Illustrated Physics
physics, drawn
Concept

Systematic error — where it appears

An error that biases every measurement in the same direction, so that repeating the experiment does not reduce it. It is what a mature precision measurement is limited by, and separating one from the signal usually means finding a parameter the two depend on differently and sweeping it.

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

Two pivots with one period, and the length between them. The period of a uniform bar 1 m long swung about a pivot, against the distance of that pivot from its centre of mass. Close to the centre the period runs away, because there is almost no restoring torque; far from it the bar behaves more and more like a simple pendulum. In between is a minimum at h = k = 0.2887 m, the radius of gyration, and because there is a minimum every period above it belongs to two pivots at once — here 0.4 m and 0.2083 m, whose product is k² and whose periods agree to 2.2e-16 seconds. Suspend the bar from those two knife edges in turn, adjust until the periods match, and the equivalent simple pendulum is the distance between them: 0.6083 m. Then g = 4π²(h+h′)/T² gives 9.8100 m/s², recovered from the drawing to 1.8e-16. The mass of the bar, the distribution of that mass, and the position of the centre of mass appear nowhere in the answer.

The length nobody has to measure

A pendulum's period depends on its length, so a pendulum will measure gravity — except that a real swinging bar has no single length, and finding where its mass effectively sits is far harder than timing it. Kater's answer was to build the instrument so that the awkward quantity cancels, leaving a distance between two knife edges that a rule can read to five figures.

mechanics · Pendulum Three centuries of finding nothing. The upper limit on η against the year it was set, on a logarithmic scale. Newton (1687) reached 1e-3; Bessel (1832) reached 2e-5; Eötvös (1922) reached 5e-9; Dicke (1964) reached 1e-11; Braginsky (1972) reached 1e-12; Eöt-Wash (2008) reached 2e-13; MICROSCOPE (2022) reached 1e-15. That is 12 orders of magnitude in 335 years, and every one of those measurements returned zero. A sequence of null results is not a sequence of failures. Each one is a statement that a principle assumed by every theory of gravity holds to a new level, and each new level excludes a class of theories that would have shown a departure there — a long-range force coupling to something other than mass-energy, a scalar partner to the graviton, a violation arising at some energy scale. The measurement is worth making again precisely because it has always come out the same way, which is what makes any departure decisive.

The fall that does not depend on what is falling

Everything falls at the same rate, and the statement has been tested for three hundred years by people looking for the exception. Twelve orders of magnitude have been added to the limit and every measurement has returned zero. The last three orders came not from a better instrument but from finding something bigger to fall towards.

astrophysics · Equivalence principle

Named alongside it

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

MeasurementPendulumCentre of massCentrifugal forceEquivalence principleEquivalent lengthFree fallGravitational massInertial massMetrologyModulationMoment of inertia

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