The length nobody has to measure

Between about 1820 and the arrival of the free-fall gravimeter, the most accurate measurements of the strength of gravity anywhere on Earth were made with a brass bar, two hardened steel edges, and a clock. They reached six significant figures. What makes that remarkable is not the clock, which was merely good, but the fact that the quantity a pendulum’s period actually depends on is one nobody could measure to anything like six figures — and the instrument was arranged so that it never had to be.

The idea is a piece of engineering rather than of physics, and it is one of the cleanest examples in mechanics of a design that removes a measurement rather than improving it.

What a pendulum’s period actually depends on #

The textbook pendulum is a point mass on a weightless string, and its period is $2\pi\sqrt{L/g}$.

Everything about the idealisation is wrong in a real instrument: the mass is not at one place, the string is not weightless, and the restoring force is only approximately the component of the weight along the arc. That is what makes the length the awkward quantity — it is not a distance anybody can point to, and the whole essay is about measuring around it rather than measuring it.

A real pendulum is a rigid body. Its restoring torque is $-Mgh\sin\theta$ with $h$ the distance from the pivot to the centre of mass, and its resistance to being turned is its moment of inertia about the pivot, which by the parallel-axis theorem is $M(k^2 + h^2)$ with $k$ the radius of gyration about the centre of mass. The period is therefore

$$T = 2\pi\sqrt{\frac{k^2 + h^2}{g h}}.$$

That expression is the small-angle one, and the small angle is where its whole simplicity comes from: the restoring torque is proportional to the displacement only because $\sin\theta$ has been replaced by $\theta$, which is the same replacement every potential admits near its own minimum.

The same object has different moments of inertia about different axes, in a ratio set entirely by where its mass sits — so a swinging body’s period depends on a distribution rather than on a length. That is what a compound pendulum makes explicit: it has an equivalent length, defined as the simple pendulum with the same period, and that equivalent length is a computed quantity rather than a measured one.

So an experimenter wanting $g$ from a period needs $k$ and $h$: the radius of gyration of a machined bar with a knife edge, adjustment screws, and whatever weights have been added to trim it, and the position of its centre of mass to the same precision. Both are computable for an idealised shape and neither is measurable to six figures on a real one. The instrument is dominated by a quantity it cannot see.

The length a compound pendulum pretends to be #

There is a way of stating the difficulty that immediately suggests its own solution. Whatever the body is, some simple pendulum has the same period, and the length of that simple pendulum is

$$L_{\text{eq}} = \frac{k^2 + h^2}{h}.$$

The expression has a minimum, at $h = k$, and that is the whole of Kater’s idea. A function with a minimum takes every value above it twice. Two pivots, one either side of the centre of mass at distances $h$ and $h'$, give the same period whenever

$$\frac{k^2 + h^2}{h} = \frac{k^2 + h'^2}{h'} \quad\Longleftrightarrow\quad h\,h' = k^2 .$$

And at such a pair the equivalent length becomes

$$L_{\text{eq}} = \frac{k^2 + h^2}{h} = \frac{h h' + h^2}{h} = h + h' .$$

The awkward quantity has cancelled. The equivalent length is the distance between the two knife edges, which is a distance between two hard steel corners on the outside of the instrument, measurable with a comparator to a micron. Then

$$g = \frac{4\pi^2 (h + h')}{T^2},$$

and neither the mass of the bar, nor its distribution, nor the position of its centre of mass appears anywhere in the answer.

Why it is not merely a trick #

It is worth being clear about what has and has not been achieved, because the cancellation can look like sleight of hand.

Nothing has been assumed about the bar. The relation $h h' = k^2$ follows from the period expression alone, and that expression holds for any rigid body whatever — a bar, a bar with weights bolted to it, a bar with a hole drilled in one end. The instrument works on an object of unknown and irregular mass distribution, which is exactly what a real one is once it has been adjusted.

