern	Pattern may have a different shape

The defining feature of SHM is therefore not simply that the object moves back and forth.

It is the relationship between restoring force and displacement.

Describing Oscillations Scientifically

Instead of saying:

The object moves back and forth.

we can describe the motion more precisely:

The object oscillates periodically about its equilibrium position with an amplitude of 4 cm and a period of 2 s.

This statement uses scientific terminology to describe:

The type of motion
The equilibrium position
The amplitude
The period

Scientific vocabulary allows oscillations to be described clearly and quantitatively.

A Complete Example

A mass attached to a spring oscillates between:

$x = - 0.08 m$

and:

$x = + 0.08 m$

It completes 10 oscillations in 5.0 seconds.

Amplitude

A = 0.08 m

Period

$T = $ \frac{5.0}{10} $$

T = 0.50 s

Frequency

$f = $ \frac{1}{T} $$

$f = $ \frac{1}{0.5} $$

f = 2.0 Hz

The system completes two full oscillations every second.

Did You Know?

Simple harmonic motion appears throughout physics because many systems behave approximately like SHM when they are displaced only a small distance from a stable equilibrium position.

This makes SHM useful for studying:

Mechanical vibrations
Sound
Waves
Molecular vibrations
Electrical oscillations
Seismic motion

Even complicated oscillating systems can often be understood by first studying the much simpler model of simple harmonic motion.

Key Vocabulary

Oscillation – Repeated motion back and forth around an equilibrium position.

Simple harmonic motion (SHM) – Oscillatory motion in which the restoring force is proportional to displacement and directed toward equilibrium.

Equilibrium position – The central position where the net restoring force is zero.

Displacement – The distance and direction of an object from equilibrium.

Restoring force – A force that acts toward the equilibrium position.

Amplitude – The maximum displacement from equilibrium.

Period – The time required for one complete oscillation.

Frequency – The number of complete oscillations per second.

Hertz (Hz) – The unit of frequency, equal to one oscillation per second.

Periodic motion – Motion that repeats after equal intervals of time.

Key Takeaways

Oscillatory motion is repeated movement around an equilibrium position.
Simple harmonic motion is a special type of oscillation.
In SHM, the restoring force is proportional to displacement and acts in the opposite direction: $F \propto - x$
A mass attached to an ideal spring is a standard example of SHM.
A simple pendulum approximately undergoes SHM when its oscillations are small.
Amplitude is the maximum displacement from equilibrium.
Period is the time for one complete oscillation.
Frequency is the number of oscillations completed each second.
At maximum displacement, speed is zero while restoring force and acceleration are greatest.
At equilibrium, speed is greatest while restoring force and acceleration are zero.
Not every oscillation is SHM.
The defining feature of SHM is the relationship between restoring force and displacement.