HookTwo days after opening, London's newest bridge was closed by people walking
The Millennium Bridge opened on 10 June 2000 — the first new crossing of the Thames in central London for a century — and something like 90,000 people walked it that first day, up to 2,000 at a time. Under them, the deck began to sway sideways, tens of millimetres at well under one hertz. Then the feedback kicked in: to keep balance on a moving deck, walkers unconsciously widened their stance and timed their steps to the sway, which meant thousands of feet pushing sideways in sync, at the deck's own natural frequency. The more it moved, the better-timed the pushing became. Two days later the bridge was closed. The fix took until February 2002 and about £5 million — not stiffening, but damping: 37 viscous dampers and more than 50 tuned mass dampers, engineered specifically to drain energy out of the oscillation faster than pedestrians could feed it in. The wobble has never returned.
That is section 3.6's first half in one story: circular motion supplies the mathematics of \(\omega\), simple harmonic motion turns it into oscillations, and forced vibrations, resonance and damping decide whether a driven system rings or settles. The second half changes scale rather than subject — internal energy, specific and latent heat, the gas laws and the kinetic theory model are what you get when the oscillating, colliding objects number \(10^{23}\) and the only sensible questions are statistical. Two required practicals (SHM with a pendulum and mass-spring; the gas laws) anchor it.
ModelCircular motion — accelerating without changing speed
Measure angles in radians (one radian when arc length equals radius; \(2\pi\) in a full turn) and circular motion becomes bookkeeping. Angular speed is angle swept per second: \(\omega=v/r=2\pi f\), in rad s\(^{-1}\). An object circling at constant speed is nevertheless accelerating, because its velocity — a vector — is continuously changing direction. That acceleration points at the centre, with magnitude \(a=v^2/r=\omega^2 r\), and Newton's second law then demands a resultant force towards the centre: \(F=mv^2/r\).
The crucial discipline: centripetal force is not a new force. It is the job description filled by real forces — friction for a cornering car, tension for a hammer thrower, gravity for the Moon, the normal force from a banked track. On a free-body diagram you draw the real forces only, then set their resultant equal to \(mv^2/r\) pointing at the centre. Draw an extra 'centripetal' arrow and the diagram is wrong.
A washing machine spins at 1,400 rpm with a drum radius of 0.25 m. Frequency: \(f=1400/60=23.3\ \text{Hz}\), so \(\omega=2\pi f=147\ \text{rad s}^{-1}\). Centripetal acceleration: \(a=\omega^2 r=147^2\times 0.25\approx 5{,}400\ \text{m s}^{-2}\) — about 550g. A wet 0.5 kg towel therefore needs an inward force of \(F=ma\approx 2.7\ \text{kN}\), which the drum wall supplies. At the perforations there is no wall, so the water gets no inward force and simply carries on in a straight line, leaving through the hole. Spin-drying is Newton's first law with a motor attached — nothing 'flings' the water out.
ModelSimple harmonic motion — the acceleration that opposes displacement
An oscillation is simple harmonic when the acceleration is proportional to the displacement from equilibrium and directed back towards it. Both clauses are compulsory, and the whole definition compresses into one line: \[a=-\omega^2 x\] The minus sign carries the second clause. From it flow the standard solutions: \(x=A\cos(\omega t)\) for a start at maximum displacement, speed \(v=\pm\omega\sqrt{A^2-x^2}\), maximum speed \(v_{max}=\omega A\) as the oscillator crosses the centre, and maximum acceleration \(a_{max}=\omega^2 A\) at the extremes — where, note, the speed is momentarily zero. Speed and acceleration peak a quarter-cycle apart.
Graphs against time make the phase relationships visible: displacement a cosine, velocity a negative sine (a quarter-cycle ahead), acceleration an inverted cosine — displacement's mirror image, which is the equation \(a=-\omega^2x\) drawn rather than written. Sketching all three, aligned vertically, is a routine exam ask; get the quarter-cycle offsets right and the marks follow.
