Illustrated Physics
physics, drawn
Concept

Fictitious force — where it appears

A term that appears in Newton's law when it is written in an accelerating frame, arising from the coordinates rather than from any interaction. Nothing is removed by calling it fictitious: the centrifugal term flattens the planet by twenty kilometres and the Coriolis term organises the weather.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

One puck, two frames. A puck slides outward from the centre of a turntable at 1.4 m/s across a table of radius 1 m, which turns through 0.55 of a revolution in the 0.71 seconds the crossing takes. On the left, in the room: a straight line, because no force acts along it. On the right, on the turntable: a spiral, because the coordinates turned underneath it. The two panels are the same numbers with the same clock — the second is the first with the angle rotated by −Ωt — so whatever is bending the path on the right is arithmetic, and it is still worth a name, because on a turning planet the right-hand panel is the one everybody lives in.

The forces that are not there

Writing Newton's law in a frame that is turning produces three extra terms. Nothing was added to the world to make them appear and nothing is removed by calling them fictitious — one of them flattens the planet, one of them turns the weather, and both are computable to four figures.

mechanics · Circular motion The deflection is a circle, not a bend. Trajectories integrated from Newton's law in a rotating frame with the Coriolis term and nothing else — no pressure gradient, no friction, no force of any kind. 0.5 m/s at 45°, 0.3 m/s at 30°. Each path closes on itself after one inertial period, checked to a millionth of its own radius, and the radius is the speed divided by the Coriolis parameter: 4.85 km, 4.11 km. The deflection usually described as a curving of the path is a complete circle, traversed clockwise in the northern hemisphere in half a pendulum day, and a body left alone in a rotating frame goes nowhere at all.

The deflection that closes on itself

The Coriolis term is usually described as bending a path to the right. Integrated rather than described, it does not bend the path — it closes it. A body left alone in a rotating frame travels a circle of radius U/f and comes back to where it started in half a pendulum day, having gone nowhere at all, and drifting buoys in every ocean draw exactly that.

mechanics · Circular motion Pushed one way, travelling another. A parcel released from rest under a steady pressure-gradient acceleration of 2.0e-4 metres per second squared pointing east, integrated for 36 hours at 60°, 30°, 10°. Without rotation it would accelerate east indefinitely. With it, the motion is an inertial circle about a mean velocity at right angles to the push, and the mean over a whole number of inertial periods is measured off each path and matches the geostrophic value a/f to two per cent. At high latitude the loops are tight and the drift is almost purely along the isobars; near the equator the parcel travels a long way down the gradient before the rotation has had time to turn it, which is why geostrophic balance is a high-latitude statement and the tropics need a different set of approximations.

The ratio that decides whether the planet is turning

Whether the rotating terms matter is not a question about size. It is one dimensionless ratio, U over fL, and it runs from ten thousand in a teacup to a millionth in the Earth's core. Where it is small the pressure gradient stops accelerating the fluid and starts balancing a force on fluid already moving across it — so the flow runs along the pressure contours instead of down them, and a weather map is a streamline plot.

mechanics · Circular motion A closed cabin accelerating at 3 m/s², and everything in it leaning the same way. A closed box accelerating to the right at 3 m/s², with water in the bottom and air above. Everything that can hang or float lines up with the effective gravity g − a, tilted 17.0° from the vertical towards the back: the water's surface and the isobars below it are perpendicular to it, a floating block stands along it, a plumb bob hangs along it — and a helium balloon on a string leans the other way along the same line, forward, into the acceleration. The balloon is not doing anything unusual. Buoyancy always points opposite to gravity, and in the cabin the effective gravity has a horizontal part, so the air's pressure gradient does too, and it pushes the balloon along it.

The balloon that leans the wrong way

When a car pulls away, everything loose in it swings back — except a helium balloon on a string, which swings forward. Nothing strange is acting on it. Buoyancy points against gravity, and inside an accelerating cabin gravity has a sideways part. Read that way, a lift cannot change how deep a boat floats, free fall abolishes floating altogether, and a centrifuge is Archimedes' principle with the word "up" pointing at the axis.

fluids · Buoyancy

Named alongside it

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

Rotating frameCoriolis forceLatitudeAngular velocityFoucault pendulumInertial oscillationReference frameApproximationArchimedes principleBeta planeBuoyancyCapillary length

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