Simple Harmonic Motion Explained: Period, Frequency, Energy, and Graphs

Overview

Start here if you want the big picture first. Simple harmonic motion, usually shortened to SHM, is a special kind of oscillation in which the restoring force points toward equilibrium and is proportional to displacement from equilibrium.

In symbols, that idea is often written as:

F = -kx

or, using Newton's second law,

a = -ω²x

These two equations say the same essential thing. The minus sign matters: it shows that force and acceleration are directed opposite to the displacement. If the object is to the right of equilibrium, acceleration is to the left. If the object is to the left, acceleration is to the right.

This is why SHM is not just “back and forth motion.” An object can move back and forth without being in SHM. To qualify as simple harmonic motion, the motion must satisfy that proportional restoring relationship.

The key quantities are:

• Displacement, x: distance from equilibrium, with direction
• Amplitude, A: maximum displacement from equilibrium
• Period, T: time for one complete oscillation
• Frequency, f: number of oscillations per second
• Angular frequency, ω: how quickly the phase changes, where ω = 2πf = 2π/T

The basic links among them are worth memorizing because they appear constantly in step by step physics solutions:

f = 1/T
T = 1/f
ω = 2πf = 2π/T

For a mass-spring system, the period depends on the mass and spring constant:

T = 2π√(m/k)

For a simple pendulum at small angles, the period is approximately:

T = 2π√(L/g)

These formulas already show an important feature of SHM: the period is determined by system properties, not by where the object happens to be during its motion. In ideal SHM, changing the starting point changes amplitude, but not period.

If you are revising from a formula sheet, it helps to pair this guide with broader equation summaries such as the GCSE Physics Equations List and Rearrangement Guide, the A-Level Physics Equations List with Definitions and Unit Checks, or the AP Physics Formula Sheet Guide: What Every Equation Means.

Core framework

This section gives you the structure behind the topic so you can move between equations, graphs, and intuition without treating them as separate tasks.

1. The defining rule of SHM

The central rule is:

acceleration is proportional to displacement and opposite in direction

That means:

a = -ω²x

This equation is often the fastest way to recognize SHM in physics questions and answers. If a problem gives you a relation where acceleration is directly proportional to negative displacement, the motion is SHM.

It also tells you what happens at important positions:

• At equilibrium (x = 0): acceleration is zero
• At maximum displacement (x = ±A): acceleration has maximum magnitude

That can feel surprising at first. Students often think acceleration should be greatest where speed is greatest, but in SHM the opposite is true.

2. Standard SHM equations

A common displacement equation is:

x = A cos(ωt)

or sometimes

x = A sin(ωt)

Both are valid. The choice depends on where the object starts when t = 0.

From displacement, we can get velocity and acceleration:

v = -Aω sin(ωt)
a = -Aω² cos(ωt)

Since x = A cos(ωt), the acceleration equation becomes:

a = -ω²x

There is also a very useful equation linking speed and position directly:

v² = ω²(A² - x²)

This is especially helpful in worked physics problems because it avoids time completely.

3. Period and frequency physics

The period tells you how long one full cycle takes. The frequency tells you how many cycles happen each second. They are reciprocals:

f = 1/T

If a spring oscillates 5 times in 2 seconds, then:

• frequency = 5/2 = 2.5 Hz
• period = 1/2.5 = 0.40 s

Angular frequency is then:

ω = 2πf = 5π rad s⁻¹

In exam settings, students often lose marks by mixing up frequency in hertz with angular frequency in radians per second. They are related, but they are not the same number unless by coincidence.

4. What the graphs mean

Understanding spring motion graphs makes SHM much easier. The displacement-time graph is sinusoidal. But the velocity-time and acceleration-time graphs are also sinusoidal, just shifted in phase.

Displacement-time graph:

• maximum at the turning points
• zero at equilibrium
• slope of the graph gives velocity

Velocity-time graph:

• maximum magnitude at equilibrium
• zero at the turning points
• is one-quarter of a cycle out of phase with displacement

Acceleration-time graph:

• zero at equilibrium
• maximum magnitude at the turning points
• is in antiphase with displacement

A simple way to remember this is to imagine releasing a mass from the right-hand turning point:

• At first, x = +A
• Its speed is 0
• Its acceleration is maximum toward the center

As it passes equilibrium:

• x = 0
• speed is maximum
• acceleration is 0

At the left-hand turning point:

• x = -A
• speed is 0 again
• acceleration is maximum back toward the center

This one sequence explains nearly every SHM graph interpretation question.

5. Energy in oscillation

Oscillation energy is another place where the topic becomes clearer when the pieces are linked. In ideal SHM, total mechanical energy stays constant while energy transfers between kinetic and potential forms.

For a spring:

Total energy: E = 1/2 kA²
Elastic potential energy: U = 1/2 kx²
Kinetic energy: K = 1/2 mv²

At the turning points:

• speed = 0
• kinetic energy = 0
• potential energy is maximum

At equilibrium:

• displacement = 0
• potential energy is minimum
• kinetic energy is maximum

This gives a powerful shortcut: wherever speed is greatest, kinetic energy is greatest. Wherever displacement magnitude is greatest, potential energy is greatest.

If damping is ignored, total energy remains constant. In real systems, friction and air resistance gradually reduce amplitude, so real oscillations often approach but do not maintain perfect SHM forever.

Practical examples

These examples show how to use the framework quickly and confidently.

Example 1: Find period and frequency for a mass-spring system

A 0.50 kg mass is attached to a spring with spring constant 200 N m⁻¹. Find the period and frequency.

Use:

T = 2π√(m/k)

Substitute values:

T = 2π√(0.50/200)

T = 2π√(0.0025)

T = 2π(0.05) ≈ 0.314 s

Now find frequency:

f = 1/T ≈ 1/0.314 ≈ 3.18 Hz

So the motion repeats about 3.18 times each second.

