Push a swing at random moments and not much happens. Push it in time with its own rhythm and the child goes higher and higher with almost no effort. That’s resonance — and to understand it properly, we first need to sort oscillations into two families: free and forced.
📘 What you need to know
A free oscillation has only internal forces acting and no energy input — the system is displaced, released, and left alone
A freely oscillating system always vibrates at its natural frequencyf0 — the frequency it chooses when allowed to oscillate freely
A forced oscillation is produced by a periodic external driving force, which replaces the energy lost to damping
A forced system vibrates at the driving frequencyf — the driver sets the rhythm, not the system
Resonance: when the driving frequency equals the natural frequency (f = f0), the amplitude of the oscillations is at its maximum
At resonance, energy is transferred from the driver to the system most efficiently
Free Oscillations
A free oscillation is the “flick it and walk away” kind. Displace the system, let go, and it does its own thing: a struck tuning fork ringing, a plucked guitar string, a pendulum after a single nudge.
What makes it free? Two things, and examiners want both:
Only internal forces act — the restoring force inside the system does all the work, with no external force interfering
No energy is put in — there is no transfer of energy to or from the surroundings while it oscillates
Strictly speaking this only ever happens in a vacuum, but anything vibrating in air still counts as free as long as no external force is driving it. And here’s the crucial property: a free oscillation always runs at the system’s own natural frequency, f0 — the frequency of the oscillation when the system is allowed to oscillate freely. Every oscillating system has one.
Forced Oscillations
In the real world, friction and air resistance (that’s damping — next page’s topic) constantly steal energy from an oscillator, so left alone it dies away. To keep it going, something has to top the energy back up: a periodic external force, often called the driving force. It does work against the resistive forces, cycle after cycle.
Oscillations kept alive this way are forced oscillations: oscillations produced by a periodic external force. And the driver is the boss — a forced system vibrates at the driving frequency, whatever that happens to be, not at its own natural frequency.
driver pushes at frequency f
forces the system to follow
system oscillates at frequency f
amplitude depends on how close f is to f0
biggest when f = f0
Resonance: When the Frequencies Match
Now bring the two ideas together. The driver pushes at frequency f; the system would like to oscillate at f0. As the driving frequency gets closer to the natural frequency, each push lands more and more in time with the motion, so the system gains more energy from the driver every cycle — and the amplitude climbs.
When the two frequencies are exactly equal, every push arrives at the perfect moment. Energy is transferred from the driver to the oscillating system most efficiently, and the amplitude reaches its maximum. That is resonance:
When the frequency of the applied force on an oscillating system equals the system’s natural frequency, the amplitude of the resulting oscillations is at its maximum.
The resonance curve: amplitude of the forced oscillations against driving frequency. The amplitude climbs as f approaches f₀, peaks sharply at f = f₀ (resonance), then drops as f moves past it.
Resonance You’ve Already Done: the Swing
Pushing a child on a swing is the textbook example, and it maps perfectly onto the vocabulary:
The swing-plus-child system has a fixed natural frequency — its own back-and-forth rhythm
Each small push is the driving force, and how often you push is the driving frequency
Push once per cycle, exactly in time with the swing, and the driving frequency equals the natural frequency — resonance, so the amplitude grows and grows and the child swings highest
Push slightly too often or too rarely and the amplitude still increases, but to a lesser extent — some pushes land at the wrong moment
The same physics is everywhere once you look: a singer shattering a wine glass by holding exactly the glass’s natural frequency, parts of a car buzzing at one particular engine speed, and — connecting back to the last page — a musician blowing across a pipe, forcing the air column until it locks onto one of its natural frequencies and sings.
🧭 Answering “free or forced?” questions
Hunt for a driver — is a periodic external force feeding energy in?
No driver → free: say that only internal forces act, there is no energy input, and the system oscillates at its natural frequency
Driver found → forced: name the external periodic force and state that the system is made to vibrate at the driving frequency
Mention amplitude and energy — if the driving frequency matches the natural frequency, add that resonance occurs and the amplitude is maximum
Quick recap: free oscillation = internal forces only, runs at f0. Forced oscillation = driven by an external periodic force, runs at the driving frequency. Resonance = driving frequency equals f0, giving maximum amplitude and the most efficient energy transfer.
WE 1
State and explain whether each of the following is a free or a forced oscillation:
(a) A guitar string plucked once and left to ring.
(b) A phone vibrating on a desk, making the desktop hum.
(c) A wine glass vibrating violently while a singer holds a steady, loud note.
Part (a)
Free — after the pluck, only internal forces act on the string and no energy is put in; it vibrates at its natural frequency
free oscillationPart (b)
Forced — the phone applies a periodic external force to the desk, which is made to vibrate at the phone’s driving frequency
forced oscillationPart (c)
Forced — the sound wave is a periodic external driving force acting on the glass; the singer’s note is at the glass’s natural frequency, so resonance gives a very large amplitude
forced oscillation (with resonance)
WE 2
A child on a swing has a natural frequency of 0.50 Hz. A parent gives the swing one gentle push every 2.0 s. Explain why the amplitude of the swing becomes very large.
Find the driving frequency
One push every 2.0 s means the driving frequency is
f = 1/T = 1 ÷ 2.0 = 0.50 HzCompare with the natural frequency
The driving frequency equals the natural frequency (0.50 Hz), so resonance occurs
Explain the energy transfer
Every push arrives in time with the swing’s motion, so energy is transferred from the parent to the swing most efficiently and the amplitude builds up
resonance → maximum amplitude
💡 Top tips
The three magic phrases for resonance marks: driving frequency = natural frequency, maximum amplitude, and energy transferred most efficiently
Never define a free oscillation as just “not forced” — mark schemes want internal vs external forces and energy transfer mentioned, plus a comment on amplitude where relevant
Remember who sets what: the driver sets the frequency of a forced oscillation; the closeness of f to f0 sets the amplitude
Musical instruments are resonance machines — the reed or your breath is the driver, and the string or air column responds hugely at its natural frequencies (the harmonics)
⚠ Common mistakes
Saying a forced system vibrates at its natural frequency — it vibrates at the driving frequency
Treating resonance as just “a loud noise” — it’s a precise condition: f = f0
Claiming nothing happens unless the frequencies match exactly — near f0 the amplitude still grows, just to a lesser extent
Thinking “free” means no forces at all — the internal restoring force is always there; it’s external forces and energy input that are absent
Up next: damping — the resistive forces that shrink oscillations, the three flavours (light, critical, heavy), and what they do to the shape of the resonance curve.
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