Experiment: The Pendulum
Oscillations and Mechanical Waves asserts that a pendulum of length \(\ell\) swinging through a small arc has period \(T=2\pi\sqrt{\ell/g}\), the same whether the arc is wide or narrow and whatever the mass and material of the bob; Rigid Bodies and Rotating Frames asserts that in the frame rotating with the Earth a Coriolis term must turn the plane of that swing at the rate \(\Omega\sin\lambda\). Both are predictions, and neither chapter shows either of them being measured. This chapter reports the measurements. It also reports what happens when the small arc is not small, which is the first place in this treatise where a linearization is tested rather than assumed, and what happens when the restoring torque comes from a twisted wire instead of from gravity, which is the instrument of Experiment: The Cavendish Torsion Balance.
The pendulum deserves a chapter of its own for a reason beyond the two predictions it settles: it is the first precision instrument in physics. Everything the measurement chapter Measurement, SI Units, and the Theory of Errors requires of an instrument — a reading that is a number, a calibration chain that terminates in base units, a stated uncertainty, and a systematic error budget written down and argued over — appears here first and appears complete. A pendulum is a length and a time and nothing else; that is what makes it an absolute instrument, in the technical sense that its scale is fixed by the metre and the second without reference to any other apparatus of the same kind. In four guises it delivered the first precision frequency standard (Section 34.1), the transducer that converts a force too weak to weigh into a visible angle (Section 34.2), the first absolute determination of the acceleration of free fall (Section 34.3), and the first laboratory proof, consulting nothing outside the room, that the Earth turns (Section 34.5).
Galileo and Huygens: the isochronous pendulum (1602–1673)
Tests Phenomenon 28.9 and Equation (28.7). Assuming Equation (28.8).
The claim under test is the one stated in Oscillations and Mechanical Waves as the isochronism of the small-amplitude pendulum: the period does not depend on the amplitude, nor on the bob. It is an approximate claim, and the interest of the seventeenth-century work is that both halves of it were established — Galileo found the approximation, Huygens found its limits and then removed them exactly, by changing the path from a circle to a cycloid.
Apparatus
Galileo's instrument is a bob on a cord of measured length hung from a fixed point, and, for the comparative observations, two such pendulums of equal length side by side carrying bobs of very different density — cork and lead — so that a dependence on the material of the bob would show as a growing difference of phase [Galilei:1638]. There is no clock: the counting of swings is the clock, and where an independent time is needed it comes from a water vessel weighed before and after, the technique Galileo used for the inclined plane.
Huygens' instrument is a pendulum clock: a bob on a flexible suspension, maintained against friction by a weight-driven escapement, with a dial that accumulates the count. Its distinguishing feature is a pair of rigid metal cheeks flanking the point of suspension, bent to arcs of a cycloid, against which the suspension wraps as the bob swings out [Huygens:1673]. The cheeks are the experiment; the rest is a clock.
Procedure
Release two equal pendulums together from unequal arcs and count the swings until they can be seen to separate: an amplitude dependence of the period appears as a slow accumulation of phase, so that a fractional period difference of \(\epsilon\) becomes a visible half-swing after about \(1/2\epsilon\) swings. The same protocol run with cork and lead bobs tests the independence of the bob. To test the square-root law, compare pendulums whose lengths stand in the ratio \(4:1\) and count the swings of each in the same interval.
With the clock, the procedure is to vary the driving weight, and with it the amplitude at which the escapement settles, and read the change in the rate; then to fit the cycloidal cheeks and repeat. The signature sought is not a small rate but a rate that stops moving when the amplitude is changed.
Observations and data
Galileo reports that pendulums of equal length keep step whether the arc is wide or narrow, and that a cork bob and a lead bob of the same suspension length stay together for a great many swings even though the cork is stopped by the air much sooner; and that the period varies as the square root of the length [Galilei:1638]. Huygens reports the residual he was looking for: the circular pendulum runs slow at large amplitude, so that a clock's rate depends on how hard it is driven, and the cycloidal constraint removes that dependence entirely [Huygens:1673].
Neither man quotes a number in a unit and neither quotes an uncertainty; both quote a comparison. The quantitative content of the two reports is therefore recovered by asking what a count of swings can resolve, and setting that against the exact period of the circular pendulum, which is the quantity Galileo's claim asserts to be constant and Huygens' measurement shows is not.
Quadrature of Equation (28.8) at fixed energy gives the period of a swing of angular amplitude \(\theta_{0}\) as
after the substitution \(\sin\left(\theta/2\right) =\sin\left(\theta_{0}/2\right)\sin\varphi\). The integral is the complete elliptic integral of the first kind of modulus \(k=\sin\left(\theta_{0}/2\right)\). That function is not among the results of Part II — Mathematical Methods, so nothing below leans on its properties: the entries of Table 34.1 are quoted as numerical values of Equation (34.1), and every algebraic statement made about the amplitude dependence uses instead the small-amplitude series \(T/T_{0}=1+\theta_{0}^{2}/16+11\theta_{0}^{4}/3072+\cdots\) obtained by the perturbation method of Oscillations and Mechanical Waves, which reproduces the table to the digits shown up to about \(60^\circ\) and fails by \(1.6\times 10^{-4}\) at \(90^\circ\).
| $\theta_{0}$ | $T/T_{0}$ | $\left(T-T_{0}\right)/T_{0}$ | $N_{1/2}$ | clock error |
|---|---|---|---|---|
| (swings) | (\(\mathrm{s}\) per day) | |||
| \(1^\circ\) | \(1.000019\) | \(1.9\times 10^{-5}\) | \(2.6\times 10^{4}\) | \(1.6\) |
| \(2^\circ\) | \(1.000076\) | \(7.6\times 10^{-5}\) | \(6.6\times 10^{3}\) | \(6.6\) |
| \(5^\circ\) | \(1.000476\) | \(4.8\times 10^{-4}\) | \(1.1\times 10^{3}\) | \(41\) |
| \(10^\circ\) | \(1.001907\) | \(1.9\times 10^{-3}\) | \(262\) | \(165\) |
| \(20^\circ\) | \(1.007669\) | \(7.7\times 10^{-3}\) | \(65\) | \(663\) |
| \(30^\circ\) | \(1.017409\) | \(1.7\times 10^{-2}\) | \(29\) | \(1.5\times 10^{3}\) |
| \(45^\circ\) | \(1.039973\) | \(4.0\times 10^{-2}\) | \(13\) | \(3.5\times 10^{3}\) |
| \(60^\circ\) | \(1.073182\) | \(7.3\times 10^{-2}\) | \(6.8\) | \(6.3\times 10^{3}\) |
| \(90^\circ\) | \(1.180341\) | \(1.8\times 10^{-1}\) | \(2.8\) | \(1.6\times 10^{4}\) |
Galileo's comparison, reduced. He hangs the two bobs on threads “four or five braccia” long [Galilei:1638]; a Florentine braccio is close to \(0.583\,\mathrm{m}\), so the suspension is between \(2.33\,\mathrm{m}\) and \(2.92\,\mathrm{m}\) and, by Equation (28.7), the period is between \(3.06\,\mathrm{s}\) and \(3.43\,\mathrm{s}\). The lead bob is stated to be more than a hundred times the mass of the cork one. What is counted is swings, and two pendulums can be seen to be in step or not to within perhaps a quarter of a swing; so agreement maintained over \(N\) swings bounds the fractional period difference by
which for the fifty to a hundred swings Galileo reports is \(2.5\times 10^{-3}\). That is the entire quantitative content of the observation, and Table 34.1 says exactly what it licenses: the mass independence is established at the \(2.5\times 10^{-3}\) level, and the amplitude independence is established at that level only for arcs of order \(10^\circ\) or smaller. The bound is not idle. Galileo states isochronism for wide arcs as well as narrow ones, and there it is simply false: between \(50^\circ\) and \(5^\circ\) the exact period differs by \(4.9\,\mathrm{\%}\), so the two pendulums would stand in antiphase after ten swings — half a minute, on a suspension of four braccia.
