Engineering

Ignoring friction, the force times the distance it moves is the same at both ends of a lever, a pulley system or a gear train. A block and tackle with four ropes supporting the load lifts it with a quarter of the pull, over four times the length of rope; a gear with three times the teeth turns a third as fast with three times the torque.

Use: Lifting stone, timber and engines without a crane; sizing a mill's gearing to its millstone (woodworking and the first machines). Friction takes 5–10 % per pulley in practice, so add a margin.

First: Archimedes' law of the lever, about 250 BC (MacTutor).

Friction can resist up to μ times the force pressing two surfaces together. Rope wrapped round a post multiplies this exponentially with the angle of wrap θ in radians. With μ = 0.3, three full turns (θ = 6π) hold a load about 285 times the holding pull.

Use: Snubbing a boat or a falling tree round a post, lowering loads, belaying: a few turns let one person hold what no person could lift (knots). It also sets how much a belt can drive before it slips.

First: Euler, 1760s (MacTutor); Amontons's laws of friction, 1699.

Stress σ is force per area, in pascals; strain ε is the stretch per length. Below its limit a material stretches in proportion, by its stiffness E: 200 GPa for steel, about 10–15 for timber along the grain. Mild steel yields at about 250 MPa.

Use: A 10 mm steel rod (78.5 mm²) yields at about 20 kN, two tonnes; at a safety factor of 4 it should carry half a tonne. The same sums size bolts, chains, hangers and rope.

First: Hooke's law, 1678 (MacTutor); Thomas Young's modulus, 1807 (MacTutor).

A beam carrying w per metre over a span L bends with a moment greatest at mid-span; the stress is highest at its top and bottom faces, a distance c from the middle. Its resistance to bending, I, grows with the cube of its depth h. Sag grows with the fourth power of the span.

Use: A 50 × 200 mm joist on edge is 16 times stiffer and 4 times stronger than laid flat. Doubling a span with the same beam quadruples the stress and multiplies the sag by 16: use more, deeper joists, or a post at mid-span (framed building).

First: Galileo's first theory of beam strength, Two New Sciences, 1638 (MacTutor); the bending theory of Euler and Daniel Bernoulli, 1744–1750; Navier put the neutral axis right, 1826 (MacTutor).

A slender post or strut fails by bowing sideways at the critical load, long before the material crushes. The load falls with the square of the effective length K L: K is 1 with both ends pinned, 0.5 with both fixed, 2 for a flagpole free at the top.

Use: Halving a post's free length, by a brace at mid-height, quadruples the load it can carry; a fat hollow tube beats a thin solid bar of the same weight. Why scaffolds, towers and frames are braced diagonally (iron-framed building).

First: Leonhard Euler, 1744 (MacTutor).

Flow through a pipe of length L and bore D at speed v loses a head of water set by the friction factor f. Below a Reynolds number of about 2,000 the flow is smooth and f = 64/Re; above about 4,000 it is turbulent and f is about 0.015–0.03, read from the Moody chart.

Use: At a fixed flow, head loss falls with the fifth power of the bore: a 32 mm pipe loses about 70 % less than a 25 mm one carrying the same water. Size mains and irrigation lines so the loss leaves enough pressure at the far end (irrigation).

First: Weisbach, 1845 (MacTutor), and Darcy, 1857; Reynolds found the change from smooth to turbulent flow, 1883 (Philosophical Transactions); Moody's chart, 1944 (Transactions of the ASME).

Water in a ditch, canal or gutter flows at a speed set by its hydraulic radius (flow area divided by wetted perimeter), the slope S and the roughness n: about 0.013 for smooth concrete, 0.02–0.03 for clean earth, more for weeds and stones.

Use: An earth ditch 1 m wide running 0.5 m deep on a slope of 1 in 1,000, with n = 0.025, moves about 0.5 m/s, 250 litres a second. Keep speeds between about 0.3 m/s (silting) and 1 m/s (scour in earth) (canals and locks).

First: Gauckler, 1867, and Manning, 1891; roughness values from the US Geological Survey (Arcement and Schneider, 1989).

The power in a flow Q (m³/s) falling a head H is ρ g Q H; a machine converts a fraction η of it: about 0.6–0.7 for a good water wheel, 0.8–0.9 for a turbine. Pumping needs the same power the other way, divided by the pump's efficiency.

Use: A stream of 50 litres a second falling 10 m holds 4.9 kW; a wheel delivers about 3 kW (water power). Lifting 1 litre a second 20 m takes 196 W of water power, about 400 W from a small pump.

First: John Smeaton's model experiments on water wheels, 1759 (Philosophical Transactions).

Heat flow Q through an area A is its U-value times the temperature difference. Each layer resists in proportion to its thickness t over its conductivity k: about 0.035–0.04 W/(m K) for mineral wool or sheep's wool, about 0.06 for straw, 0.6–1.0 for brick, 1.5–2 for concrete, 1 for glass and 50 for steel.

Use: 100 mm of wool (U about 0.35) loses a sixth of the heat of a solid brick wall (U about 2); a single pane loses about four times as much as good double glazing. Insulate the roof first, where warm air collects (plan the house).

First: Joseph Fourier, Théorie analytique de la chaleur, 1822 (MacTutor).

A body cools toward its surroundings exponentially, with a time constant τ set by its heat capacity over its surface area and the heat-transfer coefficient h. Water draws heat away many times faster than still air at the same temperature.

Use: Food cools through the danger zone fast enough only in shallow layers: divide a deep pot of stew into trays about 5 cm deep before chilling (biology: bacteria double). The same law times quenching in smithing, and explains why a person in cold water chills far faster than in cold air.

First: Isaac Newton, Scala graduum caloris, 1701 (Philosophical Transactions).

A controller measures the error e between what is wanted and what is, and acts in proportion to it, to its accumulated total, and to its rate of change. Too much proportional gain makes the system hunt and oscillate; the integral term removes a steady offset; the derivative damps overshoot.

Use: The centrifugal governor on an engine or mill, a thermostat, a float valve, an autopilot and a kiln controller are all this loop. Tune by raising the proportional gain until it just starts to oscillate, then back off (the factory).

First: Maxwell analysed the stability of governors, 1868 (Proceedings of the Royal Society); Minorsky's three-term controller for steering ships, 1922 (Journal of the American Society for Naval Engineers).