Physics

A body keeps its velocity unless a force acts; the force F needed is its mass m times the acceleration a, or the rate of change of its momentum p; and every force has an equal and opposite reaction. Force is in newtons: 1 N accelerates 1 kg by 1 m/s².

Use: Sizes every brake, rope, axle and anchor. Stopping a 1,000 kg cart from 5 m/s in 2 m needs an average force of m v²/(2d) = 6,250 N. Recoil, rowing and jet engines are the third law.

First: Isaac Newton, Philosophiæ Naturalis Principia Mathematica, 1687 (Gutenberg; MacTutor).

Every two masses attract with a force proportional to both masses and falling with the square of the distance between them. G = 6.674 30 × 10⁻¹¹ N m²/kg² (CODATA 2022), still the least precisely known constant of nature. At the Earth's surface it gives an acceleration g of about 9.81 m/s², slightly more at the poles than at the equator.

Use: Weight is mass × g: the weight to design for is 9.81 N per kilogram. The same law gives the orbits of the Moon, the planets and every satellite (astronomy).

First: Newton, 1687 (MacTutor). Cavendish first measured the attraction between lead balls in the laboratory, 1798 (Philosophical Transactions); today's value: CODATA 2022.

Work W, in joules, is a force times the distance it moves through. A moving mass carries ½mv² of energy, a raised one mgh. Power P, in watts, is work per second. Friction turns energy into heat; it never destroys it.

Use: A person can sustain about 75–100 W of useful work for hours. Lifting 1 m³ of water (1,000 kg) by 10 m takes mgh = 98 kJ: about 20 minutes of a person's steady effort, or a few seconds of a small engine. Doubling a vehicle's speed quadruples the energy its brakes must absorb.

First: Joule measured the equivalence of work and heat, about 4.2 J for each calorie, 1843–1850 (Philosophical Transactions); Helmholtz stated the conservation of energy in general, 1847.

A pendulum of length L swinging through a small arc takes the same time T for each swing, whatever its weight and nearly whatever its amplitude. A pendulum that beats seconds (T = 2 s for a swing there and back) is 0.994 m long.

Use: The first accurate clocks, accurate to seconds a day instead of minutes; a way to measure g, and so to check a standard of length (one metre, one kilogram).

First: Galileo noticed the equal time of swings about 1602; Huygens built the pendulum clock in 1656 and derived the formula in Horologium oscillatorium, 1673 (MacTutor).

A body in a fluid is pushed up by the weight of the fluid it displaces. It floats when it can displace its own weight before it is submerged: a 500 kg boat sinks until it displaces 0.5 m³ of fresh water, about 0.49 m³ of seawater.

Use: Load lines and freeboard on boats; a hydrometer reading the density of brine, wine or a battery's acid; telling a solid metal from a plated one by weighing it in air and in water.

First: Archimedes, On Floating Bodies, about 250 BC (MacTutor).

In a still fluid the pressure p at depth h is the pressure at the surface plus the weight of the column above. Every 10 m of water adds about 1 atmosphere (98 kPa); the atmosphere itself weighs as much as 10.3 m of water or 760 mm of mercury.

Use: A tank 10 m above a tap gives about 1 bar. No suction pump can lift water more than 10.3 m, and in practice about 7 m: deeper wells need the pump at the bottom (wells). A dam wall must resist a push that grows with depth.

First: Torricelli's barometer, 1643 (MacTutor); Pascal had the column carried up the Puy de Dôme, 1648, and stated the law of pressure in fluids (MacTutor).

Along a streamline of a smoothly flowing fluid, pressure, speed and height trade against one another. Water leaving a hole at depth h below the surface comes out at √(2gh), as if it had fallen from the surface: 14 m/s from 10 m.

Use: Flow from a tank or a dam outlet; the venturi meter that measures flow by the drop in pressure at a throat; the speed of the jet on a Pelton wheel. Real pipes lose some of that head to friction.

First: Daniel Bernoulli, Hydrodynamica, 1738 (MacTutor); Torricelli's law of efflux, 1643.

Warming a mass m by ΔT takes its heat capacity c times both; melting or boiling it takes a latent heat L with no change of temperature. For water c = 4.18 kJ per kg per °C, melting takes 334 kJ/kg and boiling 2,257 kJ/kg.

Use: Heating 1 litre of water from 20 °C to boiling takes 335 kJ, but boiling it away takes 2,257 kJ more: so put a lid on and keep the boil gentle. Sweat cools by the same latent heat, and so do evaporative coolers and clay water pots.

