The Foucault pendulum - the physics (and maths) involved

A Foucault pendulum demonstrates the rotation of the earth – but the details are subtle.

It's easy enough to understand the motion of a pendulum suspended at one of the Earth's poles, because in this case the point of suspension is not accelerating. At any other latitude, the point of suspension accelerates and this introduces some more complicated time-dependent forces acting on the pendulum. At the poles, the (apparent) precession period is 23.9 hours, whereas elsewhere it is longer by a factor of the reciprocal sine of the latitude. But why? And is the precession uniform? On the way, we'll derive expressions for the fictitious forces mentioned in the introduction.

This page is a physical analysis of the motion. I have kept it as simple as I could (though see footnote), but it does use vector calculus – without this tool, the analysis would be very long and awkward. Before you start on the analysis below, see the Introduction to the Foucault pendulum.

We treat the earth as rotating about its axis with angular velocity Ω relative to an inertial frame. At a point O on the surface with latitude λ, we choose a Cartesian coordinate system with the x axis horizontal and South, the y axis horizontal and East and the z axis vertical and up. In the diagram, the i,k plane (the North-South, vertical plane at the position of our pendulum) is shaded. This coordinate system is not inertial: it is rotating with the Earth, so not only the position, ro, of its origin, but also i, j and k are all varying in time

P is the point on the earth's axis closest to O and ro is the vector from P to O. If the position vector of a moving particle is r relative to O, then its position vector relative to P is rp = r + ro. Further let r be sufficiently small that the gravitational field does not vary significantly between O and r.

Now P is in an inertial frame, so the equation of motion for the particle is:

       mr"p = mG + F          (1)

where m is the mass of the particle, G is the gravitational field (i.e. force per unit mass) at O, F is the sum of all forces other than gravity, and dashes signify derivatives with respect to time.

diagram of Earth

(1) is an equation of motion in the inertial frame, and it is quite simple. However, r is a vector in a non-inertial frame. We shall now write the equation of motion in terms of r, so it will refer to motion in the rotating frame. The extra terms that will appear when we make the substitution will be the fictitious forces mentioned in the introduction.

Point O rotates with the angular velocity of the earth Ω, so

       r"o = – Ω2 ro = Ω2ro(sin λ i − cos λ k)           (2)
Consider now a pendulum consisting of a particle at the end of a string length a attached to a point with coordinates in the O frame (0,0,L). Suppose initially that the pendulum were stationary and vertical in that frame. The tension To is the only external force and:

       To = mgk           (4)

where g is the (local) apparent gravitational acceleration at O. (Note that g and G are neither parallel nor equal in magnitude because of the rotation of the earth. In Sydney, the centripetal acceleration due to the Earth's rotation (acG - g) has a magnitude of 28 mm.s-1: small compared to g ≈ 9800 mm.s-1 and therefore often neglected. In the Southern hemisphere, g points North of the centre of the Earth, as shown. See Why g is not parallel to Fg for an explanation.) For this vertical pendulum,

diagram of pendulum

      mr" = mG + mgk + mΩ2 ro(sin λ i − cos λ k)           (5).

The last term in (5) is the first of our fictitious forces, called the centrifugal force. Called fictitious because it doesn't exist: r"p is the acceleration in an inertial frame and, in (1), there is no centrifugal force. r" is the acceleration in a noninertial frame and, in such frames, Newton's first and second laws do not apply.

This 'pendulum', remember, is not moving with respect to the earth: it's a plumb bob and it just hangs vertically. So r, r' and r" are all zero. Thus for this case, (5) gives:

       G + Ω2ro(sin λ i − cos λ k) = − gk           (6).

In this equation, G is the gravitational field, −gk is the apparent gravitational acceleration, and the difference is the centripetal acceleration at that latitude, r"o = Ω2ro(sin λ i − cos λ k) = − Ω2ro.

Now consider a moving pendulum with tension T, and use Newton’s second law in an inertial frame:

       mr"p = T + mG          (7).

