Apply Simple Harmonic Motion — Apply C.1 Simple harmonic...

SECTION 1

Historical Context & Motivation

Humans have been fascinated by repetitive, back-and-forth motion for centuries. From the rhythmic swing of a pendulum in a grandfather clock to the vibration of a guitar string, oscillations are everywhere in nature. The formal study of simple harmonic motion (SHM) began when scientists noticed that many different systems — springs, pendulums, floating objects — all follow the same elegant mathematical pattern. Understanding SHM is the gateway to understanding waves, sound, light, and even quantum mechanics.

1583

Galileo and the Pendulum

Galileo Galilei observed a swinging chandelier in the Cathedral of Pisa and noticed that the period of a pendulum's swing remained roughly constant regardless of its amplitude — a property called isochronism.

1678

Hooke's Law Published

Robert Hooke formulated the relationship between the force exerted by a spring and its displacement: F = −kx. This became the foundation for analysing SHM in spring systems.

1687

Newton's Principia

Isaac Newton published his laws of motion, providing the mathematical framework (F = ma) needed to derive the equations of SHM from Hooke's Law.

1822

Fourier's Theorem

Joseph Fourier showed that any periodic motion can be decomposed into a sum of simple harmonic oscillations, making SHM the fundamental building block of all periodic phenomena.

The central question SHM addresses is deceptively simple: How do we mathematically describe any system that oscillates back and forth around a central equilibrium position? In the IB Physics C.1 topic, you will learn to apply SHM equations to predict displacement, velocity, acceleration, and energy at any moment during an oscillation. This is essential for understanding wave behaviour more broadly.

SECTION 2

Core Principles & Definitions

Simple harmonic motion occurs whenever an object experiences a restoring force that is directly proportional to its displacement from equilibrium and always directed back toward that equilibrium position. This single condition — proportional and opposite — is what defines SHM and distinguishes it from other types of oscillation.

Displacement (x)

The distance and direction of the oscillating object from its equilibrium (rest) position at any instant. Measured in metres (m). In SHM, displacement varies sinusoidally with time.

Amplitude (x₀ or A)

The maximum displacement from the equilibrium position. It represents the 'size' of the oscillation. The IB syllabus uses x₀, though many textbooks use A.

Period (T) and Frequency (f)

The period is the time for one complete oscillation (in seconds). Frequency is the number of oscillations per second (in hertz). They are reciprocals: T = 1/f.

Angular Frequency (ω)

A measure of how rapidly the oscillation cycles, defined as ω = 2πf = 2π/T. Measured in rad s⁻¹. It connects SHM to circular motion.

Restoring Force Condition

For SHM, the acceleration must be proportional to displacement and directed toward equilibrium: a = −ω²x. This is the defining equation of SHM.

✦ KEY TAKEAWAY

Think of SHM like a ball rolling in a bowl. If you push the ball to one side and release it, the curved surface always pushes it back toward the centre. The farther you push it, the stronger the push back. That proportional 'push back toward the middle' is exactly what defines SHM — whether it's a mass on a spring, a pendulum, or a vibrating molecule.

SECTION 3

Visual Explanation — Displacement, Velocity & Acceleration

The diagram below shows how displacement, velocity, and acceleration all vary sinusoidally with time in SHM. Notice that they are all the same shape (sinusoidal) but shifted in phase relative to each other. Velocity leads displacement by 90° (π/2 rad), and acceleration leads velocity by another 90°, meaning acceleration is exactly 180° (π rad) out of phase with displacement.

The three sinusoidal curves show displacement (cyan), velocity (violet), and acceleration (pink) as functions of time. Velocity leads displacement by π/2, and acceleration is exactly π out of phase with displacement — when displacement is at a maximum, acceleration is at a maximum in the opposite direction.

This phase relationship is critically important for problem-solving. At the equilibrium position (x = 0), the velocity is at its maximum and the acceleration is zero. At the extreme positions (x = ±x₀), the velocity is zero and the acceleration is at its maximum magnitude. Keep this in mind whenever a problem asks you about conditions at specific points in the oscillation.

SECTION 4

Mathematical Framework

The IB C.1 topic requires you to use several interconnected equations. Each equation flows logically from the defining condition of SHM and connects displacement, velocity, acceleration, and energy. Let's build them step by step.