Huygens' measurement, reduced. The signature he reports is that a circular clock's rate follows its driving weight, and stops doing so once the cheeks are fitted [Huygens:1673]. In numbers: an escapement that lets the amplitude wander between \(4^\circ\) and \(6^\circ\) — a small drift, and less than the change produced by a week's accumulation of dust on the pallets — moves the fractional period excess from \(3.05\times 10^{-4}\) to \(6.86\times 10^{-4}\), a change of \(3.81\times 10^{-4}\), that is, of \(33\,\mathrm{s}\) a day. With the cheeks fitted the same drift moves the rate by nothing at all, because Equation (34.6) contains no amplitude. That is a null result of the strongest kind: not a small residual, but a term absent by construction.
A body sliding without friction under gravity along an arc of a cycloid, cusps upward, reaches the lowest point of the arc in the same time from every starting height. Unlike the circular pendulum, whose period grows with amplitude, the cycloidal pendulum is isochronous exactly and at every amplitude, up to the cusps; and a pendulum can be made to travel a cycloid by letting its suspension wrap on two cycloidal cheeks [Huygens:1673]. This is the first case in physics of a nonlinear problem solved exactly by choosing the constraint rather than approximating the motion. Rests on Proposition 28.12 and Equation (28.7).
Derivation. Derives Phenomenon 34.2. Take the cycloid generated by a circle of radius \(a\) rolling on the underside of a horizontal line, in the parametrization
whose lowest point is \(\theta=0\) and whose cusps are at \(\theta=\pm\pi\). Differentiating, \(\dd x=a\left(1+\cos\theta\right)\dd\theta\) and \(\dd y=a\sin\theta\,\dd\theta\), so that
Measuring the arc \(s\) from the lowest point gives \(s=4a\sin\left(\theta/2\right)\), and the height above that point is
The height is therefore exactly quadratic in the arc length, with no higher corrections — which is the whole content of the choice of curve. The potential energy per unit mass along the path is \(gy=gs^{2}/8a\), so the tangential equation of motion is
a harmonic equation with no approximation made anywhere, whose period
is independent of \(s\) and hence of the amplitude, up to the full half-arc \(s=4a\). Comparing with \(T=2\pi\sqrt{\ell/g}\), the cycloidal pendulum swings like a circular one of length \(\ell=4a\). That a suspension of that same length, wrapping on cycloidal cheeks generated by circles of radius \(a\), does travel this curve is the geometric fact that the evolute of a cycloid is a congruent cycloid: the radius of curvature of Equation (34.3) is \(\rho=\dd s/\dd\left(\theta/2\right)=4a\cos\left(\theta/2\right)\), equal to \(4a\) at the vertex and falling to zero at the cusps, exactly as the free length of the wrapping suspension does.
∎Interpretation
Two results, not one. Galileo's is that a pendulum has a proper frequency, set by its length and by the local gravity and by nothing else — not by the amplitude, not by the mass, not by the material. That is what makes a swinging weight a standard rather than a curiosity, and it is the observational content of the harmonic approximation of Oscillations and Mechanical Waves: the mass cancels from \(\ell\ddot{\theta}=-g\sin\theta\) before any linearization is performed, which is why the cork and the lead keep step, while the amplitude survives in \(\sin\theta\) and drops out only when the equation is linearized.
Huygens' result is the sharper one, because it measures the error of that linearization instead of assuming it away. The circular pendulum's period grows with amplitude as \(T\left(A\right)=T\left(1+A^{2}/16+O(A^{4})\right)\), derived in Oscillations and Mechanical Waves; at an amplitude of \(7^\circ\), that is about one part in \(10^{3}\), and a clock whose amplitude wanders by that much gains or loses of order \(81\,\mathrm{s}\) a day (Table 34.1). A minute and a half a day is not a subtlety for a seventeenth-century navigator, and the cycloidal cheeks are the response: not a smaller error but a structurally zero one. The trade is instructive and recurs throughout this treatise — Huygens bought exactness in the amplitude at the cost of a new systematic, since the cheeks must be cut to the right curve and the suspension must be perfectly flexible, and neither condition is better satisfied in practice than the small-angle condition it replaced.
Primary references
[Galilei:1638] [Huygens:1673]. Huygens' Horologium Oscillatorium also contains the theory of the compound pendulum used in Section 34.3 and the proposal for a universal length standard discussed in Section 34.4.
Coulomb: the torsion pendulum (1784)
Tests Phenomenon 30.60 and Equation (30.55). Assuming Equations (28.2) and (30.21).
Replace gravity by a twisted wire and the pendulum becomes something else: an instrument whose restoring constant can be made as small as the wire is fine, and which therefore responds visibly to forces far below anything a balance can weigh. Coulomb established the law of that restoring torque and the scaling of its constant with the geometry of the wire, in the memoir that also gave him the instrument he used the following year on the electrostatic force.
Apparatus
A fine metal wire, hung vertically from a torsion head at the top of a tall case, carrying at its lower end a horizontal needle or bar of known dimensions — so that its moment of inertia \(I\) about the wire axis is computable from measured lengths and masses. The case is graduated in angle and closed against draughts; the torsion head can be turned through a read angle, so that a known twist may be imposed on the wire independently of the needle's position. Wires of several materials, and of systematically varied length \(L\) and diameter \(d\), are used in turn [Coulomb:1784].
Procedure
Two measurements, run on each wire. First, the free torsional oscillation: displace the needle, release it, and time many periods, repeating at several amplitudes. Second, the static twist: turn the torsion head and read the equilibrium deflection of the needle, comparing the angle applied at the top with the angle taken up at the bottom. Then vary \(L\) and \(d\) and repeat both. Nothing in either measurement requires a force standard, and that is the point of the instrument.
Observations and data
The restoring torque is proportional to the angle of twist and to nothing else: the torsional period is independent of the amplitude, and the static deflection is linear in the applied twist. The constant of proportionality depends on the wire alone — not on the body hung from it — and varies as the fourth power of the diameter and inversely as the length [Coulomb:1784].
Coulomb, like Galileo, states his results as proportionalities rather than as numbers with units. Put into the modern form, they are two: the restoring torque is \(-\kappa\varphi\) with \(\kappa\) independent of \(\varphi\), and \(\kappa\propto d^{4}/L\) with the same two exponents for iron and for brass. Table 34.2 gives the second of these in numbers, and the paragraph after it checks the first against a published instrument; the derivation of both from linear elasticity is Phenomenon 34.3 below, and the two equations used here, Equations (34.8) and (34.9), are established there.
| material | $d$ | $L$ | $\kappa$ | $T$ |
|---|---|---|---|---|
| (\(\mathrm{mm}\)) | (\(\mathrm{m}\)) | (\(\mathrm{N}\,\mathrm{m}/\mathrm{rad}\)) | (\(\mathrm{s}\)) | |
| iron | \(0.05\) | \(1.0\) | \(5.02\times 10^{-8}\) | \(116\) |
| iron | \(0.10\) | \(1.0\) | \(8.03\times 10^{-7}\) | \(28.9\) |
| iron | \(0.20\) | \(1.0\) | \(1.29\times 10^{-5}\) | \(7.2\) |
| iron | \(0.40\) | \(1.0\) | \(2.06\times 10^{-4}\) | \(1.8\) |
| iron | \(0.10\) | \(2.0\) | \(4.02\times 10^{-7}\) | \(40.9\) |
| copper | \(0.10\) | \(1.0\) | \(4.64\times 10^{-7}\) | \(38.0\) |
The amplitude independence is the sharper of the two findings, and its reach is worth stating in numbers because it is what separates this instrument from the gravity pendulum of Section 34.1. Table 34.1 shows the gravity pendulum's period rising by \(18\,\mathrm{\%}\) between a vanishing arc and \(90^\circ\). Over the same range — indeed over several complete revolutions of the needle — Coulomb finds no change in the torsional period at all, and Equation (34.8) says why: the constitutive relation is linear in the shear strain, and the strain a torsion pendulum reaches is tiny. Three full turns of the \(0.10\,\mathrm{mm}\) iron wire of Table 34.2 give a surface shear strain of only \(\gamma=\left(d/2\right)\varphi/L=9.4\times 10^{-4}\) — a tenth of a per cent. The angle is enormous and the deformation is not, which is why a wire may be twisted through three revolutions and still answer with a linear torque.