Nor is the adjustment a matter of computing where the conjugate point is. The two knife edges are fixed, and a small weight is slid along the bar until the two periods agree. That moves both $k$ and $h$ at once, in a way nobody needs to model: the criterion is an equality of two measured times, and the experimenter watches it converge.

The result is an instrument whose output depends on exactly two measurements — a length between two edges and a time — and both of those are the two quantities that nineteenth-century metrology was best at.

The systematic that does not cancel #

Two things do not cancel, and both were understood at the time.

The first is amplitude. The period expression above already assumes small oscillations, and a real pendulum swings through a degree or two.

The small-angle error is second order, which is what makes it survivable: at two degrees the fractional error in $\sin\theta$ is a hundredth of a per cent, and in the period a quarter of that. But it does not cancel between the two swings of a reversible pendulum, because both are made at the same amplitude — so it is a systematic that has to be corrected rather than differenced away, and Kater’s method is silent about it.

That is the useful distinction between a systematic and a limitation: a systematic that can be measured and corrected costs precision only through the accuracy of the correction. Two degrees of amplitude known to five per cent contributes four parts per million, and the amplitude decays during the run, so the correction is applied to a mean.

The second is the knife edges. They are not lines; they have a radius of curvature of a few microns and the pendulum rolls rather than pivots, which shifts the effective pivot point by roughly that radius. That is a part in $10^5$ of the length, and it is dealt with by reversing the pendulum, by using edges of equal radius, and — in the best work — by measuring the radius.

Counting swings, and the trick that multiplies the clock #

The timing half of the instrument deserves its own arithmetic, because reaching six figures on a two-second interval with a nineteenth-century clock is not obvious.

The straightforward route is to count. Timing a single swing to a hundredth of a second gives five parts in a thousand and is useless; timing ten thousand swings — about five and a half hours — to the same hundredth of a second gives five parts in ten million, because the error is divided by the count while the interval is multiplied by it. Nothing about the clock has improved; the measurement has simply been made long.

The refinement that made it practical is better than counting. Set the pendulum swinging beside a clock pendulum of very nearly, but not exactly, the same period, and watch them through a slit: they drift into and out of step, and the interval between successive coincidences is a beat period. If the two differ by one part in ten thousand, the beat is ten thousand periods long, and timing the beat to one period determines the difference to one part in ten thousand of itself — which is one part in a hundred million of the period.

That is a measurement of a small difference rather than of a large quantity, and it is the same move as every null method: what is timed is the moment two things agree, which can be caught to a fraction of a swing by eye, rather than the length of an interval, which cannot. The unknown pendulum’s period is then the clock’s, which is known, plus a difference measured far more precisely than either.

So both halves of Kater’s instrument are the same idea applied twice. The length is obtained by a cancellation that removes the quantity nobody can measure; the period is obtained by a comparison that removes the clock’s own accuracy from the problem and leaves only its stability.

Building it wrong on purpose #

The two periods can never be made exactly equal, because the adjustment is by a screw and the timing has a resolution. Bessel’s treatment of the residue is the part of the design that is genuinely clever.

Writing the two periods as $T_1$ and $T_2$ and the two distances as $h_1$ and $h_2$,

$$\frac{8\pi^2}{g} = \frac{T_1^2 + T_2^2}{h_1 + h_2} + \frac{T_1^2 - T_2^2}{h_1 - h_2},$$

which is exact. The first term contains the sum of the distances, which is the measured quantity; the second contains their difference, and it is the only term carrying the mismatch.

So the pendulum is built deliberately lopsided: the centre of mass is placed close to one knife edge and far from the other, and the term carrying the one imperfect adjustment is divided by a large number. The instinct to make a precision instrument symmetric would have made this one six times worse.

That inversion is the transferable idea, and it recurs wherever an instrument has one adjustment that cannot be made exactly. Find the term the imperfection enters through, and design so that its coefficient is small. It is the same move as choosing a bridge circuit whose output is a difference, or a spectrometer that measures a null rather than a level.

