A worked context from start to finish

One context, written at all three grades

This page works one New Zealand context all the way through, so you can see the same material written at each grade. The context is base isolation in earthquake-resistant buildings — the technology under Te Papa Tongarewa, Parliament House and Wellington Hospital. Your own assessment will use whatever context your school sets; copy the method, not the content.

Step 1 — Plan the Level 8 physics first

Feature of the context	Level 8 physics idea	Quantitative?
A building sways back and forth after a jolt	Simple harmonic motion — restoring force proportional to displacement	Yes — $T = 2\pi\sqrt{m/k}$
Some buildings shake violently in one quake and barely move in another	Resonance — driving frequency matching natural frequency	Yes — compare $f_0$ with quake frequencies
Rubber-and-lead bearings under the foundations	Natural frequency depends on stiffness: lowering $k$ lowers $f_0$	Yes
The lead core in each bearing	Damping — energy removed from the oscillation	Qualitative
The building must still not move much in wind	A design trade-off between period and stiffness	Qualitative

Five features, four Level 8 ideas, three quantitative. That is enough for Excellence before anything is written.

Step 2 — Achieved: relate the physics to the context

"A tall building behaves like an oscillating system undergoing approximately simple harmonic motion: when it is displaced sideways, the structure provides a restoring force roughly proportional to the displacement, so it oscillates about its rest position with a natural frequency $f_0$. Every building has such a natural frequency, determined by its mass and its stiffness, related by $T = 2\pi\sqrt{m/k}$. Base isolators are flexible bearings placed between the building and its foundations. Damping removes energy from an oscillation, reducing the amplitude over time."

Named Level 8 ideas, each related to a part of the context: Achieved.

Step 3 — Merit: explain how or why

"The damage a building suffers depends on resonance. An earthquake drives the ground back and forth over a spread of frequencies, and if the dominant driving frequency is close to the building's natural frequency $f_0$, energy is transferred efficiently into the oscillation on every cycle and the amplitude builds up to a large value. If the two frequencies are far apart, most cycles push the building out of step with its own motion and little energy accumulates.

Base isolators work by moving $f_0$ away from the frequencies an earthquake contains. Since $f_0 = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}$, and the mass of the building cannot be reduced, the only available option is to reduce the effective stiffness $k$ of the connection to the ground. The rubber bearings do exactly that: they are far more flexible horizontally than a concrete foundation, so the whole building-on-bearings system has a much smaller $k$ and therefore a much longer natural period — typically 2 to 3 seconds instead of the 0.5 seconds of a fixed-base building of the same height. Most of the energy in a shallow earthquake arrives at periods well under 1 second, so the isolated building is driven far from resonance."

That explains how and why the physics produces the effect: Merit.

Step 4 — Excellence: link the ideas into a coherent picture

"The three ideas are not separate applications but one design problem with one free variable.

The resonance condition sets the goal: keep the building's natural frequency away from the frequencies the ground supplies. The SHM relationship $T = 2\pi\sqrt{m/k}$ says how that can be achieved, and immediately rules out two of the three quantities — the mass is set by the building itself, and the driving frequencies are set by the geology of the fault. Only the stiffness can be chosen, which is why base isolation exists at all and why it takes the form of a deliberately floppy layer rather than a stronger structure. The counter-intuitive design — making the connection to the ground weaker — follows directly from the algebra.

But lowering $k$ alone creates a second problem, and this is where damping becomes necessary rather than optional. A system with a low stiffness and very little damping oscillates for a long time once started, and its displacement amplitude can be large — up to several hundred millimetres at the base. That is unacceptable in a real building: the services crossing the isolation plane, the lifts and the surrounding footpaths all have limited clearance. The lead core inside each bearing deforms plastically as the bearing shears, converting the oscillation's kinetic energy into heat, so the amplitude is limited and the motion dies away within a few cycles instead of persisting.

Damping also broadens the resonance response. An undamped system responds enormously at exactly $f_0$ and hardly at all elsewhere; a damped one responds moderately over a wider band. For a building that is a favourable trade: the driving frequency of a future earthquake is not known in advance, so a design that is very sensitive to one particular frequency is a gamble, while one that is moderately insensitive across a range is robust.

Finally, the same physics sets a limit on how far the approach can be taken. Lowering $k$ further would push $T$ higher still and improve the isolation from short-period shaking, but it would also make the building more responsive to long-period ground motion — which is exactly what deep or distant large earthquakes produce, and what soft sedimentary basins such as the Hutt Valley amplify. A very soft isolation system would also let the building sway noticeably in a strong nor'wester, since wind loading is effectively a low-frequency drive. The chosen period of around 2–3 seconds is therefore not an arbitrary engineering figure; it is the value that sits above the common shallow-earthquake band and below the long-period band, with damping added to cover the uncertainty at both ends.

Taken together, the picture is coherent: resonance defines the danger, the SHM relationship identifies stiffness as the only lever, damping fixes the problem that pulling that lever creates, and the interaction of the three sets the specific period the engineers aim for."

Ideas that require each other, used to explain a real design choice and its limits: Excellence.