Run the relation backwards and it calibrates. The one torsion instrument in this treatise whose constants are published is Cavendish's: two balls of \(0.730\,\mathrm{kg}\) on an arm of half-length \(a=0.914\,\mathrm{m}\), so that \(I=2ma^{2}=1.22\,\mathrm{kg}\,\mathrm{m}^{2}\), and torsional periods of \(9.0\times 10^{2}\,\mathrm{s}\) and \(4.5\times 10^{2}\,\mathrm{s}\) for the two suspensions he used [Cavendish:1798]. Equation (34.9) turns those into \(\kappa=5.9\times 10^{-5}\,\mathrm{N}\,\mathrm{m}/\mathrm{rad}\) and \(2.4\times 10^{-4}\,\mathrm{N}\,\mathrm{m}/\mathrm{rad}\) without any force ever being applied. Inverting Equation (34.8) with the copper modulus asks, for the first of these, a wire of diameter
and the fourth root makes the answer nearly indifferent to the assumed length — halving \(L\) gives \(0.28\,\mathrm{mm}\). Both are ordinary diameters for drawn wire. The dynamical calibration and the elastic formula therefore agree on the same instrument, which is the consistency Coulomb's memoir asserts and Experiment: The Cavendish Torsion Balance depends on.
A wire twisted about its own axis resists with a torque proportional to the angle, with a constant \(\kappa\) that scales as the fourth power of the wire's diameter and inversely as its length, the remaining factor being a property of the material [Coulomb:1784]. Halving the diameter therefore makes the suspension sixteen times softer, which is what allows a torsion pendulum to be tuned over orders of magnitude in sensitivity by a change of wire alone, and what makes it the transducer of choice for forces at the limit of detection. Rests on Phenomenon 30.60 and Equation (30.55).
Derivation. Derives Phenomenon 34.3. Twist a circular shaft of radius \(R\), length \(L\) and shear modulus \(\mu\) through a total angle \(\phi\). Plane cross-sections stay plane and rotate rigidly, so a filament at radius \(r\) is sheared by \(\gamma=r\phi/L\) and carries the shear stress \(\tau=\mu\gamma=\mu r\phi/L\) — the linear elastic constitutive relation of Continuum Mechanics and Elasticity. Integrating its moment over the section,
so that \(\kappa=\mu J/L\propto d^{4}/L\), which is the observed scaling. Two consequences follow at once. The torque is strictly linear in \(\phi\), so the equation of motion of the suspended body is \(I\ddot{\varphi}=-\kappa\varphi\) and the period \(T=2\pi\sqrt{I/\kappa}\) is amplitude-independent, as measured; and, running the argument backwards,
delivers the torque scale of the instrument from a computed moment of inertia and a timed period. No force is ever applied to calibrate it. That is what makes the torsion balance absolute, and it is the reason the same instrument can be turned on gravitation (Experiment: The Cavendish Torsion Balance) and on electrostatics [Coulomb:1785] without either measurement borrowing a scale from the other.
∎Set swinging and left alone, a torsion pendulum on a fine fibre decays far more slowly than any gravity pendulum of comparable period: quality factors of \(10^{3}\) and beyond are ordinary for a metal fibre, and much higher values are reached with fused silica. The consequence is practical and dominates the design of every experiment built on the instrument — the equilibrium position cannot be obtained by waiting for the oscillation to stop, because at a period of minutes and a quality factor of \(10^{3}\) the amplitude takes of order a day to fall by a factor \(\ee\). Cavendish already read successive turning points instead of waiting [Cavendish:1798], and every later torsion experiment has done some version of the same thing [Boys:1895] [Heyl:1930] [Gundlach:2000]. Rests on Equation (34.9) and Definition 28.19.
Derivation. Derives Phenomenon 34.4. With a linear damping torque the equation of motion is \(I\ddot{\varphi}+b\dot{\varphi}+\kappa\varphi=0\), that is \(\ddot{\varphi}+2\beta\dot{\varphi}+\omega_{0}^{2}\varphi=0\) with \(\beta=b/2I\) and \(\omega_{0}^{2}=\kappa/I\). The underdamped solution is \(\varphi=A\ee^{-\beta t}\cos\left(\omega t+\delta\right)\) with \(\omega=\sqrt{\omega_{0}^{2}-\beta^{2}}\), so the stored energy, being quadratic in the amplitude, decays as \(\ee^{-2\beta t}\). Defining the quality factor of Oscillations and Mechanical Waves as \(Q=\omega_{0}/2\beta\), the amplitude falls by a factor \(\ee\) in the time
that is, after \(Q/\pi\) periods. For a torsion period of a few minutes and \(Q\sim10^{3}\) this is of order \(10^{5}\) seconds, about a day, which is the assertion above. The same \(Q\) fixes the sharpness of the resonance, \(\Delta\omega\approx\omega_{0}/Q\).
∎It fixes one thing more, and that one decides how a torsion balance should be built. The dissipation that damps the ringdown and the fluctuation that shakes the fibre are the same physical process seen twice, so \(Q\) also sets the instrument's noise floor — and the floor is worth having in closed form.
A suspension of torsion constant \(\kappa\), moment of inertia \(I\), resonant angular frequency \(\omega_{0}=\sqrt{\kappa/I}\) and quality factor \(Q\), in equilibrium at temperature \(T\) — the one statement in this chapter in which \(T\) is a temperature and not a period — is driven by a fluctuating torque whose one-sided power spectral density is white across the band of the instrument and equal to
\(b\) being the damping coefficient of Equation (34.10). The smallest torque distinguishable in a measurement bandwidth \(\Delta f\) is accordingly
At fixed \(I\) and \(T\) the floor falls only as the fourth root of the torsion constant, but as the square root of the quality factor: softening the fibre is a weak lever and raising \(Q\) is a strong one. Rests on Equations (34.9) and (34.10).
Derivation. Derives Proposition 34.5. Add a stochastic torque to the equation of motion used for the ringdown, \(I\ddot{\varphi}+b\dot{\varphi}+\kappa\varphi=\tau(t)\). The response to a torque of angular frequency \(\omega=2\pi f\) is \(\chi\left(\omega\right) =\left(\kappa-I\omega^{2}+\ii b\omega\right)^{-1}\), so
the integral being elementary for a quadratic denominator. In thermal equilibrium the suspension holds \(\tfrac{1}{2}\kappa\avg{\varphi^{2}}=\tfrac{1}{2}k_{B}T\) by equipartition (Part XII — Statistical Mechanics), that is \(\avg{\varphi^{2}}=k_{B}T/\kappa\); comparing the two expressions gives \(S_{\tau}=4k_{B}Tb\). This is the rotational form of Nyquist's relation and an instance of the fluctuation–dissipation theorem [Kubo:1957]: the coefficient \(b\) that damps the ringdown is the same coefficient that drives the fluctuation, because dissipation and fluctuation are one process seen twice. Writing \(b=2I\beta=I\omega_{0}/Q\) from the definition of \(Q\) used in Equation (34.10), and \(I\omega_{0}=\sqrt{\kappa I}=\kappa/\omega_{0}\), gives the second form of Equation (34.11) and then Equation (34.12).
Put Cavendish's numbers in it. With \(\kappa=5.9\times 10^{-5}\,\mathrm{N}\,\mathrm{m}/\mathrm{rad}\), \(I=1.22\,\mathrm{kg}\,\mathrm{m}^{2}\) and a torsional period of \(9.0\times 10^{2}\,\mathrm{s}\), so that \(\omega_{0}=7.0\times 10^{-3}\,\mathrm{rad}/\mathrm{s}\), and taking \(Q=10^{3}\) at a room temperature \(T=293\,\mathrm{K}\), Equation (34.11) gives \(S_{\tau}=1.4\times 10^{-25}\,\mathrm{N}^{2}\,\mathrm{m}^{2}/\mathrm{Hz}\). Averaging for a day is a bandwidth \(\Delta f=1.2\times 10^{-5}\,\mathrm{Hz}\), whence \(\tau_{\min}=1.3\times 10^{-15}\,\mathrm{N}\,\mathrm{m}\) — about \(5\times 10^{-9}\) of the gravitational torque of \(2.8\times 10^{-7}\,\mathrm{N}\,\mathrm{m}\) that the instrument had to resolve, as Experiment: The Cavendish Torsion Balance reconstructs it. Thermal noise was irrelevant to Cavendish by nine orders of magnitude; it is a real limit for the modern instruments [Gundlach:2000] [Schlamminger:2008], whose signals are many orders of magnitude smaller. Two cautions belong with Equation (34.11) and are not derived here. It assumes viscous damping, a torque proportional to \(\dot{\varphi}\); a real fibre dissipates internally, which puts more noise below the resonance than Equation (34.11) predicts and less above it, and is the same physics as the frequency dependence of \(\kappa\) taken up in Phenomenon 34.6. And the relation applies to each dissipation channel separately, with its own \(b\): gas damping, clamp losses and readout damping each contribute a fluctuating torque of their own, so a measured \(Q\) that lumps them together is the right number to use only when the fibre dominates.