The two other things a pendulum was used to weigh #

The instrument is worth a paragraph of context, because its accuracy was not an end in itself.

A pendulum measures the local value of g, and the local value varies: with latitude, because a rotating Earth requires a centripetal acceleration that comes out of the weight, and with altitude, and with what is underneath. Kater’s survey of Britain in the 1820s produced a map of g accurate enough to see the difference between standing on granite and standing on sediment, and the technique became the foundation of gravity prospecting: an ore body is denser than its surroundings, and it shows up as a few parts per million in g.

Inside a sealed box no experiment distinguishes standing on the ground from accelerating in free space, which is why the quantity a pendulum measures is an acceleration rather than a force. That matters for what the measurement means: $g$ is not a property of the Earth alone but of the frame the instrument sits in, and a pendulum in a lift reads the difference.

It was also used the other way round, as a length standard. The seconds pendulum — the one whose half-period is a second — has an equivalent length of 994 millimetres at London, and in 1824 that length was written into British law as the definition of the yard, to be recovered by experiment if the physical standard were destroyed. The clause was repealed in 1855, for a reason worth recording: a length defined by a pendulum is a length that depends on where the pendulum is standing.

The pendulum’s true potential is $1 - \cos\theta$, and the flattening of that curve near the top is where the period runs away. A total-energy line across it meets the curve at the turning points, and as the energy approaches the maximum those points slide toward the vertical and the time between them grows without bound — which is the amplitude dependence stated as a picture rather than as a series.

The metre that was nearly a pendulum #

The clause in British law had a French predecessor and a more consequential rejection, and the reason it was rejected is the same fact this essay’s last section ends on.

When the French Academy set out in 1791 to define a universal unit of length, the seconds pendulum was one of two serious candidates. It had everything to recommend it: the apparatus was cheap, the measurement was within reach of any competent observatory, and the definition needed nothing but a clock and a rule.

It was rejected because a pendulum’s length depends on where it is swung. Gravity varies with latitude by about half a per cent between the equator and the poles, and with altitude and local geology besides — so a seconds pendulum in Paris and one in Cayenne are different lengths, and a definition would have had to name a place as well as a procedure. Naming a place made the unit political rather than natural, which was precisely what the exercise was trying to avoid.

The alternative chosen was a ten-millionth of the quadrant of the meridian, which required a survey from Dunkirk to Barcelona lasting seven years and conducted through a revolution and a war. It is hard to argue that the cheaper option was not tempting.

The irony is that the meridian turned out to have the same defect in a subtler form: the Earth is not a figure anybody had measured well enough, and the metre bar that resulted was about a fifth of a millimetre short of its own definition. Both candidates were attempts to tie a unit to the planet, and both failed for the same reason — the planet is not a standard, it is an object with properties that have to be measured.

One number, and the sixty-five years it was wrong #

The reversible pendulum has one limitation that no amount of care with it addresses, and its consequences were felt for most of a century.

The instrument is an absolute one: it gives $g$ from a length and a time with nothing to compare against. But it is slow, delicate and awkward to transport, so in practice one such measurement was made at a chosen site and every other gravity measurement in the world was made relative to it, by instruments that compare rather than determine — much easier to carry, and much more precise as comparators than as absolute devices.

The chosen site was Potsdam, and the absolute measurement made there with reversible pendulums in the first decade of the twentieth century became the datum for the world’s gravity network. Every gravity value published anywhere for the next sixty years was that number plus a measured difference.

It was too high by about fourteen parts per million. The error was in the absolute measurement — the knife edges, the flexure of the support, the entrained air, all the terms this essay has listed as corrections — and it was invisible from inside the network, because every measurement in the network was a difference and every difference was right. Only a second absolute measurement, made with a different technique, could reveal it, and the falling-body instruments of the 1960s did.

