# Foucault pendulum

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## Foucault pendulum:

see pendulum**pendulum,**

a mass, called a bob, suspended from a fixed point so that it can swing in an arc determined by its momentum and the force of gravity. The length of a pendulum is the distance from the point of suspension to the center of gravity of the bob (see center of mass).

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## Foucault pendulum

A pendulum or swinging weight, supported by a long wire, by which J. B. L. Foucault demonstrated in 1851 the rotation of Earth on its axis. Foucault used a 62-lb (28-kg) iron ball suspended on about a 200-ft (60-m) wire in the Pantheon in Paris. The upper support of the wire restrains the wire only in the vertical direction. The bob is set swinging along a meridian in pure translation (no lateral or circular motion). In the Northern Hemisphere the plane of swing appears to turn clockwise; in the Southern Hemisphere it appears to turn counterclockwise, the rate being 15 degrees times the sine of the local latitude per sidereal hour. Thus, at the Equator the plane of swing is carried around by Earth and the pendulum shows no apparent rotation; at either pole the plane of swing remains fixed in space while Earth completes one rotation each sidereal day. *See* Pendulum

## Foucault Pendulum

a pendulum used to demonstrate the diurnal rotation of the earth. A Foucault pendulum is a massive bob suspended on a wire or string whose upper end is secured—for example, by means of a universal joint—in such a way that the pendulum may swing in any vertical plane. If the pendulum is deflected from the vertical and released with zero initial velocity, the plane of swing will remain fixed with respect to the stars—that is, with respect to the inertial frame of reference associated with the stars—because the force of gravity and the force of the tension of the wire that act on the bob always lie in the plane of swing and cannot cause that plane to rotate. However, an observer located on the earth and rotating together with the earth will see that the plane of swing slowly rotates, with respect to the earth’s surface, in the direction opposite that of the earth’s rotation. This effect confirms the diurnal rotation of the earth.

At the north pole or the south pole, the plane of swing of a Foucault pendulum rotates through 360° in a sidereal day, that is, through 15° in a sidereal hour. At a point on the earth’s surface whose geographic latitude is equal to ϕ, the horizontal plane rotates about the vertical with an angular velocity of ω sin ϕ, where ω is the angular velocity of the earth. Therefore, the apparent angular speed of rotation (in degrees per sidereal hour) of the plane of swing of a Foucault pendulum at latitude ϕ has the value ω_{L} = 15° sin ϕ; that is, the smaller the value of ϕ, the lower the angular speed of rotation of the plane of swing. At the equator, the plane of swing does not rotate. In the southern hemisphere, the plane of swing is observed to rotate in the direction opposite that observed in the northern hemisphere.

The motion of a Foucault pendulum is studied theoretically by introducing the Coriolis force to take into account the diurnal rotation of the earth. More precise calculations show that the pendulum wire does not move in a single plane but describes a conical surface and that, when the pendulum is released from the point of maximum deflection, it always passes to the right of the equilibrium position in the northern hemisphere. For ω_{L}, a refined calculation yields the value

ω_{L}= 15°[1 – 3/8(*a*/*l*)^{2}]sin ϕ

where *a* is the amplitude of swing and *I* is the length of the wire. The larger the value of *I*, the smaller the additional term that reduces the angular speed of rotation. Therefore, for experimental demonstrations, it is advisable to use a Foucault pendulum with the longest possible wire, that is, a wire several tens of meters long. The first such pendulum, which was installed by J. B. L. Foucault in the Panthéon in Paris in 1851, had a length of 67 m. The length of the Foucault pendulum in St. Isaac’s Cathedral in Leningrad is 98 m.

### REFERENCES

Bukhgol’ts, N. N.*Osnovnoi kurs teoreticheskoi mekhaniki*, part 1. Moscow, 1972. Chapter 4, section 39.

Verin, A.

*Opyt Fuko*. Leningrad-Moscow, 1934.