Now we need to relate the acceleration r"p in the inertial frame to the r", the (apparent) acceleration in the frame on the surface of the earth. We write

       r = rxi + ryj + rzk

remembering that the i,j,k coordinates vary: they rotate with Ω = − Ω cos λ i − Ω sin λ k. Now rp = ro + r so

      rp = ro + rxi + ryj + rzk

so    r'p = r'o + (rx)'i + (ry)'j + (rz)'k + rx i' + ry j'+ rzk'

Now i' is the motion of a point at i relative to O due to the rotation Ω of the frame so i' = Ω X i and similarly j' = Ω X j and k' = Ω X k, so

       r' = {(rx)' i + (ry)' j + (rz)' k} + Ω X r

which, by definition, = r' + Ω X r and so

      r"p = r"o + r" + 2 Ω X r' + Ω X (Ω X r)           (8).

For a pendulum small compared with the earth, r is much less than ro (typically a million times) so the last term in (8) is much less r"o which, as we have seen is tens of mm.s−2. So, neglecting this last term, and noting that r"o= Ω X (Ω X ro):

      r"p = r" + Ω X (Ω X ro) + 2 Ω X r'              (8').

where the first term is the apparent acceleration in the frame, the second is the centripetal acceleration and the last is the Coriolis acceleration. (6), (7) and (8) together then give the equation of motion of the pendulum in the frame of the earth's surface:

       m r" = T − mgk − 2m Ω X r'           (9).

Note the significance of the terms: the first is the string tension, the second is the apparent weight in the rotating frame and the third term, which depends on the velocity of the pendulum and on the earth's rotation, is the other fictitious force, the Coriolis force. What has happened to the centrifugal force? Note the approximation made above: for altitudes small compared with the radius of the earth, the centripetal acceleration is not only small (tens of mm.s−2) but always has the same amplitude and direction in the earth frame. It is therefore 'included' in g. One could say that the 'centrifugal force' is the difference between G, the gravitational field, and g, the apparent gravitational acceleration. The Coriolis force depends on r', the velocity in the earth frame and so cannot be thus 'included'. Of course, it is very small, because Ω is only 2π radians per day, or 0.000073 s−1. Which is why the pendulum precesses so slowly.

Resolving (9) in coordinates fixed with respect to the earth gives:

       mx" = Tx + 2mΩ sin λ y'                 (9.1),

       my" = Ty − 2mΩ (sin λ x' + cos λ z')       (9.2)

and    mz" = Tz − mg + 2mΩ cos λ y'           (9.3).

For a swing of small angle, z << a, motion in the z (vertical) direction is small and so z' and z" are approximately zero. Equation (9.3) therefore becomes:

       Tz = mg − 2mΩ cos λ y'           (9.3*),

whose last term is very small.

For small oscillations (x, y, z all << a), the components of the tension T in the string are

       Tx = − T x/a , . . . Ty = − T y/a, . . . Tz = T(L − z)/a           (10)

(where, to first order, Tz = T = mg.) Let us define the constant ω2 = g/a and substitute (10) and (9.3*) in (9.1) and (9.2), taking advantage of z' approx = 0 :

       x" − 2Ω sin λ y' + ω2x = 0           (9.1*)

       y" + 2Ω sin λ x' + ω2y = 0           (9.2*).

Now let us define the complex variable ζ = x + iy = |ζ|exp(i φ) (which represents the position of the pendulum in the horizontal plane). Multiplying (9.2*) by i and adding it to (9.1*) gives the differential equation for motion in the horizontal plane:

       ζ" + 2iΩ sin λ . ζ' + ω2ζ = 0           (11).

This is a version of the equation of the harmonic oscillator, but note that the coefficient of the the first derivative of ζ is purely imaginary and so it does not contribute damping.

It may be verified by substitution that the solution to (11) is:

       ζ = {Aexp(iωt) + Bexp(−iωt)} exp(−iΩt sin λ)           (12)

where ω2 has been neglected in comparison with Ω2 and where the complex constants A and B depend on the initial conditions of position and velocity. (Two simple cases are A = +/-B, both real, corresponding to motion passing through the equilibrium point. Note, however, that if the pendulum is released from a point of maximum amplitude, it never passes through ζ = 0.)