First: Joseph Black, lectures at Glasgow and Edinburgh from the 1760s, published 1803 (1806 edition, Internet Archive).

No engine working between a hot source and a cold sink can turn more than 1 − T(cold)/T(hot) of the heat into work, with both temperatures in kelvin; the rest must be rejected. Run backwards, the same law says a heat pump can deliver several units of heat for each unit of work. Entropy, which this limit defines, never decreases in an isolated system.

Use: A steam engine between 150 °C (423 K) and 30 °C (303 K) can never beat 28 %: hence higher pressures and temperatures in steam and turbines. A heat pump moving heat from 0 °C outdoors to 35 °C indoors could in theory deliver about 9 units of heat per unit of electricity; real ones deliver about 3, still three times a heater (refrigeration).

First: Sadi Carnot, Réflexions sur la puissance motrice du feu, 1824 (Gutenberg); Clausius named entropy and stated the second law, 1850–1865 (MacTutor).

Every surface radiates power in proportion to the fourth power of its absolute temperature; σ = 5.670 × 10⁻⁸ W/(m² K⁴) and the emissivity ε is near 1 for most non-shiny surfaces. The hotter it is, the shorter the wavelength of its peak: dull red at about 600 °C, yellow-white above 1,200 °C.

Use: Why a kiln or forge loses heat so fast at high temperature and needs thick insulation; judging a fire's or metal's temperature by its colour (smithing); and why the Earth, radiating at about −18 °C in the infrared, is warmed by gases that absorb infrared.

First: Stefan from measurements, 1879 (MacTutor); Boltzmann derived it from thermodynamics, 1884 (MacTutor); Wien's displacement law, 1893.

A wave's speed v is its frequency f times its wavelength λ. Sound travels at 343 m/s in air at 20 °C and about 1,480 m/s in water; light and radio at 299,792,458 m/s in a vacuum, the value that now defines the metre.

Use: Count seconds from lightning to thunder and divide by 3 for the distance in kilometres. A radio signal at 7 MHz has λ = 43 m, so a quarter-wave aerial is about 10.7 m long (radio).

First: Rømer showed that light takes time to travel, 1676; the metre has been defined by the speed of light since 1983 (BIPM SI Brochure).

Light bends at a surface so that the refractive index times the sine of the angle is the same on both sides; n is 1.00 for air, 1.33 for water and about 1.5 for glass. A thin lens of focal length f images an object at distance u at distance v. Beyond a critical angle (about 42° inside glass) light cannot leave at all and is totally reflected.

Use: Grinding lenses for spectacles (their power in dioptres is 1/f in metres), magnifiers, telescopes and microscopes; total internal reflection is what keeps light inside an optical fibre.

First: Ibn Sahl, 984; Willebrord Snell, 1621, published by Descartes, 1637 (MacTutor); Huygens explained it with waves, Treatise on Light, 1690 (Gutenberg).

Diffraction blurs every image: a telescope of aperture D cannot separate points closer than about 1.22λ/D radians, and a microscope cannot resolve detail finer than λ divided by twice its numerical aperture NA. With visible light the best light microscope reaches about 0.2 µm.

Use: A light microscope shows bacteria (1–2 µm) but not most viruses (20–300 nm), which needed the electron microscope. A telescope twice as wide shows detail twice as fine. Since 2000, fluorescence methods have beaten the limit for labelled molecules (Nobel Prize in Chemistry 2014).

First: Ernst Abbe, 1873 (MacTutor); Lord Rayleigh's criterion, 1879 (MacTutor).

The current I through a conductor is proportional to the voltage V across it, and its resistance R is the ratio. Power delivered is V × I, and the power lost as heat in a wire is I²R.

Use: Wire sizing and the reason grids run at high voltage: sending 10 kW at 230 V needs 43 A, at 11,000 V under 1 A, so the same wire wastes about 2,300 times less. Fuses and breakers are sized on the same current.

First: Georg Simon Ohm, Die galvanische Kette, 1827 (MacTutor).

The voltage induced in a coil of N turns is proportional to how fast the magnetic flux Φ through it changes. In a transformer the same changing flux links two coils, so their voltages stand in the ratio of their turns.

Use: Every generator spins a coil past a magnet; every transformer raises voltage for transmission and lowers it for use. Only alternating current can be transformed, which is why the grid is AC.

First: Michael Faraday, 1831 (Experimental Researches in Electricity, Gutenberg; MacTutor).

A charge q moving with velocity v through electric field E and magnetic field B feels this force. For a wire of length L carrying current I across a field B, the force is B × I × L, at right angles to both.