The moral is the one that goes with every relative network. A comparison chain is only as good as the one absolute value it hangs from, and its internal consistency says nothing about that value at all. Sixty-five years of gravity data were internally consistent to a part in a million and collectively wrong by fourteen.

Where the model stops #

The support is not rigid. The pendulum reacts against whatever it is hung from, and a stand that yields by micrometres shortens the period measurably. Kater’s own results were revised when it was realised that his support was flexing; the modern treatment measures the stand’s compliance and corrects for it, or uses two pendulums swinging in antiphase so that the reaction cancels.

Air is not nothing. Buoyancy reduces the effective weight, entrained air adds to the effective inertia, and damping shortens the observed period slightly. All three are corrections at the $10^{-4}$ level, all three are computed, and the entrained-air term is the one that cannot be computed reliably from theory and has to be measured by swinging at two pressures.

The bar is treated as one rigid body. It is not; it flexes elastically as it swings, and the flexure changes the effective moment of inertia. This is a part in $10^6$ effect for a stiff bar and it is the reason precision pendulums were made short and thick rather than long and elegant.

Everything in this essay lives inside a harmonic fit: the period is a property of the curvature at the bottom of the potential, and every correction is a departure from that fit. Comparing the pendulum’s potential with two unlike ones makes the point that the fit is generic — the same approximation, the same leading correction, and the same eventual failure whatever the well is.

And the whole instrument has been superseded. Absolute gravimeters now drop a corner cube in a vacuum and count interference fringes, reaching a part in $10^9$. The pendulum’s descendants survive as relative instruments, and the reason is worth noting: the pendulum’s answer depends on a length that must be transferred from a standard, and the falling-corner-cube instrument’s depends on a wavelength and a clock, both of which are now defined rather than measured.

What the pictures cannot show #

The period curve is drawn for a uniform bar, and a real Kater pendulum is not uniform — it carries a large bob near one end and a small adjustable one near the other, precisely to place the centre of mass off centre. The shape of the curve is the same and the numbers on the axes are not, and no figure here shows the actual instrument.

A precision pendulum is damped by air and by its knife edges, so its amplitude falls during a measurement — and since the period depends on amplitude, a decaying swing has a drifting period. The damping is read off the ratio of successive peaks, and it enters the correction as well as the error budget. That is the practical reason such instruments were run in evacuated cases.

Nor can any of these figures show the timing. Everything above treats the period as known; in practice it is obtained by counting several thousand swings against a clock and dividing, and the entire six-figure accuracy of the method rests on the fact that a small timing error divided by a large number of periods is a very small error in one period. That averaging is the other half of the instrument and it is invisible in a figure of a curve.

Where this ladder goes next #

Three rungs of this ladder have been about the pendulum as a system: the approximation that makes it simple, the exact period when the approximation is dropped, and the three ways it comes to rest. This rung is about the pendulum as an instrument, and the change of view is the point: what matters is no longer which quantities the system contains but which of them the answer contains.

The habit worth carrying away is the design question. Which of the quantities in this measurement can actually be measured well, and can the apparatus be arranged so that only those appear? Kater could time to a part in $10^6$ and measure a length between two steel edges to a part in $10^5$, and could not find a centre of mass to better than a part in $10^3$. The instrument he built contains a time and a length and does not contain a centre of mass, and it is six figures accurate for exactly that reason.

One last comparison makes the point about design rather than precision. A modern absolute gravimeter is a more accurate instrument by three orders of magnitude, and it is accurate for the same kind of reason rather than by being better made: it reduces the measurement to a wavelength and a clock, both of which are defined constants, and contains no property of the falling body whatever — not its mass, not its shape, not where its centre of mass is. Kater’s pendulum removed one awkward quantity from the answer and the corner cube removed all of them, and the intellectual step is identical.

What is left on this ladder is the escapement — the mechanism that keeps a clock pendulum swinging, which must deliver energy without disturbing the period it is there to preserve, and which is the place where the amplitude correction above stops being a correction and becomes the design problem.