ModelTwo standard oscillators and the energy see-saw
AQA's two workhorse systems each earn their period formula. A mass on a spring obeys \(T=2\pi\sqrt{m/k}\): stiffer spring, faster; more mass, slower; and — the detail examiners probe — independent of both amplitude and gravitational field strength, so it keeps identical time in orbit. A simple pendulum obeys \(T=2\pi\sqrt{l/g}\) for small swings (the small-angle approximation is the standard 'limitation' mark): longer, slower; and independent of the bob's mass, for the same reason all masses free-fall alike — the restoring force and the inertia both scale with m and cancel.
Energy in undamped SHM is a see-saw with a constant total \(\tfrac{1}{2}kA^2\): all potential at the extremes, all kinetic at the centre, trading twice per cycle. Plotted against displacement, kinetic and potential energy are two parabolas — one inverted — summing to a flat line. Damping drags that line downward cycle by cycle as the oscillator does work against resistive forces; the amplitude decays and the energy leaves as heat, which is precisely the effect the Millennium Bridge engineers purchased 37 viscous dampers to maximise.
DataRequired practical 7 — SHM with a pendulum and a mass-spring system
Both halves share one measurement strategy: never time one oscillation. Time 20 or more and divide, so a human reaction-time uncertainty of about 0.2 s is shared across the count and shrinks to around 0.01 s per period. Place a fiducial marker at the equilibrium position and count transits there, because that is where the oscillator moves fastest and the timing judgement is sharpest — timing at the extremes, where it lingers, is the classic error. Count from zero, not one.
For the pendulum, vary length l (measured to the centre of mass of the bob) and keep the amplitude small; plot \(T^2\) against \(l\). Linearising matters: \(T=2\pi\sqrt{l/g}\) squares to \(T^2=(4\pi^2/g)\,l\), a straight line through the origin whose gradient gives \(g=4\pi^2/\text{gradient}\). For the mass-spring system, vary the suspended mass and plot \(T^2\) against \(m\); the gradient is \(4\pi^2/k\). In both, the largest uncertainties are timing (attacked by counting many oscillations and repeating) and, for the pendulum, the length measurement to the bob's centre — a ±2 mm judgement inside a 300 mm length is smaller than it looks in percentage terms, which is exactly the kind of comparison Paper 3 asks you to make.
CaseForced vibrations, resonance and damping — the bridge, properly explained
Left alone after a push, a system oscillates freely at its natural frequency. Driven by a periodic force, it is forced to oscillate at the driver's frequency — and the amplitude of its response depends on how the two frequencies compare. Drive far below the natural frequency and the system placidly follows the driver; far above, and it barely responds; drive at the natural frequency and energy transfer from driver to oscillator is at its most efficient and the amplitude grows towards whatever limit damping sets. That peak response is resonance.
Damping reshapes the whole picture: more damping makes the resonance peak lower and broader and shifts it slightly below the natural frequency. Engineers pick the damping level to suit: critical damping returns a displaced system to equilibrium in the shortest time without oscillating — car suspension, analogue meter needles — while lighter damping is tolerated where some ring-down is acceptable. The Millennium Bridge is the full syllabus in one structure: pedestrians were a periodic driver near the deck's lateral natural frequency; synchronisation locked the driving to the sway (each cycle of larger amplitude recruited better-timed footsteps, feeding energy in faster); and the cure was not to raise the natural frequency by stiffening but to add viscous and tuned mass dampers, cutting the peak of the resonance curve below anything feet can excite. The same physics runs constructive uses — a radio circuit tuned to resonate at one station's frequency, the resonant cavities of musical instruments, MRI — and destructive ones, like unbalanced machinery shaking itself loose.
ModelInternal energy, specific heat and latent heat
The internal energy of a substance is the sum of the randomly distributed kinetic and potential energies of its molecules — kinetic in their motion, potential in the intermolecular bonds. Heat the substance or do work on it and internal energy rises; the split between raising temperature and changing state is the whole content of two equations.
