Push a child on a swing at random moments and mostly you just annoy them. Push once per swing, at exactly the right instant, and with tiny nudges they end up soaring. You have not pushed harder — you have pushed in time. Match the rhythm of a system to the rhythm it already wants, and energy pours in. That is resonance, and it tunes radios, shatters wine glasses and occasionally brings down bridges.
📘 What you need to know
A free oscillation has only internal forces acting and no energy input
A free oscillation happens at the system’s natural frequencyf0
A forced oscillation is produced by a periodic external force, and happens at the driving frequencyf
Resonance occurs when f = f0, and the amplitude is then a maximum
At resonance, energy is transferred from the driver to the oscillator most efficiently
An amplitude–driving frequency graph (a resonance curve) peaks at f0
Free oscillations
Pull a mass on a spring down and let go. Nobody is doing anything to it any more. It swings at the one frequency it likes best.
Only internal forces act — no external driving force
No energy is put in and none is taken out by the surroundings
It oscillates at its natural frequencyf0, which is fixed by the system itself (its mass, stiffness, length, tension…)
Forced oscillations
Now grab the top of the spring and shake it up and down, over and over. That is an external periodic driving force, and the mass has no choice: it settles into oscillating at your frequency, not its own.
The driven mass copies the driver’s frequency, whatever that frequency is. What it does not copy is the amplitude — that depends on how close f is to f0.
Here’s the sentence that gets the mark: a forced oscillation happens at the driving frequency, not the natural one. Students lose marks writing “it vibrates at its natural frequency because it’s being pushed”. No — the driver is in charge of frequency. The natural frequency only decides how big the response is.
What resonance actually is
Slowly turn up the driving frequency and watch the amplitude of the driven mass.
When f is well belowf0, the mass just follows the driver lazily. Small amplitude
As fapproachesf0, each push arrives just as the mass is ready for it. Energy accumulates and the amplitude climbs
At f = f0 the pushes are perfectly timed. Energy transfer is most efficient and the amplitude is at its maximum. This is resonance
Beyond f0 the driver starts pushing while the mass is still coming back. The pushes fight the motion, and the amplitude falls
The condition for resonancedriving frequency f = natural frequency f0→ maximum amplitude, maximum energy transfer
Every oscillating system has a curve like this. The sharper the peak, the fussier the system is about being driven at exactly the right frequency.
Why the timing matters so much
A push only feeds energy in if it acts in the direction the system is already moving. At resonance every single push does that, so energy builds cycle after cycle and the amplitude grows to a large steady value. Off resonance, the driver spends half its time pushing against the motion and takes energy back out again.
Identical pushes both times. The only difference is when they arrive — and that is worth an enormous difference in amplitude.
Driver at f = f0
every push helps
Energy transfer most efficient
so
Maximum amplitude
The swing, properly: the swing has a fixed natural frequency set by its length. Your pushes are the driving force. Push once per cycle and the driving frequency equals the natural frequency — resonance — and the child goes highest. Push slightly too fast or too slow and they still swing, just far less.
🔎 Free or forced? Ask three questions
Is there a periodic external force? If yes, it is forced. If no, it is free.
Is energy being supplied? A free oscillation gets none once it has been released.
What frequency does it settle at? Free → f0. Forced → the driving frequency f.
Is it resonating? Only if the driving frequency has been matched to f0.
WE 1
State and explain whether each of the following is a free or a forced oscillation. (a) A guitar string is plucked once and left to ring. (b) A singer holds a note and a nearby wine glass shatters. (c) A rear-view mirror buzzes only when the engine idles at one particular speed.
(a) plucked guitar stringFree. Once released there is no external periodic force and no energy input, so it rings at its own natural frequency.
(b) singer and wine glassForced. The sound wave is a periodic external force driving the glass. It shatters because the singer’s frequency matches the glass’s natural frequency — resonance.
(c) buzzing mirrorForced. The engine’s vibration drives the mirror. It buzzes at just one engine speed because only then does the driving frequency equal the mirror’s natural frequency.
Mark schemes want internal / external forces and energy input. Avoid writing “a free oscillation is not forced” — it says nothing.
WE 2
A washing machine drum spins and its panel vibrates violently, but only when the drum turns at 630 revolutions per minute. Determine the natural frequency of the panel, and explain your reasoning.
Step 1 — convert the driving rate into a frequencyf = 630 / 60f = 10.5 HzStep 2 — interpret “violently”
The amplitude is a maximum, so the panel is at resonance.
Step 3 — apply the resonance condition
At resonance f = f₀, so
f₀ = 10.5 HzThe huge amplitude at one special speed is the fingerprint of resonance. Away from 630 rpm the panel is still forced — just barely moving.
WE 3
A string of length 0.50 m is fixed at both ends, and waves travel along it at 150 m s⁻¹. A loudspeaker beside it plays a pure tone. State two frequencies at which the string will resonate, and explain why it does not resonate at 200 Hz.
Step 1 — the string’s natural frequencies are its harmonicsf₁ = v / 2L = 150 / (2 × 0.50)f₁ = 150 HzStep 2 — harmonics are whole multiples of f₁f₂ = 300 Hz (also 450 Hz, 600 Hz…)
150 Hz and 300 HzStep 3 — why not 200 Hz?
200 Hz is not a whole multiple of 150 Hz, so no standing wave fits the fixed ends. The driving frequency matches no natural frequency, so there is no resonance.
A string doesn’t have one natural frequency — it has a whole set of them. Resonance happens at every single one.
💡 Top tips
Define a free oscillation using internal forces only and no energy input — that is what the mark scheme wants.
A forced system oscillates at the driving frequency. Always.
Resonance is f = f0, and the payoff is maximum amplitude and most efficient energy transfer. Say both.
“It vibrates hardest at one particular speed” is exam code for resonance.
A system with many natural frequencies (a string, a pipe) resonates at every harmonic.
⚠ Common mistakes
Saying a forced oscillator vibrates at its natural frequency — it vibrates at the driving frequency
Thinking resonance means a bigger push. The push is the same size; it is just better timed
Defining a free oscillation as “one that isn’t forced” — no marks
Confusing amplitude with frequency: at resonance the amplitude is maximum, not the frequency
Assuming resonance always breaks things. Most of the time it is useful: radios, MRI, musical instruments
Forgetting that a real oscillator loses energy, so the amplitude at resonance settles rather than growing forever
Quick recap: A free oscillation has only internal forces and no energy input, so it swings at its natural frequencyf0. A forced oscillation is driven by a periodic external force and swings at the driving frequencyf. When f = f0 you get resonance: energy transfers most efficiently and the amplitude peaks. Plot amplitude against driving frequency and you get a curve with its maximum at f0.
One loose end. In that resonance curve, why doesn’t the peak shoot up to infinity? Because something always drains energy away — friction, air resistance, a resistive force of some kind. That drain is called damping, it flattens the peak and broadens it, and it is the last page of this sub-section.
Resonance not ringing true?
Book a free meeting and we’ll work through free versus forced oscillations, resonance curves and past-paper questions together.
