The Foucault Pendulum and Earth’s Rotation

In 1851, Léon Foucault hung a 67-meter wire from the dome of the Panthéon in Paris, attached a heavy iron ball to its end, and gently set it swinging. Over the next several hours, observers watched the plane of oscillation slowly rotate — not because any force was twisting the pendulum, but because the Earth beneath their feet was turning. This elegant demonstration gave humanity its first direct, laboratory-scale proof of Earth’s rotation, without telescopes or astronomical observation.

Why the Plane Stays Fixed in Space

Newton’s first law tells us that a body free of net torque will preserve its angular momentum — it will keep swinging in the same plane relative to the distant stars. The Foucault pendulum is an almost ideal realization of this: the forces acting on the bob (gravity downward, string tension along the wire) produce no net torque about the vertical axis that would rotate the oscillation plane. So the plane truly is fixed in the inertial frame. What rotates is the Earth underneath it. Foucault’s genius lay in recognizing this and turning it into an experiment anyone could witness.

Equations of Motion in the Rotating Frame

An observer standing on Earth lives in a rotating, non-inertial frame. To write Newton’s second law in this frame, two fictitious accelerations must be added:

$$ddot{mathbf{r}} = mathbf{g} – 2boldsymbol{Omega} times dot{mathbf{r}} – boldsymbol{Omega} times (boldsymbol{Omega} times mathbf{r})$$

The second term is the Coriolis acceleration and the third is the centrifugal acceleration. For a laboratory-scale pendulum the centrifugal term is roughly $Omega^2 R approx 0.034,text{m/s}^2$ — about 0.3% of $g$ — and is absorbed into an effective gravitational field. The dominant new effect is Coriolis. Projecting onto horizontal coordinates $x$ (east) and $y$ (north) and keeping only the Coriolis term:

$$ddot{x} = -omega_0^2 x + 2Omega_z dot{y}$$
$$ddot{y} = -omega_0^2 y – 2Omega_z dot{x}$$

where $omega_0 = sqrt{g/L}$ is the natural angular frequency of the pendulum and $Omega_z = Omega_text{Earth}sinvarphi$ is the vertical component of Earth’s rotation vector at geographic latitude $varphi$. This system can be solved by writing $zeta = x + iy$ and seeking solutions of the form $e^{ialpha t}$; the result shows that the pendulum oscillates in a plane that itself rotates at angular rate $Omega_z$.

Precession Rate and Latitude Dependence

The time for one complete rotation of the oscillation plane is:

$$T_text{prec} = frac{2pi}{Omega_z} = frac{T_text{Earth}}{sinvarphi} = frac{24,text{h}}{sinvarphi}$$

At the North Pole the pendulum rotates clockwise (as seen from above) at exactly the same rate as the celestial sphere appears to rotate — one revolution per sidereal day. At the equator, the vertical component of $boldsymbol{Omega}$ vanishes, so the Coriolis force acts entirely in the horizontal plane without twisting the oscillation plane, and no precession is observed.

Pendulum Period and the Need for Length

The oscillation period of a simple pendulum is:

$$T = 2pisqrt{frac{L}{g}}$$

For a 67-meter pendulum this gives $T approx 16.4,text{s}$. To observe precession, the pendulum must complete many swings before air resistance damps the motion to imperceptibility. This is why Foucault pendulums are always long (tens of meters), heavy (tens of kilograms), and hung in still indoor air. A short, light pendulum would simply stop before the rotation became visible. The length also matters because a longer pendulum has a longer period and thus a lower amplitude loss per swing, giving more time to accumulate observable precession.

Initial Conditions and the Charron Ring

If the pendulum is given any sideways flick at release, its path becomes elliptical rather than linear, and the ellipse itself rotates — masking or distorting the Foucault precession. Foucault’s original solution was to tie the bob back with a thread, wait for it to hang perfectly still, then burn the thread with a flame so the bob was released without any lateral impulse. Modern installations use a Charron ring: a ring that holds the bob at one point of its arc; when the ring is released (mechanically or by burning a thin cord), the bob swings freely in a clean vertical plane. Controlling initial conditions is as important as the physics itself.

Historical Background

Léon Foucault (1819–1868) first tested the idea in his cellar using a 2-meter pendulum in January 1851. Satisfied by what he saw, he approached the French government; with Napoleon III’s backing, he staged the public demonstration at the Panthéon on 3 February 1851. The news spread rapidly — within months, Foucault pendulums were installed in observatories and public halls across Europe and America. Foucault himself went on to invent the gyroscope the same year, exploiting the same principle — the conservation of the direction of angular momentum in the inertial frame. He also measured the speed of light with a rotating mirror and discovered eddy currents in conductors.

Where to See a Foucault Pendulum Today

Foucault pendulums are a staple of science museums worldwide. Notable installations include the Science Museum in London, the Smithsonian National Air and Space Museum in Washington D.C., the Musée des Arts et Métiers in Paris (which holds Foucault’s original wire and bob), and numerous university physics buildings. Each swings at a precession rate set by its city’s latitude, offering every visitor a direct, unhurried view of the planet’s rotation — the same spectacle that astonished Paris in 1851.

References

1. Foucault, L. (1851). Démonstration physique du mouvement de rotation de la Terre au moyen du pendule. Comptes Rendus de l’Académie des Sciences, 32, 135–138.
2. Landau, L. D., & Lifshitz, E. M. (1976). Mechanics (3rd ed., §39: Motion in a non-inertial frame). Pergamon Press.
3. Goldstein, H., Poole, C., & Safko, J. (2002). Classical Mechanics (3rd ed., Ch. 4: The Kinematics of Rigid Body Motion). Addison-Wesley.
4. Amir-Moéz, A. R. (1961). Introduction to the theory of the Foucault Pendulum. The Mathematics Teacher, 54(5), 329–332.
5. Kamerlingh Onnes, H. (1879). Nieuwe bewijzen voor de aswenteling der aarde (Doctoral dissertation). University of Groningen.