Use: The torque of every electric motor, the deflection in a moving-coil meter and a loudspeaker, and the beam steering of the old cathode-ray tube.

First: Hendrik Lorentz, 1895 (MacTutor); Faraday's first motor, 1821.

Electric charge is the source of the electric field; there are no magnetic charges; a changing magnetic field makes an electric field (Faraday); and currents and changing electric fields make magnetic fields. Together they predict waves travelling at c = 1/√(μ₀ε₀), which is the measured speed of light.

Use: Light, radio, microwaves and X-rays are one thing at different wavelengths (the spectrum). Every aerial, waveguide, radar and wireless link is designed from these four lines.

First: James Clerk Maxwell, A dynamical theory of the electromagnetic field, 1865 (Philosophical Transactions; MacTutor); written as these four by Heaviside, 1884; radio waves made and detected by Hertz, 1887 (MacTutor).

The speed of light is the same for every observer, so moving clocks run slow by the factor shown, and nothing with mass reaches c. Mass is energy: 1 gram is 9 × 10¹³ J, about what a large power station makes in a day.

Use: Nuclear energy is mass turning into energy (below). A GPS satellite at 3.9 km/s runs about 7 microseconds a day slow from its speed alone.

First: Albert Einstein, 1905: Zur Elektrodynamik bewegter Körper (Annalen der Physik) and Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? (Annalen der Physik).

Gravity is the curvature of space and time. The left side describes the curvature (Λ is the cosmological constant); the right side, the density of mass, energy and momentum that causes it. Clocks run faster higher in a gravitational field, light bends past the Sun, and accelerating masses send out gravitational waves.

Use: GPS satellites' clocks gain about 45 microseconds a day from weaker gravity and lose about 7 from their speed; uncorrected, the net 38 would put positions out by kilometres within a day (Ashby, 2003). The same equations predicted black holes and the expanding universe.

First: Einstein, November 1915; full account 1916 (Annalen der Physik; Relativity, Gutenberg). Gravitational waves first detected 2015 (LIGO).

Light is absorbed and emitted in packets, photons, each with energy h × f, where h = 6.626 070 15 × 10⁻³⁴ J s exactly. A photon of red light carries about 1.8 eV, ultraviolet at 254 nm about 4.9 eV: enough to break the bonds of DNA.

Use: Why ultraviolet sterilises water and burns skin while red light does neither; why a solar cell needs photons above its band gap (1.1 eV for silicon) and wastes the excess as heat, which caps a single-junction cell at about 30 % (Shockley and Queisser, 1961).

First: Max Planck, 1900–1901 (Annalen der Physik); Einstein's light quanta and the photoelectric effect, 1905 (Annalen der Physik).

The wave function ψ of a particle evolves under its energy operator Ĥ. Solved for the electron in a hydrogen atom it gives only the energies −13.6 eV/n², and for every atom a fixed set of shells: the structure of the periodic table.

Use: Explains chemical bonding, the colours of flames and stars (the spectrum), why some solids conduct and others insulate, and so the transistor (electronics), the laser and the LED.

First: Erwin Schrödinger, 1926 (Annalen der Physik; MacTutor).

Each unstable nucleus has a fixed chance of decaying in each second, so half of any sample is left after one half-life, a quarter after two, about a thousandth after ten. Carbon-14 has a half-life of 5,730 years, iodine-131 of 8 days.

Use: Dating wood, bone and charcoal up to about 50,000 years old by their carbon-14; knowing that fallout iodine is nearly gone after 80 days; protection by time, distance and shielding, since a dose falls with the square of the distance from a small source.

First: Rutherford and Soddy, 1902 (Philosophical Magazine); radiocarbon dating, Libby, 1949.

A nucleus weighs less than its parts; the missing mass Δm is its binding energy. Splitting uranium-235 releases about 200 MeV and 2–3 neutrons that can split more nuclei, a chain reaction. Fissioning 1 g of it gives about 8 × 10¹⁰ J, as much as burning about 3 tonnes of coal. Fusing hydrogen into helium, which powers the Sun, releases more still per kilogram.

Use: Nuclear power and its hazards; why a reactor needs a moderator, control rods and cooling that never fails (energy without the carbon bill). In December 2022 a laboratory fusion target first gave out more energy than the laser light put into it: 3.1 MJ from 2.05 MJ (NIF).

First: Hahn and Strassmann found barium among the products, 1938–1939 (Naturwissenschaften); Meitner and Frisch explained it as fission, 1939 (Nature).