DEFINING CONDITION OF SHM

a = −ω²x

Where a = acceleration (m s⁻²), ω = angular frequency (rad s⁻¹), and x = displacement from equilibrium (m). The negative sign shows acceleration is always directed toward equilibrium.

DISPLACEMENT AS A FUNCTION OF TIME

x = x₀ sin(ωt) or x = x₀ cos(ωt)

Use cosine if the object starts at maximum displacement (x = x₀ at t = 0). Use sine if the object starts at equilibrium (x = 0 at t = 0). Here x₀ is the amplitude.

VELOCITY AS A FUNCTION OF DISPLACEMENT

v = ±ω√(x₀² − x²)

This is especially useful when you know the displacement and need to find the speed. At x = 0, v = ωx₀ (maximum speed). At x = ±x₀, v = 0. The ± indicates the object can be moving in either direction.

ENERGY IN SHM

E_T = ½mω²x₀² = ½kx₀²

The total energy is constant and equals the maximum kinetic energy or the maximum potential energy. At any displacement: E_K = ½mω²(x₀² − x²) and E_P = ½mω²x².

💡 IB Exam Tip

On the IB exam, always check whether the problem starts the oscillation at maximum displacement or at equilibrium. This tells you whether to use sine or cosine. Also, make sure your calculator is in radians mode when working with ωt.

SECTION 5

Energy in Simple Harmonic Motion

One of the most elegant features of SHM is the continuous exchange between kinetic energy and potential energy. As the object moves through equilibrium, all its energy is kinetic. As it reaches maximum displacement, all its energy is potential. At every point in between, the total mechanical energy remains constant (assuming no damping). The diagram below illustrates this energy exchange.

_TE_P = ½mω²x²E_K = ½mω²(x₀²−x²)E_Total = ½mω²x₀²Max KE, Zero PEMax PE, Zero KEMax PE, Zero KE

Kinetic energy (cyan, solid curve) is maximum at x = 0 and zero at x = ±x₀. Potential energy (amber, dashed parabola) is zero at x = 0 and maximum at the extremes. The total energy (red, horizontal dashed line) remains constant throughout the motion.

Summary of key quantities at special positions in SHM

Position	Displacement	Velocity	Acceleration	KE	PE
Equilibrium	0	Maximum (±ωx₀)	0	Maximum	0
Extreme (+x₀)	+x₀	0	−ω²x₀ (max, toward eq.)	0	Maximum
Extreme (−x₀)	−x₀	0	+ω²x₀ (max, toward eq.)	0	Maximum

SECTION 6

Worked Example

A horizontal mass-spring system oscillates with simple harmonic motion. The mass is 0.50 kg, the spring constant is 200 N m⁻¹, and the amplitude of oscillation is 0.040 m. Find: (a) the period, (b) the maximum speed, (c) the speed when the displacement is 0.020 m, and (d) the total energy of the system.

Mass-Spring SHM Problem

Step 1 — Identify Given ValuesMass m = 0.50 kg, spring constant k = 200 N m⁻¹, amplitude x₀ = 0.040 m.

Step 2 — Find Angular Frequency (ω)For a mass-spring system, ω = √(k/m) = √(200/0.50) = √400 = 20 rad s⁻¹.

ω = 20 rad s⁻¹

Step 3 — Calculate the Period (T)T = 2π/ω = 2π/20 = 0.314 s ≈ 0.31 s.

T ≈ 0.31 s

Step 4 — Calculate Maximum Speed (v_max)The maximum speed occurs at equilibrium (x = 0). Using v_max = ωx₀ = 20 × 0.040 = 0.80 m s⁻¹.

v_max = 0.80 m s⁻¹

Step 5 — Speed at x = 0.020 mUsing v = ω√(x₀² − x²) = 20 × √(0.040² − 0.020²) = 20 × √(0.0016 − 0.0004) = 20 × √0.0012 = 20 × 0.03464 = 0.693 m s⁻¹ ≈ 0.69 m s⁻¹.

v ≈ 0.69 m s⁻¹

Step 6 — Total EnergyE_T = ½kx₀² = ½ × 200 × (0.040)² = ½ × 200 × 0.0016 = 0.16 J. Alternatively, E_T = ½mω²x₀² = ½ × 0.50 × 400 × 0.0016 = 0.16 J. Both methods agree.