∎Interpretation
The torsion pendulum is the pendulum turned into a force transducer, and Equation (34.9) is why it is an absolute one: a mass distribution, a length and a time fix the torque scale, with no comparison against any other force. That is the property Experiment: The Cavendish Torsion Balance depends on — Cavendish's gravitational torque is read as \(\kappa\theta\) with \(\kappa\) obtained from the instrument's own period — and it is why the same design has served, essentially unchanged, for the electrostatic force law [Coulomb:1785], for the gravitational constant [Cavendish:1798] [Boys:1895] [Heyl:1930] [Gundlach:2000], and for the tests of the equality of inertial and gravitational mass [Eotvos:1922] [Roll:1964] [Braginsky:1972] [Schlamminger:2008] that The Equivalence Principle and Classical Tests reports.
The limitation belongs here too, because it is a property of the instrument and not of any one experiment using it.
Coulomb's law of torsion is very good but not exact. A real fibre is anelastic: its response lags the applied twist, so that the effective \(\kappa\) measured from a slow static deflection differs from the one inferred from the period of a fast oscillation, by a fractional amount that grows as the quality factor falls. Kuroda showed that this biases the time-of-swing determinations of the gravitational constant upward, by an amount of order \(1/\pi Q\) [Kuroda:1995] — some hundreds of parts per million for an ordinary metal fibre, which is the size of the discrepancy that Experiment: The Cavendish Torsion Balance reports among modern determinations and does not resolve. Rests on Equations (34.9) and (34.10).
Derivation. Derives Phenomenon 34.6. Only one thing has to be assumed about the material, and it is an observation rather than a model: over the narrow band a torsion balance uses, a metal fibre dissipates with a loss angle that is very nearly independent of frequency. Everything else follows from causality.
Let \(\hat\chi(\omega)\) be the torsional compliance, defined by \(\hat\theta(\omega)=\hat\chi(\omega)\hat\tau(\omega)\) with the convention \(\theta(t)=\Re\left[\hat\theta\,\ee^{-\ii\omega t}\right]\) of Theorem 17.78, and of SI dimension \(\mathrm{rad}/\mathrm{N}/\mathrm{m}\). The work done per cycle by a torque of amplitude \(\abs{\hat\tau}\) is \(\pi\abs{\hat\tau}^{2}\Im\hat\chi(\omega)\), since the cycle average of \(\tau\dot\theta\) is \(\tfrac{1}{2}\omega\abs{\hat\tau}^{2}\Im\hat\chi\), so dissipation requires \(\Im\hat\chi>0\) for \(\omega>0\); and the loss-angle hypothesis is
with the same \(Q\) as the ring-down of Equation (34.10). To leading order in \(1/Q\) the numerator may be evaluated with \(\Re\hat\chi\) replaced by its value \(\chi_{0}\) at the working frequency, and \(\chi\) being real in the time domain makes \(\Im\hat\chi\) odd, so \(\Im\hat\chi(\omega)=\left(\chi_{0}/Q\right)\sgn\omega\).
A constant is not integrable at infinity, so use the once-subtracted form of Equation (17.92), symmetrized over the half line by the parity of the real and imaginary parts. For any reference frequency \(\omega_{r}\),
With \(\Im\hat\chi=\chi_{0}/Q\) on the half line the substitution \(u=\omega'^{2}\) removes the \(\omega'\) in the numerator and leaves an elementary integral,
the upper limit contributing nothing because the ratio tends to one. Hence \(\Re\hat\chi(\omega)=\chi_{0}\left[1-\left(2/\pi Q\right) \ln\left(\omega/\omega_{r}\right)\right]\), and inverting to the stiffness \(\kappa=1/\Re\hat\chi\), to the same order in \(1/Q\),
From Equations (17.92) and (34.13) (a frequency-independent loss angle inserted into the subtracted dispersion relation). The fibre is therefore stiffer the faster it is twisted, by \(2/\pi Q\) per \(\ee\)-fold of frequency — which is the first assertion of Phenomenon 34.6, with the fractional discrepancy between a static deflection held for a time \(t_{s}\) and an oscillation at \(\omega_{0}\) equal to \(\left(2/\pi Q\right)\ln\left(\omega_{0} t_{s}\right)\): for \(Q=10^{3}\) and a deflection read after a few hundred seconds, some parts in \(10^{3}\).
Now the bias. The time-of-swing determination of \(G\) is Equation (33.12): with \(\omega_{1}\) and \(\omega_{2}\) the torsional frequencies in the two source-mass configurations, and \(C_{1}\neq C_{2}\) the geometrical gradients, the equation of motion of Equation (33.5) gives \(I\omega_{i}^{2}=\kappa-GC_{i}\) and the fibre constant cancels in the difference. It cancels, however, only if it is the same constant at the two frequencies. With Equation (34.16) it is not, and
The two frequencies differ only by the gravitational gradient, a fractional shift of order \(10^{-4}\), so \(\ln\left(\omega_{2}/\omega_{1}\right) =\tfrac{1}{2}\ln\left(\omega_{2}^{2}/\omega_{1}^{2}\right) \approx\left(\omega_{2}^{2}-\omega_{1}^{2}\right)/2\omega_{0}^{2}\) with \(\omega_{0}^{2}=\kappa/I\), whence
which to leading order is \(G\left(C_{1}-C_{2}\right)/\pi Q\). The experimenter divides the left side of Equation (34.17) by \(C_{1}-C_{2}\) and calls the result \(G\), so what is obtained is
From Equations (34.17) and (34.18) (substituting the residual stiffness difference and dividing by the geometrical factor). an upward bias, independent of every geometrical factor and of the sense of the difference. That is Equation (33.14), and it is the second assertion of Phenomenon 34.6: at \(Q=2\times 10^{3}\) it is \(160\) parts per million, and at the \(Q\approx10^{3}\) of a bare metal fibre, twice that.
∎Two things are quoted rather than derived. The first is the loss mechanism: Equation (34.13) is an empirical description of metals over a limited band, and why internal friction in a drawn wire should be so nearly frequency-independent is materials physics that this treatise does not carry. The second is the range of validity of Equation (34.16), which cannot hold down to \(\omega=0\): the logarithm diverges, and a genuinely static compliance is finite. What the model is entitled to describe is the difference between two frequencies within the band, which is all Equation (34.19) uses, and a static deflection only in the weaker sense of a measurement completed in a finite time \(t_{s}\), which is why the discrepancy above was quoted with \(\ln\left(\omega_{0} t_{s}\right)\) rather than with an infinite one. Kuroda's paper [Kuroda:1995] makes the same restriction.
Primary references
[Coulomb:1784]. The instrument's application to the electrostatic force is [Coulomb:1785]; its application to gravitation is Experiment: The Cavendish Torsion Balance, following [Cavendish:1798]; the anelasticity systematic is [Kuroda:1995].
Kater: the reversible pendulum (1818)
Tests Equations (28.7) and (28.9). Assuming Phenomenon 28.9.
A simple pendulum measures \(g\) in principle and not in practice: no real pendulum is a point mass on a massless string, and the correction from the actual mass distribution is precisely as uncertain as the mass distribution is. Kater's reversible pendulum removes that difficulty completely, by exploiting a theorem of Huygens to make the awkward quantity drop out of the answer. The result is the first absolute determination of the acceleration of free fall, and the ancestor of every absolute gravimeter since.
Apparatus
A rigid bar, of brass or iron, carrying two knife edges facing each other and separated by a fixed measured distance, with the two edges placed asymmetrically about the centre of mass; between them, a heavy weight that can be slid along the bar and clamped, and a smaller slider for fine trimming. The bar rests on hard flat planes so that either knife edge in turn serves as the pivot, and the whole hangs from a massive support — the compliance of the support is a systematic in its own right. The comparison clock is an astronomically regulated pendulum clock, and the distance between the knife edges is measured against a length standard [Kater:1818].
