• Every oscillating system has its own natural frequency (f₀) — the rate it swings/vibrates at when left alone.
• Resonance happens when an external driving frequency (f) matches f₀ exactly — the amplitude shoots up to a maximum.
• In real life, damping (resistive forces like friction/air resistance) always removes energy, so oscillations die down over time — but the frequency stays constant even as amplitude shrinks.
• There are three types of damping: light (slow exponential decay, still oscillates), critical (fastest return to rest, no oscillation), and heavy (slow return to rest, no oscillation).
• Free oscillations = no external force, system just vibrates at f₀. Forced oscillations = a periodic external force keeps replacing lost energy, and the system is forced to move at the driver's frequency.
• A resonance curve (amplitude vs driving frequency) peaks sharply at f = f₀. More damping → the peak gets lower, wider, and shifts slightly left.
• Damping can also occur through plastic deformation of ductile materials — the material absorbs the oscillation's kinetic energy by permanently stretching (e.g. a climbing rope).

Picture pushing a friend on a swing. The swing has its own natural rhythm — say, it swings back and forth once every 2 seconds no matter how hard you push it (amplitude doesn't change that rhythm, only how high it goes). That built-in rhythm is called the natural frequency, symbol f₀.

Now imagine you give it a tiny push every single time it swings back towards you — timed exactly with that 2-second rhythm. Each push adds a bit more energy at exactly the right moment, so the swing goes higher and higher. The frequency at which you are pushing is called the driving frequency, symbol f.

When your pushing frequency (f) becomes exactly equal to the swing's natural frequency (f₀), something dramatic happens: the amplitude increases hugely compared to any other push rate. This maximum-amplitude condition is resonance.

Why does this happen physically? At resonance, the driving force and the oscillator's velocity are always in phase — the push always happens exactly when the object is moving in the same direction as the push. This means energy is transferred from the driver to the oscillating system most efficiently possible. If the driving frequency doesn't quite match, some pushes end up fighting against the motion instead of helping it, so energy transfer is less efficient and the amplitude increase is smaller.

Real examples of resonance

• An organ pipe — air resonates down the column, setting up a stationary (standing) wave.
• Glass shattering from a high-pitched sound at just the right frequency.
• A radio tuned so its electrical circuit resonates at the same frequency as a specific broadcast signal.
• A child on a swing being pushed at the natural rhythm.

This is a classic "graphical method" experiment. The goal: find the mass of an unknown object without weighing it directly — by using how a spring-mass system oscillates.

The setup

• Hang a spring from a clamp stand, attach known masses (100 g, 200 g... up to 500 g) one at a time.
• For each mass: pull the spring down to a fiducial mark, release, and time 10 oscillations (repeated 3 times for accuracy) to find the average period T.
• Plot a graph of T² (y-axis) against m (x-axis) — this gives a straight line through the origin.

Why T² against m? (The derivation)

This comes from three SHM relationships combined together:

Finding the unknown mass

Once the calibration line is plotted from the known masses, measure T for the unknown mass the same way, square it to get T², then read across the graph horizontally to the line, and drop straight down to the x-axis — that x-value is the unknown mass. You then check it against digital scales to verify.

In the real world, no oscillator swings forever. Resistive forces — friction, air resistance — constantly steal a little energy from the system on every cycle. This gradual loss of energy (and therefore amplitude) is called damping.

Key fact that trips people up: damping reduces the amplitude of the oscillations, but it does not change the frequency. A child on a swing that's slowing down still swings back and forth in the same amount of time each cycle — it just doesn't go as high. The peaks and troughs on a displacement-time graph stay evenly spaced in time even as they shrink in height.

The Three Types of Damping

Free Oscillations

A free oscillation happens when a system is displaced and then simply left alone — no external force keeps pushing it, no energy is fed in from outside. Strictly speaking this only truly happens in a vacuum (since air resistance is always an external resistive force), but in physics problems, "vibrating freely in air" still counts as a free oscillation as long as nothing is driving it periodically.

Crucial fact: a free vibration always oscillates at its own natural frequency, f₀. There's no external rhythm dictating it — it just does its own thing.

Forced Oscillations

Left alone, a real oscillator's amplitude dies away due to damping. To keep it going — to sustain the oscillation — you need to keep feeding energy back in to replace what's lost. This is done with a periodic external driving force. The resulting motion is a forced oscillation.