While temperature changes, \(Q=mc\Delta\theta\): the specific heat capacity c is the energy to warm 1 kg by 1 K (for water, a hefty 4,200 J kg\(^{-1}\) K\(^{-1}\), which is why oceans moderate climates and why a kettle works hard). The value can be measured electrically, or by a continuous-flow method — liquid streaming past a heater at a known rate, with the advantage that heat losses cancel when you run twice at different flow rates. During a change of state, \(Q=ml\): the specific latent heat l is the energy per kilogram to melt or vaporise with no temperature change at all. The added energy raises molecular potential energy — prising molecules apart against their attractions — not their kinetic energy, so the thermometer sits still while the substance transforms. That plateau on every heating curve is the single most-asked feature in this half of the section.
A 3 kW kettle holds 0.25 kg of water at 18 °C. Energy to reach 100 °C: \(Q=mc\Delta\theta=0.25\times 4200\times 82\approx 8.6\times 10^4\ \text{J}\), which at 3,000 J s\(^{-1}\) takes about 29 s. Now leave it running until the water has all boiled away: \(Q=ml=0.25\times 2.26\times 10^6\approx 5.7\times 10^5\ \text{J}\) — six and a half times the energy that did all the heating, and over three further minutes at full power, delivered without the temperature moving a single degree past 100 °C. Vaporisation is expensive; that is also why sweating cools you and why steam scalds so much worse than boiling water.
ModelIdeal gases — three old laws, one equation, one absolute scale
Three experimental laws describe a fixed mass of gas: at constant temperature, \(pV\) is constant (Boyle); at constant pressure, volume is proportional to absolute temperature (Charles); at constant volume, pressure is proportional to absolute temperature. Cool an ideal gas and both pressure and volume head for zero at the same extrapolated point, −273.15 °C — absolute zero, the anchor of the Kelvin scale. Every gas calculation runs in kelvin: add 273 to the Celsius value, always, because 20 °C is 293 K and using 20 wrecks every ratio.
The three laws fuse into the equation of state for n moles: \(pV=nRT\) with \(R=8.31\ \text{J mol}^{-1}\text{K}^{-1}\), or per molecule, \(pV=NkT\) with the Boltzmann constant \(k=1.38\times 10^{-23}\ \text{J K}^{-1}\) — the two constants related by \(k=R/N_A\). 'Ideal' is a defined fiction — a gas that obeys this equation exactly — but real gases at modest pressure and well above their boiling points sit close enough that the fiction earns its keep, from tyre pressures to weather balloons.
DataRequired practical 8 — Boyle's law and Charles's law
Boyle: a column of air is trapped in a vertical glass tube by oil; a pump raises the pressure, read from a gauge, and the column's length gives the volume (uniform bore, so length stands in for volume). The discipline is slowness: compressing a gas warms it, and Boyle's law holds only at constant temperature, so change the pressure in small steps and wait after each for the trapped air to return to room temperature — and for the oil film to finish draining down the tube wall before you read the length. Plot p against 1/V: a straight line through the origin is the law's signature; a curve through the raw p-V data proves nothing by eye.
Charles: a shorter air column is sealed in a capillary tube by a bead of liquid, immersed in a water bath. Heat gradually, stir thoroughly, and record column length against temperature. Length against θ (in °C) gives a straight line which, extrapolated backwards to zero length, crosses the temperature axis in the region of −273 °C — a school bench's own estimate of absolute zero, and a rare chance to extrapolate honestly: the line is real between roughly 10 °C and 90 °C, and the extrapolation is an inference, which is exactly the language to use when a question asks how confident the estimate is. Main uncertainties in both: thermal equilibrium (stir, wait), parallax when reading column lengths against a scale, and gauge calibration.