Procedure
Swing the pendulum about one knife edge and time it; invert it, swing it about the other, and time it again. Move the adjustable weight and repeat, iterating until the two periods agree. When they do, the distance between the knife edges — a length that can be measured directly to high accuracy — is the length of the simple pendulum having that period, and
follows with no other quantity involved. The period is compared with the clock by the method of coincidences: the experimental pendulum is set swinging in front of the clock pendulum with a mark visible only when the two are aligned, and the slow drift of the coincidences measures the difference of periods far more precisely than either could be timed alone.
One trap must be avoided by construction, and it is the reason the two knife edges are deliberately placed at unequal distances from the centre of mass; the derivation below identifies it.
Observations and data
Kater does not report a value of \(g\). He reports a length: the length of the pendulum vibrating seconds in the latitude of London, reduced to sea level, to vacuum, and to a stated temperature [Kater:1818]. That is the natural output of the method, since Equation (34.20) with \(T=2\,\mathrm{s}\) makes the two equivalent, and it is the form in which the seconds pendulum was proposed as a length standard — the proposal Section 34.4 takes up. His figure is quoted in inches; editorial rule 3 of this treatise admits SI only, and the conversion at the exact \(25.4\,\mathrm{mm}\) to the inch gives \(0.994138\,\mathrm{m}\). That is the figure customarily quoted from the memoir, and it is quoted here on that authority: no copy of the 1818 Philosophical Transactions paper is held in this treatise's evidence store, so its digits have not been read off the original. Nothing below turns on the last two of them. Table 34.3 carries that through to \(g\) and compares it with the modern picture; Table 34.4 is the error budget, rebuilt here in the terms of Measurement, SI Units, and the Theory of Errors and [JCGM:2008] [Taylor:1997] rather than reproduced from the memoir.
| quantity | value | |
|---|---|---|
| result | length of the seconds pendulum, London | \(0.994138\,\mathrm{m}\) |
| period $T$, exact by definition | \(2\,\mathrm{s}\) | |
| implied $g=\pi^{2}L$ | \(9.81175\,\mathrm{m}/\mathrm{s}^{2}\) | |
| figures quoted, as a relative resolution | \(2.6\times 10^{-7}\) | |
| accuracy actually achieved | $\sim10^{-5}$ | |
| modern | conventional $g_{n}$ (exact) | \(9.80665\,\mathrm{m}/\mathrm{s}^{2}\) |
| latitude formula at \(51.48^\circ\) | \(9.8118\,\mathrm{m}/\mathrm{s}^{2}\) | |
| Kater minus the latitude formula | \(-9\times 10^{-5}\,\mathrm{m}/\mathrm{s}^{2}\) | |
| the same, relative | \(-9\times 10^{-6}\) |
The gap between the fourth and fifth rows of Table 34.3 is the honest part of the entry. Kater prints seven significant figures, which asserts a resolution of \(2.6\times 10^{-7}\); nothing in the apparatus supports that, and the reason is Table 34.4. Every correction he must apply is between \(6\times 10^{-6}\) and \(1.4\times 10^{-4}\) of the answer, and a correction is never known to better than a few per cent of itself. The quadrature sum of the corrections is \(2.4\times 10^{-4}\); taking the residual uncertainty of each as a tenth of the correction gives an overall \(u_{g}/g\approx2\times 10^{-5}\), by Theorem 2.8. That is the number the last row of Table 34.3 should be compared against, and against it the agreement is unremarkable — which is the correct verdict, and a stronger statement about the method than a spuriously perfect one would be.
| correction | mechanism | $\abs{\delta g}/g$ |
|---|---|---|
| finite arc | period excess $\theta_{0}^{2}/16$ at $\theta_{0}=1.5^\circ$, doubled into $g$ | \(8.6\times 10^{-5}\) |
| air buoyancy | restoring torque reduced by $\rho_{\text{air}}/\rho_{\text{brass}}$ | \(1.4\times 10^{-4}\) |
| entrained air | added moment of inertia of the co-moving air [Stokes:1851] | $\sim1.4\times 10^{-4}$ |
| thermal expansion | $\alpha\,\Delta\vartheta$ on the knife-edge separation | \(9.5\times 10^{-5}\) |
| clock rate | comparison clock in error by \(1\,\mathrm{s}\) a day | \(2.3\times 10^{-5}\) |
| knife-edge radius | rolling rather than pivoting, $r/L$ | \(10^{-5}\) |
| free-air reduction | $2h/R_{\oplus}$ to sea level | \(6.3\times 10^{-6}\) |
| in quadrature | \(2.4\times 10^{-4}\) |
For any rigid body swinging about a horizontal axis there is a second, parallel axis — on the far side of the centre of mass, along the same perpendicular — about which the body swings with exactly the same period; and the distance between the two axes is the length of the simple pendulum that keeps that period. The relation is symmetric: each axis is the other's centre of oscillation. Huygens established it in the Horologium Oscillatorium [Huygens:1673], and it converts an otherwise intractable measurement of \(g\) into the measurement of one length and one time. Rests on Equations (28.7) and (28.9).
Derivation. Derives Phenomenon 34.8. Let the body have mass \(m\), centre of mass \(C\), and radius of gyration \(k\) about the horizontal axis through \(C\) parallel to the pivot, so that its moment of inertia about that axis is \(mk^{2}\). Suspended from a point at distance \(h\) from \(C\), it has, by the parallel-axis theorem, moment of inertia \(I=m\left(k^{2}+h^{2}\right)\) about the pivot, while gravity exerts the moment \(-mgh\sin\vartheta\). The rotational law of Newtonian Dynamics gives \(I\ddot{\vartheta}=-mgh\sin\vartheta\), whose small-amplitude period is
So the body swings exactly like a simple pendulum of length \(L\left(h\right)\), the equivalent length. Now ask which pairs of suspension distances give equal periods. Setting \(L\left(h_{1}\right)=L\left(h_{2}\right)\) and clearing denominators,
so either \(h_{1}=h_{2}\), or \(h_{1}h_{2}=k^{2}\). In the second case, with the two pivots on opposite sides of \(C\), their separation is
which is the assertion: the two axes are conjugate, and the distance between them is the equivalent length. Note what has vanished. Neither \(m\) nor \(k\) appears in Equation (34.23); the mass distribution of the bar, its knife edges, its adjustable weight and its clamps have all dropped out, and Equation (34.20) needs only the measured separation and the measured period.
The first root of Equation (34.22) is the trap. Two knife edges placed at equal distances from the centre of mass also give equal periods, for a trivial reason, and their separation \(2h\) is not the equivalent length — using it in Equation (34.20) yields a value of \(g\) that is simply wrong, with nothing in the data to signal the error. This is why the two distances are made deliberately unequal, so that the observed equality of periods can only be the conjugate root.
∎Interpretation
Equation (34.20) contains a length and a time, and nothing else. That is the whole claim, and it is a strong one: the measurement is absolute, in the same sense as the torsion calibration Equation (34.9) — it does not compare the local \(g\) against \(g\) somewhere else, and it needs no previously calibrated instrument of any kind. The reversible pendulum accordingly became the standard of absolute gravimetry for more than a century, and the worldwide network of \(g\) values it established is the empirical base for the latitude dependence that Rigid Bodies and Rotating Frames derives from the centrifugal term and the flattening of the Earth.
The residual difficulties are all systematic and all mechanical: knife edges are not lines, supports are not rigid, air is neither absent nor inert, and the bar is not at the temperature at which it was measured. Each is a correction of the same order as the accuracy sought, which is the recurring shape of a precision measurement and the reason Measurement, SI Units, and the Theory of Errors insists that an uncertainty statement is part of a result rather than a comment on it. The method was displaced only when a genuinely different one became available: dropping a corner cube in vacuum and reading its fall interferometrically, which replaces the mechanical corrections with optical ones and reaches some \(2\times 10^{-8}\,\mathrm{m}/\mathrm{s}^{2}\) [Niebauer:1995]. The free-fall instrument is treated with the kinematics it tests, in Experiment: Free Fall and Projectile Motion.
Primary references
Kater's memoir on the length of the pendulum vibrating seconds in the latitude of London, read to the Royal Society in 1818, is the primary source for this experiment [Kater:1818]. The theorem the method rests on is Huygens' [Huygens:1673]; the modern successor instrument is [Niebauer:1995].
