Concept

Time constant — where it appears

The time a system relaxing exponentially takes to cover all but 1/e — about 37 per cent — of the remaining distance to its final state. It is a ratio of a storage to a loss, such as heat capacity over heat conductance or capacitance times resistance, and after five time constants less than one per cent is left.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

A gridiron of steel and brass drawn to the length that holds. The rods of a gridiron pendulum, to scale, for a seconds pendulum of 0.994 m: steel rods (dark) hang from the suspension and carry cross-bars from which brass rods (light) push upward, and from their tops the next steel rods hang down again, until the last carries the bob. Steel expands downward, lowering the bob; brass expands upward, raising it. They cancel when the steel's total length times its expansion equals the brass's: 2.52 m of steel against 1.52 m of brass, a ratio of 0.605. That is why gridirons are built of so many rods — the steel alone is 2.53 times the pendulum's length, folded into 3 stages. With zinc instead of brass, which expands 1.58 times as much, 1.61 m of steel would do. Real gridirons double every rod, side by side, so that the bob hangs symmetrically.

The pendulum that must not grow when it warms

A pendulum clock keeps time by the length of its pendulum, and every material grows when it is warmed. A steel rod a metre long lengthens by a hundredth of a millimetre for each degree, and the clock loses half a second a day — a minute a fortnight between a cold winter and a warm summer, far worse than anything the clockmaker's escapement could cause. The cure, found in the 1720s, was to let the pendulum grow in two directions at once: one part downward, another upward, in exactly the proportion that leaves the bob where it was. The same trick, with different metals, kept the best clocks to a hundredth of a second a day for two centuries, and its limits are set by heat that does not spread evenly or quickly.

mechanics · Pendulum

The height at which hot air stops being light enough. Lift per cubic metre of hot air against height in the standard atmosphere, with the inside at 100 °C and 120 °C, and (dashed) what a 2800 m³ balloon carrying 550 kg plus its envelope needs per cubic metre, 0.241 kg. As the balloon climbs, the outside air thins, and although it also cools — which improves the temperature ratio — the density wins: lift per cubic metre falls by about a tenth for every kilometre. The balloon can climb only to where the curve crosses its need: 6.0 km with the envelope at 120 °C. Above the tropopause the outside temperature stops falling and the lift falls faster still; a hot-air balloon that reaches the stratosphere needs an envelope enormous compared with its load.

The balloon a cold morning lifts

A hot-air balloon is open at the bottom, so the air inside is at the same pressure as the air outside and lighter only because it is hotter. Its lift is the outside air's density times one minus the ratio of the two absolute temperatures — a small number, a third of a kilogram per cubic metre at best — and since the fabric cannot be heated much above a hundred degrees, the lift depends as much on the cold outside as on the heat inside. That is why balloons are the size of houses, why they fly at dawn, why each has a ceiling, and why the pilot must burn for where the balloon will be half a minute from now.

fluids · Buoyancy

Named alongside it

The objects these essays reach for when they reach for this one.

Archimedes principle Buoyancy Centre of oscillation Clock Density Gridiron pendulum Heat-transfer The ideal gas law Invar Lapse rate Pendulum Standard atmosphere

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