ModelKinetic theory — pressure and temperature from first principles
The gas laws are empirical — summaries of measurements. Kinetic theory is a model: assume molecules are numerous, identical, in constant random motion, negligibly small compared with their separation, colliding elastically, quickly, and exerting no forces between collisions — then derive the laws. AQA expects the derivation itself, and it runs in five moves. One molecule of mass m travelling at \(v_x\) hits a wall and rebounds: momentum change \(2mv_x\). It returns after crossing the box and back, time \(2L/v_x\), so the average force it exerts is \(mv_x^2/L\). Sum over N molecules using the mean square speed; share motion equally among three perpendicular directions, so \(\overline{v_x^2}=\overline{c^2}/3\); divide by wall area. Out drops \[pV=\tfrac{1}{3}Nm\overline{c^2}\] Set that against the empirical \(pV=NkT\) and the two match only if \(\tfrac{1}{2}m\overline{c^2}=\tfrac{3}{2}kT\). That line is the payoff of the whole section: temperature is a measure of mean molecular kinetic energy — the same for every gas at the same temperature, regardless of mass. Brownian motion, the jittering of smoke grains under bombardment, is the observational evidence that the colliding molecules of the model are really there.
How fast is the air in this room? For nitrogen at 20 °C (293 K), the molecular mass is \(m=28\times 1.66\times 10^{-27}=4.65\times 10^{-26}\ \text{kg}\). From \(\tfrac{1}{2}m\overline{c^2}=\tfrac{3}{2}kT\), the root mean square speed is \[c_{rms}=\sqrt{\frac{3kT}{m}}=\sqrt{\frac{3\times 1.38\times 10^{-23}\times 293}{4.65\times 10^{-26}}}\approx 510\ \text{m s}^{-1}\] — roughly twice the cruising speed of an airliner. Run the same numbers for helium (\(m=6.64\times 10^{-27}\ \text{kg}\)) and \(c_{rms}\approx 1{,}350\ \text{m s}^{-1}\): same temperature, same mean kinetic energy of \(\tfrac{3}{2}kT=6.1\times 10^{-21}\ \text{J}\), so the lighter molecule must move faster — part of why helium leaks from balloons and, over geological time, from the atmosphere itself.
VocabularyKey terms the mark scheme pays for
TrapsMisconceptions that cost marks
ExamWhat examiners want
Two mechanical errors cost more marks in 3.6 than any physics: a calculator left in degrees when ω demands radians, and a gas temperature left in Celsius. Convert to kelvin before touching pV = nRT — 20 °C is 293 K, and examiners' reports flag this slip every single year. Definitions are marked clause-by-clause: SHM needs both 'proportional to displacement' and 'directed towards equilibrium' (or the minus sign in a = −ω²x doing that job), and resonance needs driving frequency equal to natural frequency plus maximum amplitude or maximum rate of energy transfer.
The kinetic theory derivation of \(pV=\tfrac{1}{3}Nm\overline{c^2}\) is one of the few pieces of extended bookwork the specification expects you to reproduce, and it is marked line by line: momentum change at the wall, time between wall collisions, force on the wall, the averaging over N molecules, and the factor of three from resolving motion into three directions. Learn it as five numbered moves and it is the most predictable six marks on the paper. Listing the model's assumptions is quick AO1 credit — but questions increasingly ask which assumption fails in a real gas (molecular volume and intermolecular forces, at high pressure and low temperature), which is the AO3 twist.
On the 6-mark levels-of-response staples — explain resonance and how damping controls it, or interpret a heating curve — build the causal chain in order rather than scattering facts: driver frequency, natural frequency, energy transfer per cycle, amplitude, then what damping changes. For the required practicals, Paper 3 Section A rewards the specific techniques by name: timing 20 oscillations from a fiducial marker at the centre of the swing, plotting T² (not T) to linearise, waiting for thermal equilibrium and stirring in the gas experiments, and describing the absolute-zero extrapolation as an estimate whose reliability rests on the linearity of the measured region. AQA weights application over recall — above 40% of marks are AO2 — so every formula in this section should arrive attached to a situation.