The seconds pendulum and the metre
Before it was an instrument for measuring \(g\), the pendulum was a candidate for defining the metre. The proposal is Huygens' — a universal measure taken from the length of the pendulum that beats seconds, so that any observer anywhere could reconstruct the unit from a clock [Huygens:1673] — and it is the earliest attempt at what Measurement, SI Units, and the Theory of Errors calls a definition by reproducible procedure rather than by artefact. It was not adopted, and the reason it was not adopted is a measurement made with pendulums.
A pendulum beating seconds — one swing per second, hence a period of \(2\,\mathrm{s}\) — is about \(0.9936\,\mathrm{m}\) long at the conventional acceleration \(9.80665\,\mathrm{m}/\mathrm{s}^{2}\), within about \(0.7\,\mathrm{\%}\) of the metre. But it is not the same length everywhere: because \(g\) runs from about \(9.780\,\mathrm{m}/\mathrm{s}^{2}\) at the equator to about \(9.832\,\mathrm{m}/\mathrm{s}^{2}\) at the poles, as Rigid Bodies and Rotating Frames derives from the centrifugal term and the flattening of the Earth, the seconds pendulum is some \(5\,\mathrm{mm}\) longer at the pole than at the equator, a spread of about one part in \(190\). A length unit defined this way would therefore have depended on where it was realized. Rests on Equation (28.7) and Axiom 2.2.
Derivation. Derives Phenomenon 34.9. Inverting \(T=2\pi\sqrt{\ell/g}\) with \(T=2\,\mathrm{s}\) gives
numerically \(\ell=0.99362\,\mathrm{m}\) at \(g=9.80665\,\mathrm{m}/\mathrm{s}^{2}\), and \(\ell=0.99092\,\mathrm{m}\) and \(\ell=0.99619\,\mathrm{m}\) at the equatorial and polar values quoted above — a difference of \(5.3\,\mathrm{mm}\). The near-coincidence with the metre is a historical accident of the same kind as the near-coincidence of the metre with the ten-millionth of the meridian quadrant, and carries no physical content; the spread does. Latitude dependence of this size was well established by pendulum surveys before the metre was defined, so the defect was known in advance rather than discovered afterwards.
∎The metre was defined instead from the Earth's meridian, then from a platinum–iridium artefact, then from a krypton wavelength, and now from the speed of light and the second [BIPM:2019]. The pendulum's descendant in the modern system is not the metre but the second: a counted oscillation of a system whose frequency is a property of nature rather than of an artefact. What changed between Huygens' proposal and the present definition is not the idea but the oscillator — from a weight on a wire, whose frequency depends on where you stand, to an atomic transition, which does not.
Foucault: the pendulum at the Panthéon (1851)
Tests Phenomenon 29.61. Assuming Equations (28.7) and (29.66).
That the Earth rotates was, by 1851, not in doubt; but every argument for it was astronomical, and therefore rested on observations of things outside the laboratory. Foucault produced the missing argument: an experiment performed in a closed room, consulting nothing beyond its walls, whose result is the rotation rate of the Earth.
Apparatus
A heavy sphere — of order \(28\,\mathrm{kg}\), of lead clad in brass — hung on a fine steel wire some \(67\,\mathrm{m}\) long from the crown of the dome of the Panthéon in Paris, at a latitude close to \(48.85^\circ\). The suspension is designed to be as nearly isotropic as possible, since any asymmetry in the support gives the two horizontal directions slightly different frequencies and thereby drives the swing off a straight line. A pointer fixed to the bob writes in a low ring of damp sand laid on the floor beneath, so that successive extremes of the swing are recorded as a physical mark. Foucault had already run the experiment twice on a smaller scale, first privately in a cellar with a wire of a couple of metres and then publicly at the Paris Observatory, before moving it to the Panthéon where the long wire makes the effect unmistakable within a single session [Foucault:1851].
Procedure
The release is the delicate part, and it is done with a cord: the bob is drawn aside, tied off to a fixed point with a light thread, the system is allowed to come completely to rest, and the thread is then burned through. Pulling the bob aside by hand and letting go imparts an uncontrolled sideways impulse, which turns the intended straight swing into a narrow ellipse; and an elliptical swing precesses on its own account, at a rate set by the area of the ellipse and the anharmonicity of the motion, which is indistinguishable in the data from the effect being sought. Burning the thread releases the bob from rest along the line of the thread, with no transverse velocity. Thereafter the experiment is left alone and the direction of the swing plane is read off the sand at intervals.
Observations and data
The plane of the swing turns steadily, and it turns clockwise seen from above — the same sense in every trial, in Paris. The rate is close to \(11.3^\circ\) an hour, so that a full circuit takes about \(31.8\,\mathrm{h}\) [Foucault:1851]. The sense reverses in the southern hemisphere and the rate falls to zero on the equator, both of which follow from the derivation below and neither of which is a free parameter.
Table 34.5 collects the apparatus, the prediction and the reading, and then does with them the thing that makes this an absolute measurement: it inverts the observed rate for \(\Omega\) and recovers the length of the sidereal day from a weight, a wire and a floor.
| quantity | value | |
|---|---|---|
| apparatus | wire length $\ell$ | \(67\,\mathrm{m}\) |
| bob mass | \(28\,\mathrm{kg}\) | |
| latitude $\lambda$ | \(48.85^\circ\) | |
| swing amplitude (working) | \(3\,\mathrm{m}\) | |
| predicted | $g$ at the station | \(9.8095\,\mathrm{m}/\mathrm{s}^{2}\) |
| swing period $T_{0}=2\pi\sqrt{\ell/g}$ | \(16.42\,\mathrm{s}\) | |
| $\Omega$ (sidereal) | \(7.2921\times 10^{-5}\,\mathrm{rad}/\mathrm{s}\) | |
| precession rate $\Omega\sin\lambda$ | \(5.491\times 10^{-5}\,\mathrm{rad}/\mathrm{s}\) | |
| the same, in degrees per hour | \(11.32^\circ\) | |
| one full turn | \(1.144\times 10^{5}\,\mathrm{s}\) | |
| the same, in hours | \(31.79\,\mathrm{h}\) | |
| plane turned per swing period | \(9.0\times 10^{-4}\,\mathrm{rad}\) | |
| advance of the trace per swing period | \(2.7\,\mathrm{mm}\) | |
| observed | precession rate | \(11.3^\circ\) per hour |
| one full turn | \(31.8\,\mathrm{h}\) | |
| assigned uncertainty on the rate | \(1.2\,\mathrm{\%}\) | |
| recovered | sidereal day, from the rate and $\lambda$ | \(8.64(10)\times 10^{4}\,\mathrm{s}\) |
| sidereal day, adopted | \(8.6164\times 10^{4}\,\mathrm{s}\) |
Two features of the table carry the argument. The predicted and observed rates agree, and the recovered sidereal day sits within a fifth of its own uncertainty of the astronomical value — which is a weak test of the astronomy and a strong one of the mechanics, since nothing astronomical entered the measurement. And the numbers are small: \(2.7\,\mathrm{mm}\) of advance per swing at the working amplitude, on a trace made in sand. That is why the wire had to be \(67\,\mathrm{m}\) long. Shorten it and the rate is unchanged — the precession does not know \(\ell\) — but the swings come faster, and the advance per swing shrinks with the period, so the effect stays detectable only for as long as the pendulum keeps swinging. The advance per swing is \(A\,\Omega\sin\lambda\,T_{0}\) and therefore grows as \(A\sqrt{\ell}\): a \(2\,\mathrm{m}\) pendulum turns its plane by \(11.3^\circ\) in an hour just as the Panthéon one does, but at a \(0.2\,\mathrm{m}\) amplitude each of its \(2.8\,\mathrm{s}\) swings advances the trace by \(31\,\mu\mathrm{m}\) — ninety times less — and it is stopped by air in an hour or two.
The one systematic that can imitate the effect is not air, and not the support: it is the shape of the orbit.
Let a pendulum of length \(\ell\) traverse, in a frame that is not rotating, an ellipse of semi-axes \(a\) and \(b\) about the vertical, with \(a,b\ll\ell\). Then the anharmonicity of the suspension alone makes the ellipse precess, in the sense in which it is traversed, at
while the frequency of the motion around the ellipse is \(\omega=\omega_{0}\left[1-\left(a^{2}+b^{2}\right)/16\ell^{2}\right]\). Both reduce to known results at the two ends of the family. \(b\rightarrow0\) recovers the arc correction \(\omega=\omega_{0}\left(1-\theta_{0}^{2}/16\right)\) of Table 34.1, with \(\theta_{0}=a/\ell\). And \(a=b\) recovers the conical pendulum: the bob's total angular rate about the vertical is then \(\omega+\Omega_{\text{ell}} =\omega_{0}\left(1+a^{2}/4\ell^{2}\right)\), which is the expansion of the exact conical relation \(\omega_{\text{con}}^{2}=g/\left(\ell\cos\vartheta\right)\) at \(a=\ell\sin\vartheta\). Rests on Equations (28.8) and (34.38).