Here's the important bit: a forced oscillation always vibrates at the same frequency as the driver — not necessarily at f₀. Only when the driving frequency happens to equal f₀ do you get resonance (maximum amplitude). Otherwise, the system is forced along at whatever rate the driver dictates, usually with a smaller amplitude.

If you plot the amplitude (A) of the resulting oscillations against the driving frequency (f), you get a resonance curve. This single graph tells you almost everything about how a system responds to being driven at different rates.

• When f < f₀: the amplitude of oscillations is increasing as f approaches f₀.
• At the peak, where f = f₀: amplitude is at its absolute maximum — this is resonance.
• When f > f₀: the amplitude starts falling again as the driving frequency moves away from f₀.

Damping & Resonance — How the Curve Changes

Damping always removes energy, so it makes sense that adding more damping to a system reduces how "extreme" its resonance response is. As damping increases:

• The amplitude of resonance vibrations decreases — the peak of the curve gets lower.
• The resonance peak broadens — the curve becomes flatter and wider, meaning a bigger range of driving frequencies still produce a reasonably large amplitude (resonance becomes "less sharp").
• The peak shifts slightly to the left of f₀ (i.e. the frequency at which maximum amplitude occurs becomes very slightly less than f₀) — but only noticeably so under heavy damping.
• Note: the actual natural frequency f₀ of the oscillator itself never changes — damping doesn't alter f₀, it only alters how the amplitude responds around it.

So far we've talked about damping from resistive forces like friction and air resistance. But there's another important way energy can be removed from an oscillating system: through plastic deformation of a material.

Damping and amplitude are inversely proportional — more damping means less amplitude, always.

Ductile vs Brittle materials

A ductile material (like copper, gold, or silver) can be stretched a long way before it snaps — it undergoes a large amount of plastic deformation (permanent stretching) before it finally breaks. A brittle material snaps suddenly with very little permanent stretching beforehand.

When a ductile material is used as part of an oscillating system, the kinetic energy of the oscillator gets transferred into permanently deforming the material — rather than the energy simply being lost as heat via friction. This still counts as damping, because it reduces the amplitude of the oscillations, but the mechanism is different: the energy goes into stretching/deforming the material's structure.

• I can define natural frequency and driving frequency correctly.
• I can state the condition for resonance and explain why amplitude is maximum there (in terms of energy transfer).
• I can give at least two real-world examples of resonance.
• I understand and can describe Core Practical 16, including why T² is plotted against m and how the gradient relates to k.
• I can define damping and explain that it reduces amplitude but not frequency.
• I can describe and sketch displacement-time graphs for light, critical, and heavy damping.
• I can choose the correct type of damping for a given real-world scenario (e.g. weighing scales, car suspension, door closers).
• I can distinguish free oscillations from forced oscillations, with correct definitions.
• I know that free oscillations occur at f₀, and forced oscillations occur at the driving frequency f.
• I can sketch a resonance curve (amplitude vs driving frequency) and label the peak at f₀.
• I can describe all three effects of increasing damping on a resonance curve (lower, broader, shifted left).
• I understand how plastic deformation of ductile materials can act as a damping mechanism.
• I can explain the climbing rope example in terms of energy, force, and damping.
• I never confuse resistive force with restoring force.

This phrase alone won't get you marks. Examiners want a proper reference to internal vs external forces and energy input — always mention that a free oscillation has no external driving force and no energy input, while a forced oscillation has both.

These are opposite jobs. Resistive force = opposes motion, causes damping (e.g. friction, air resistance). Restoring force = brings the system back towards equilibrium (e.g. spring tension, gravity component on a pendulum). Mixing these up is one of the most common ways to lose marks in this topic.

It doesn't! Damping only affects amplitude. The time period (and therefore frequency) of a lightly damped oscillator stays constant even as its amplitude shrinks. Don't describe a damped displacement-time graph as having "increasing time period" — the peaks stay evenly spaced.

When asked how increased damping affects a resonance curve, you need all three: (1) amplitude/peak height decreases, (2) the curve broadens, (3) the peak shifts slightly to a frequency below f₀. Examiners often award one mark per correct feature mentioned.

Remember it's T² against m, not T against m — this is what makes the graph a straight line through the origin. If you (or a question) plots T against m instead, you'd get a curve, not a straight line, and the gradient method wouldn't work directly.

Don't just say "because the frequencies match" — examiners want the mechanism: at resonance, the driving force does work on the system in phase with its motion, so energy is transferred from the driver to the oscillator most efficiently, meaning maximum energy (and therefore maximum amplitude) builds up.