E_T = 0.16 J

SECTION 7

SHM vs Other Types of Oscillation

Not all oscillations are simple harmonic. In real life, friction and air resistance cause oscillations to lose energy over time — this is called damped oscillation. When an external periodic force drives the system, we get forced oscillation. Understanding pure SHM first is essential because it provides the idealised baseline from which these more complex behaviours can be understood.

Comparison of pure SHM and damped oscillation

Feature	Simple Harmonic Motion	Damped Oscillation
Amplitude over time	Constant — does not change	Decreases exponentially
Total energy	Conserved	Decreases over time (dissipated as heat)
Restoring force	Proportional to displacement only	Proportional to displacement plus a velocity-dependent drag force
Period	Independent of amplitude	Slightly affected at high damping
Real-world example	Idealised frictionless spring	Car shock absorber, pendulum in air

✦ KEY TAKEAWAY

SHM is like the 'perfect world' version of oscillation — no energy lost, no friction, perfectly symmetrical. Real oscillations are messier, but the SHM model captures the essential physics. Just as learning to drive on a flat, empty road prepares you for complex traffic, mastering SHM equations prepares you to tackle damping, resonance, and wave mechanics.

SECTION 8

Connection to Waves & Advanced Topics

Simple harmonic motion is not just an isolated topic — it is the fundamental building block of wave theory. When you study transverse and longitudinal waves later in the IB course, you will see that each particle in a wave medium undergoes SHM. The displacement equation x = x₀ sin(ωt) for a single oscillator extends to the travelling wave equation y = y₀ sin(kx − ωt), where k is the wave number. Understanding SHM deeply will make wave topics much more intuitive.

How SHM concepts extend into wave theory

Concept	SHM (C.1)	Waves (C.2–C.5)
What oscillates	A single object about a fixed point	Many particles, each in SHM, with a phase difference between neighbours
Key equation	x = x₀ sin(ωt)	y = y₀ sin(kx − ωt)
Energy transfer	KE ↔ PE within one system	Energy propagates through space via the medium
ω (angular frequency)	Describes how fast one object oscillates	Same meaning, but connects to wavelength via v = fλ

In more advanced physics (HL topics and university-level courses), SHM connects to resonance (when a driving frequency matches the natural frequency of a system), standing waves, and even quantum mechanics where the quantum harmonic oscillator is one of the most important models. The time you invest in mastering C.1 will pay dividends throughout your physics career.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

A mass on a spring passes through its equilibrium position. At this instant, is the acceleration zero, at its maximum, or somewhere in between? Explain your reasoning using the defining equation of SHM.

PROBLEM 2 — BASIC CALCULATION

A pendulum completes 30 full oscillations in 60 seconds. Calculate: (a) its period T, (b) its frequency f, and (c) its angular frequency ω.

PROBLEM 3 — INTERMEDIATE

A 0.25 kg mass on a spring oscillates with SHM of amplitude 0.060 m and period 0.40 s. Calculate the speed of the mass when its displacement from equilibrium is 0.030 m.

PROBLEM 4 — APPLIED

A car's suspension spring compresses 0.05 m when a 60 kg person sits on it. If the car hits a bump and the spring oscillates vertically in SHM, estimate the period of the resulting oscillation. (Use g = 9.8 m s⁻².)

PROBLEM 5 — CRITICAL THINKING

A mass oscillates in SHM with amplitude x₀. At what displacement (as a fraction of x₀) is the kinetic energy exactly equal to the potential energy? Show your reasoning algebraically.

SUMMARY

Lesson Summary

Simple harmonic motion occurs whenever a restoring force is proportional to displacement and directed toward equilibrium, giving the defining equation a = −ω²x. Displacement varies sinusoidally with time as x = x₀ sin(ωt) or x₀ cos(ωt), depending on initial conditions. The angular frequency ω = 2πf = 2π/T connects period, frequency, and the rate of oscillation.

In SHM, kinetic and potential energy exchange continuously while the total mechanical energy E_T = ½mω²x₀² remains constant. The speed at any displacement is given by v = ω√(x₀² − x²). Velocity is maximum at equilibrium and zero at the extremes; acceleration is the opposite. SHM is the foundation of wave behaviour — every particle in a wave undergoes SHM, linking this topic directly to the rest of the IB Physics Wave behaviour unit.