Derives Proposition 34.10. Let \(x\) and \(y\) be horizontal Cartesian coordinates of the bob with origin at the lowest point of the sphere, \(\rho^{2}=x^{2}+y^{2}\), and \(\omega_{0}^{2}=g/\ell\). The constraint puts the bob at height \(z=\ell-\sqrt{\ell^{2}-\rho^{2}} =\rho^{2}/2\ell+\rho^{4}/8\ell^{3}+O(\rho^{6})\), and differentiating it gives \(\dot{z}=S/\ell+O(\rho^{3})\) with \(S:=x\dot{x}+y\dot{y}\). Both pieces are needed: the quartic term of the potential and the \(\dot{z}^{2}\) the constraint generates are of the same order, and keeping only one of them is the commonest way to get the coefficient wrong. To fourth order in the displacement,
The Euler–Lagrange equation for \(x\) has \(\pp\Lag/\pp\dot{x}=m\dot{x}+\left(m/\ell^{2}\right)Sx\) and \(\pp\Lag/\pp x=\left(m/\ell^{2}\right)S\dot{x}-m\omega_{0}^{2}x -\left(m\omega_{0}^{2}/2\ell^{2}\right)\rho^{2}x\); the term \(\left(m/\ell^{2}\right)S\dot{x}\) cancels between the two sides, and with \(\dot{S}=\dot{x}^{2}+\dot{y}^{2}+x\ddot{x}+y\ddot{y} =v^{2}-\omega_{0}^{2}\rho^{2}\) to the order retained, both coordinates obey the same equation,
and likewise for \(y\), with \(v^{2}=\dot{x}^{2}+\dot{y}^{2}\).
Insert the unperturbed ellipse \(x=a\cos\omega t\), \(y=b\sin\omega t\) with \(\omega\approx\omega_{0}\). Then \(\rho^{2}=\left(a^{2}+b^{2}\right)/2 +\left[\left(a^{2}-b^{2}\right)/2\right]\cos2\omega t\) and \(v^{2}=\omega_{0}^{2}\left\{\left(a^{2}+b^{2}\right)/2 -\left[\left(a^{2}-b^{2}\right)/2\right]\cos2\omega t\right\}\), so
The perturbation is thus a constant plus a modulation at twice the frequency, and the modulation acts differently on a cosine and a sine. Removing the secular term as in Theorem 28.72: with \(x_{0}=a\cos\omega t\) the resonant part of \(-\varepsilon x_{0}\) is \(-a\left(\varepsilon_{0} +\varepsilon_{2}/2\right)\cos\omega t\), because \(\cos2\omega t\cos\omega t=\tfrac{1}{2}\left(\cos\omega t +\cos3\omega t\right)\); with \(y_{0}=b\sin\omega t\) it is \(-b\left(\varepsilon_{0}-\varepsilon_{2}/2\right)\sin\omega t\), because \(\cos2\omega t\sin\omega t=\tfrac{1}{2}\left(\sin3\omega t -\sin\omega t\right)\). Hence
From Equations (34.27) and (34.28) (removing the secular terms of the two components separately). The two components keep their amplitudes — the vertical angular momentum \(m\left(x\dot{y}-y\dot{x}\right)\) is conserved exactly, because Equation (34.26) is axisymmetric, and at this order it is \(mab\omega\), so the perturbation can redistribute phase but not amplitude — and acquire a growing phase difference.
That difference is a rotation of the axes. Put \(\psi=\omega_{x}t\) and \(2\delta=\left(\omega_{y}-\omega_{x}\right)t\), so that \(x=a\cos\psi\) and \(y=b\sin\left(\psi+2\delta\right)\) with \(\delta\) small over any interval short compared with its own growth time. The same ellipse turned bodily through a small angle \(\Theta\) and traversed with phase \(\psi'=\psi+\eta\) has components \(x=a\cos\psi'-\Theta b\sin\psi'\) and \(y=\Theta a\cos\psi'+b\sin\psi'\); expanding both to first order in \(\Theta\) and \(\eta\) and matching the coefficients of \(\cos\psi\) and \(\sin\psi\) in each component gives \(a\eta+\Theta b=0\) from the first and \(\Theta a+b\eta=2\delta b\) from the second, whence
Now put in the numbers. From Equation (34.29), \(\omega_{y}^{2}-\omega_{x}^{2}=-\varepsilon_{2} =3\omega_{0}^{2}\left(a^{2}-b^{2}\right)/4\ell^{2}\), so \(\omega_{y}-\omega_{x}=3\omega_{0}\left(a^{2}-b^{2}\right)/8\ell^{2}\) and \(2\delta=\left(\omega_{y}-\omega_{x}\right)t\); substituting in Equation (34.30),
which is Equation (34.25). The difference of squares cancels, so the rate depends on the area of the ellipse alone, and it changes sign with \(b\), that is with the sense of traversal.
The frequency around the ellipse is the rate of \(\psi'\), namely \(\omega=\omega_{x}+\dot{\eta} =\omega_{x}-\Omega_{\text{ell}}b/a =\omega_{x}-3\omega_{0}b^{2}/8\ell^{2}\). With \(\varepsilon_{0}+\varepsilon_{2}/2 =\omega_{0}^{2}\left(5b^{2}-a^{2}\right)/8\ell^{2}\) from Equation (34.28), so that \(\omega_{x}=\omega_{0}\left[1+\left(5b^{2}-a^{2}\right) /16\ell^{2}\right]\),
the second statement of the proposition. Both limiting checks are now computations rather than assertions. At \(b=0\), Equation (34.32) is \(\omega_{0}\left(1-\theta_{0}^{2}/16\right)\) with \(\theta_{0}=a/\ell\), and \(\Omega_{\text{ell}}\) vanishes: a plane swing does not precess. At \(a=b\) the modulation \(\varepsilon_{2}\) vanishes identically, the two components share the frequency \(\omega_{x}=\omega_{0}\left(1+a^{2}/4\ell^{2}\right)\), and that single number is the whole angular rate of the bob about the vertical, which is what the proposition asserts as \(\omega+\Omega_{\text{ell}}\) — Equations (34.31) and (34.32) sum to it. The matching Equation (34.30) is singular at \(a=b\), as it must be, since a circle has no axes whose orientation could be tracked; the rate Equation (34.31) is not, and the circular case is settled by the direct computation just given.
∎Equation (34.25) is what the burnt thread is for. At the Panthéon, \(\omega_{0}=0.3827\,\mathrm{rad}/\mathrm{s}\) and \(\ell^{2}=4489\,\mathrm{m}^{2}\), so an ellipse of semi-major axis \(3\,\mathrm{m}\) precesses at the Foucault rate itself once its minor semi-axis reaches
and contributes \(1\,\mathrm{\%}\) of it at \(b=5.7\,\mathrm{mm}\). Since \(b\simeq v_{\perp}/\omega_{0}\) for a release from the end of the swing, holding the systematic at \(1\,\mathrm{\%}\) means releasing the bob with a transverse velocity below \(2.2\,\mathrm{mm}/\mathrm{s}\), against a peak speed of \(a\omega_{0}=1.15\,\mathrm{m}/\mathrm{s}\): a fractional tolerance of \(2\times 10^{-3}\) on the release. No hand meets that, and a burnt thread does. Two further properties of Equation (34.25) are what let the residue be diagnosed rather than merely feared. It is proportional to the area of the ellipse, so it decays as the swing decays, while the Foucault rate does not; and it changes sign with the sense in which the ellipse is traversed, while the Foucault rate is clockwise in Paris whatever the release. Repeating the run with the swing started along a different azimuth, and fitting the rate against amplitude rather than quoting a single number, separates the two.