Equation (12) is readily interpreted: if the earth were not rotating (Ω = 0), the solution is just that for a simple pendulum with period t = 2π/ω whose elliptical path is given by

Subject to the approximations already made, the axis of this ellipse does not vary with time. The same situation arises at the equator (where λ = 0). With Ω not equal to 0 and at latitude λ not equal to 0, the complex vector ζs rotates (in the frame of the earth) with angular frequency − Ω sin λ, i.e. with a period 2π/(Ω sin λ) = Tearth/(sin λ) where Tearth is the period of rotation of the earth*. Before we check the direction of precession, the author congratulates readers who have persevered and offers in felicitation this puzzle: why does the longest day of the year have neither the earliest sunrise nor the latest sunset? A clue is to be found early in the introductory page. Now back to the precession: Ω is positive (the earth rotates anti-clockwise as seen from the North Pole); λ is negative in the southern hemisphere, so − Ω sin λ is positive: the pendulum should precess anti-clockwise (from above) in the frame of the earth when viewed from the Southern hemisphere.

Complications

As usual, the real situation is more complicated: a finite pendulum which does not swing in a plane traces an ellipse whose axis precesses (in the direction of the orbit) even in a Newtonian frame. If the orbit of a pendulum passes through the origin, we see from symmetry that it should not precess in a Newtonian frame. To demonstrate the precession due to the Earth's rotation, the pendulum should move slowly over a small angle. Any asymmetries in the support, launching or drag of the pendulum which would give rise to elliptical orbits must be removed. This is one reason why Foucault pendula are usually long, and firmly and symmetrically mounted.

Finally, a complication that applies in principle to all pendulums, including those of clocks. Taking x and y components of equation (8) shows an asymmetry in the x and y directions:

       x" − 2Ω sin (λ) y'+((g/a)−Ω2 sin (λ)2) x = 0, and

       y" + 2Ω sin (λ) x'+((g/a)−Ω2) y = 0

Thus, at positions other than the poles, there is a small difference in the pendulum frequencies for the NS and EW directions. Ω/2π is of course about once per day, and for Foucault pendula, (g/a)1/2/2π is of order once every several seconds, so this variation is small. For a grandfather clock, (g/a)1/2/2π is faster and so the variation with orientation is smaller still. It is occasionally said that one might detect the difference in the timekeeping of a good pendulum clock when it is rotated between a NS and an EW swing. However, other effects, such as the angle of the floor, may be larger.

* Special cases. The special cases when the pendulum swings NS or EW are of course simpler. If one assumes that the precession rate is constant (we have just seen that this is not quite true), one can estimate the precession rate by considering one of these special cases. This is done in Richard Feynman's Lectures on Physics (vol 1).

Consider a time at which the pendulum swings North-South. At the Northern end of the swing, the pendulum bob is further from the axis (Southern hemisphere) by a distance Z.sin λ, where Z is the amplitude of the swing, so the support point overtakes the pendulum bob with a relative horizontal speed ZΩ = 2πA.sin λ/Tearth. Divide the circumference 2πA of the envelope of the pendulum's path relative to the earth by this speed and one has an estimate of the period of precession Tearth/sin λ.

Thanks to Norman Phillips of Merrimack USA, Jacques Gilbert of Canada and Cleon Teunissen of the Netherlands for suggesting improvements to an earlier version of this discussion.

Links to other sites with information on the Foucault pendulum:

A review of Umberto Eco's novel "Foucault's Pendulum". In my opinion, the central joke is indeed funny and had me laughing out loud several times towards the end. It is also filled with interesting scholarly references. However I am not convinced it was funny enough nor sufficiently well researched to justify the tedium regularly encountered on the way.


1996. Modified 9/8/04 J.Wolfe@unsw.edu.au, phone 61- 2-9385 4954 (UT + 10, +11 Oct-Mar).
School of Physics, University of New South Wales, Sydney, Australia.

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