Example 2: Find maximum speed

An oscillator has amplitude 0.080 m and angular frequency 12 rad s⁻¹. Find the maximum speed.

In SHM:

v_max = Aω

Substitute:

v_max = 0.080 × 12 = 0.96 m s⁻¹

Maximum speed occurs at equilibrium, not at the turning points.

Example 3: Find acceleration at a given displacement

A particle moves in SHM with angular frequency 5 rad s⁻¹. What is its acceleration when displacement is 0.12 m?

Use:

a = -ω²x

a = -(5²)(0.12)

a = -25 × 0.12 = -3.0 m s⁻²

The negative sign shows the acceleration is directed toward equilibrium.

Example 4: Use the energy-position-speed relation

An oscillator has amplitude 0.20 m and angular frequency 8 rad s⁻¹. Find the speed when displacement is 0.12 m.

Use:

v² = ω²(A² - x²)

v² = 8²(0.20² - 0.12²)

v² = 64(0.0400 - 0.0144)

v² = 64(0.0256) = 1.6384

v = 1.28 m s⁻¹

This method is efficient because it avoids finding time first.

Example 5: Read the graphs conceptually

If a mass is at maximum positive displacement, what are the signs or values of velocity and acceleration?

• displacement: positive maximum
• velocity: zero
• acceleration: negative maximum magnitude

This kind of graph-reading question is common in physics homework help and exam preparation because it tests understanding rather than substitution.

Example 6: Pendulum note

For a simple pendulum, SHM is only a good approximation for small angular displacements. If the angle becomes too large, the motion is still oscillatory, but the standard SHM model becomes less accurate. That distinction matters in lab work and in higher-precision questions.

If you are writing up pendulum data or comparing measured and theoretical values, it is useful to review uncertainty handling and graph presentation in the Physics Lab Report Checklist: Sections, Graphs, Uncertainty, and Common Mistakes and the Significant Figures Rules in Physics: How to Round, Multiply, and Report Results.

Common mistakes

This section helps you avoid the errors that make SHM seem inconsistent when the issue is usually a sign, graph, or definition mistake.

1. Treating any oscillation as SHM

Not every repeated motion is simple harmonic. The restoring force must be proportional to displacement and directed toward equilibrium.

2. Forgetting the minus sign in a = -ω²x

The minus sign is not decorative. It shows the restoring nature of the motion. Without it, the model would predict acceleration away from equilibrium.

3. Mixing up amplitude and displacement

Amplitude is the largest possible displacement. Displacement is the current position relative to equilibrium. The object can have displacement 0.03 m while the amplitude is 0.10 m.

4. Assuming speed is greatest at the ends

At the turning points, the object changes direction, so speed is zero. Maximum speed occurs at equilibrium.

5. Confusing frequency and angular frequency

Frequency is measured in hertz. Angular frequency is measured in radians per second. Remember:

ω = 2πf

6. Misreading graph phase relationships

Displacement, velocity, and acceleration do not peak together. Velocity is a quarter-cycle out of phase with displacement, and acceleration is opposite in phase to displacement.

7. Using pendulum SHM formulas at large angles without caution

The small-angle approximation matters. If the angle is large, the simple formula for period is only approximate.

8. Ignoring units

Spring constant should be in N m⁻¹, mass in kg, length in m, time in s, and angular frequency in rad s⁻¹. Unit checks are a fast way to catch errors before they spread.

9. Forgetting that ideal energy conservation assumes no damping

Real oscillators lose energy. If amplitude is decreasing over time, the motion is damped and the simplest constant-energy SHM model is no longer a perfect description.

Students who struggle with these distinctions often benefit from comparing SHM graphs with other familiar graph-based topics, such as exponential charge and discharge curves in Capacitors and RC Circuits Explained with Charging and Discharging Graphs or trajectory graphs in the Projectile Motion Calculator Guide: Equations, Graphs, and Worked Examples. Seeing what makes SHM sinusoidal helps sharpen your graph intuition across physics.

When to revisit

Come back to this topic whenever you need to move from memorizing formulas to actually using them. SHM is worth revisiting at several points in a course because the same framework supports many different questions.

Revisit this guide when:

• you start solving spring or pendulum problems
• you meet sinusoidal graphs and need to interpret phase differences
• you study wave motion, where oscillation ideas reappear
• you are preparing for AP, IB, GCSE, A-Level, or introductory college exams
• you are analyzing lab data from oscillation experiments
• you need a quick reference for physics formulas involving period, frequency, and energy

A practical way to use this article during revision is to test yourself in a fixed order:

1. State the condition for SHM: a = -ω²x.
2. Define amplitude, period, frequency, and angular frequency.
3. Sketch displacement-time, velocity-time, and acceleration-time graphs for one cycle.
4. Mark where speed is maximum and where acceleration is maximum.
5. Explain how kinetic and potential energy change during one oscillation.
6. Solve one problem using the period formula and one using v² = ω²(A² - x²).

If you can do those six things without guessing, your understanding is usually strong enough to handle most standard SHM questions.

For exam-focused revision, it can help to pair this concept guide with broader checklists such as the IB Physics Revision Guide: Topic-by-Topic Formula and Concept Checklist. The goal is not just to remember an equation, but to know what the equation says about motion, graphs, and energy.

In short, the most useful habit is to treat SHM as one connected system: restoring force creates acceleration, acceleration shapes the motion, the motion creates the graphs, and the graphs reveal the energy changes. Once those links are clear, simple harmonic motion stops being a set of disconnected formulas and becomes one of the most manageable topics in introductory physics.