A long pendulum, released from rest and left undisturbed, does not keep its plane of swing with respect to the room. The plane turns at the rate \(\Omega\sin\lambda\), where \(\Omega\) is the Earth's sidereal angular velocity and \(\lambda\) the latitude: clockwise seen from above in the northern hemisphere, anticlockwise in the southern, one turn per sidereal day at either pole, and not at all on the equator. The room may be sealed and nothing outside it consulted, so the experiment measures \(\Omega\) from within the laboratory [Foucault:1851]. The rate depends on the latitude and not at all on the length of the pendulum, on its mass, or on the amplitude of the swing — which is what distinguishes the effect from every mechanical imperfection that could mimic it. Rests on Phenomenon 29.61 and Equation (28.7).
Derivation. Derives Phenomenon 34.11. Work in the frame rotating with the Earth at angular velocity \(\vect{\Omega}\), and use local axes at latitude \(\lambda\) with \(x\) east, \(y\) north and \(z\) up, so that \(\vect{\Omega}=\Omega\left(0,\cos\lambda,\sin\lambda\right)\). The centrifugal term is time-independent and merely redefines the local vertical and the effective \(g\), as Rigid Bodies and Rotating Frames shows; absorbing it, the equation of motion of the bob is
For small horizontal excursions the vertical velocity is second order, so \(\dot{\vect{r}}\simeq\left(\dot{x},\dot{y},0\right)\) and
The vertical component is taken up by the tension and does not move the bob; only the vertical component of \(\vect{\Omega}\) survives in the horizontal motion. Writing \(\Omega_{z}=\Omega\sin\lambda\) and using the restoring acceleration \(-\omega_{0}^{2}\left(x,y\right)\) with \(\omega_{0}^{2}=g/\ell\) from Oscillations and Mechanical Waves,
Combine them into \(\zeta=x+\ii y\):
Now substitute \(\zeta=\ee^{-\ii\Omega_{z}t}\eta\), which is precisely the statement that \(\eta\) is the motion referred to axes turning at the rate \(\Omega_{z}\). Differentiating, \(\dot{\zeta}=\ee^{-\ii\Omega_{z}t}\left(\dot{\eta} -\ii\Omega_{z}\eta\right)\) and \(\ddot{\zeta}=\ee^{-\ii\Omega_{z}t}\left(\ddot{\eta} -2\ii\Omega_{z}\dot{\eta}-\Omega_{z}^{2}\eta\right)\); substituting into Equation (34.37), the terms in \(\dot{\eta}\) cancel identically and there remains
In the rotating axes the bob is an ordinary plane pendulum, of frequency \(\sqrt{\omega_{0}^{2}+\Omega_{z}^{2}}\), which for \(\Omega_{z}\ll\omega_{0}\) is \(\omega_{0}\) to a fractional accuracy of \(\left(\Omega_{z}/\omega_{0}\right)^{2}/2\) — for the Panthéon pendulum, of order \(10^{-9}\). Its plane is fixed in those axes; therefore in the room the plane turns at \(-\Omega_{z}\), that is, at \(\Omega\sin\lambda\) in the clockwise sense seen from above when \(\lambda>0\). Numerically, with the sidereal day \(86164\,\mathrm{s}=2\pi/\Omega\) and \(\lambda=48.85^\circ\), the period of one full turn is
about \(31.8\,\mathrm{h}\), or \(11.3^\circ\) an hour — the observed value. Two features of the result carry the physical weight. First, \(\ell\), \(m\) and the amplitude are absent, so the effect cannot be confused with a property of the particular pendulum. Second, only \(\sin\lambda\) enters: the horizontal component of \(\vect{\Omega}\) drops out entirely, having been absorbed by the tension, which is why the experiment reports the component of the Earth's rotation along the local vertical and not the whole of it.
∎Interpretation
The measurement is local in the strict sense: the apparatus is a weight, a wire and a floor, and the result is the rotation rate of the planet they stand on. This is the sharpest available form of the statement of Rigid Bodies and Rotating Frames that rotation, unlike uniform motion, is detectable from inside a sealed laboratory; and it is evidence about the Earth obtained without astronomy, which is what made the public demonstration decisive for people whom the astronomical argument had not moved.
There is a second reading of the same data, and it is the one that matters later in this treatise. The swing plane is not held fixed by any force — Equation (34.38) contains no torque about the vertical — yet after one sidereal day, during which the station has been carried once around its latitude circle, the plane has turned through \(\Omega\sin\lambda\times2\pi/\Omega=2\pi\sin\lambda\) and not through \(2\pi\). The deficit is
which is exactly the solid angle subtended at the centre of the Earth by the spherical cap the latitude circle bounds — a purely geometric quantity, independent of \(\ell\), of \(m\), of \(g\) and of \(\Omega\) itself. That the deficit is the holonomy of the Levi-Civita connection of the sphere, so that the pendulum is performing parallel transport (Definition 13.146) rather than merely producing the same number, is a theorem this treatise does not yet carry: it needs the identification of the holonomy around a closed curve with the integral of the Gaussian curvature over the region enclosed, which Differentiable Manifolds, Tensors, and Curvature does not prove. Taken with that theorem, the Foucault pendulum is the oldest laboratory measurement of a geometric phase and belongs with the fibre-bundle language of Lie Groups, Lie Algebras, and Fibre Bundles; taken without it, Equation (34.40) is an arithmetical coincidence that this chapter has checked and not explained.
Foucault himself drew the operational conclusion and built the instrument that follows from it: if a swinging mass keeps its plane against a rotating frame, a spinning mass will keep its axis, which is the gyroscope [Foucault:1852]. The optical descendant is Sagnac's interferometer [Sagnac:1913], in which the rotation appears as a fringe shift rather than a slow turning; ring-laser and fibre gyroscopes are its industrial form, and they measure \(\Omega\sin\lambda\) in seconds where the pendulum needs hours.
Primary references
[Foucault:1851]. The gyroscope sequel is [Foucault:1852]; the Coriolis term whose consequence this is was published by Coriolis in 1835 [Coriolis:1835]; the optical successor is [Sagnac:1913].
The pendulum as the first precision instrument
| Experiment | Measures | Result |
|---|---|---|
| Galileo, Huygens 1602–1673 | period | the period is set by the length and by $g$ alone, to \(2.5\times 10^{-3}\) in the seventeenth-century data; the amplitude dependence is removed exactly on a cycloid (Table 34.1) |
| Coulomb 1784 | torque | restoring torque linear in the twist, with $\kappa\propto d^{4}/L$; the torque scale follows from a moment of inertia and a period (Table 34.2) |
| Kater 1818 | $g$ | an absolute value of the local free-fall acceleration from one length and one time: \(9.81175\,\mathrm{m}/\mathrm{s}^{2}\) at London, good to about \(2\times 10^{-5}\) (Table 34.3) |
| Foucault 1851 | $\Omega\sin\lambda$ | the swing plane turns at about \(11.3^\circ\) an hour at Paris, one circuit in about \(31.8\,\mathrm{h}\), whence a sidereal day of \(8.64(10)\times 10^{4}\,\mathrm{s}\) (Table 34.5) |
Table 34.6 makes the common structure visible. In each case a quantity that cannot be weighed, timed or seen directly — an elastic constant, the strength of gravity, the rotation of a planet — is converted into the period or the azimuth of an oscillation, and the oscillation is counted. Counting is the one laboratory operation that can be done without error, which is why every subsequent increase in precision in this treatise has taken the same form: the Josephson and quantum-Hall standards of Experiment: The Quantum Hall Effect, the atomic clocks behind Experiment: Precision Spectroscopy and Atomic Clocks, and the fringe counting of Experiment: Gravitational Waves are all frequency measurements wearing different hardware.
The lineage is also literal. The suspension that isolates a gravitational-wave interferometer's mirrors is a pendulum, chosen for exactly the reason Coulomb's wire was chosen — a low resonant frequency decouples the mass from ground motion above it — and its thermal noise is governed by the quality factor of Equation (34.10) through the same fluctuation–dissipation argument [Kubo:1957] that gives Equation (34.12), which is why such suspensions are made of fused silica and not of metal: the lever that matters is \(Q\). The torsion pendulum of Section 34.2 remains the instrument of choice for gravitation at laboratory scale [Gundlach:2000] [Schlamminger:2008] and hence for the least well determined of the constants, as Experiment: The Cavendish Torsion Balance reports. And the reversible pendulum's successor, the free-fall gravimeter, still measures \(g\) the way Kater did in principle: a length and a time [Niebauer:1995].