Foucault pendulum

It was the morning hours of January 3, 1851, when Jean Bernard Léon Foucault became the first person — without looking up at the sky — to observe that the Earth rotates. The 32-year-old French experimental physicist was observing the oscillations of a two-meter-long string pendulum in the basement of his Paris home. He had experimented for many months to suspend this pendulum so that it could move almost frictionlessly. And so he noticed that the plane of oscillation of the pendulum (that is, the plane in which the pendulum swings) appeared to rotate with respect to the floor of the cellar, by 11 degrees of arc per hour. It was the birth of the first terrestrial experiment that proved the rotation of the Earth without “a look at the sky”. Since an external force acting on the pendulum could be excluded, it was therefore not the pendulum but the floor of the cellar (i.e. the earth) that changed its direction.

Later, Foucault demonstrated the experiment at the Paris Observatory with a 12-meter-long pendulum and then (every Thursday from March 31, 1851) at the Paris Panthéon, the French Hall of Fame with the tombs of famous scientists, artists and statesmen, with a 67-meter-long pendulum and a 29-kg pendulum bob. The famous Italian writer Umberto Eco was inspired by this pendulum to write his world-famous novel “The Foucault Pendulum”.

So it is clear: The oscillation plane of the Foucault pendulum only appears to rotate. In fact, it maintains its direction while one circles the pendulum on the globe. This phenomenon is comprehensible e.g. by the following mental experiment, if one imagines the pendulum suspended at the north pole. Here the pendulum always swings in the same direction, while an observer on the rotating earth moves once in a circle on a day below it. The direction of swinging consequently changes by 15 degrees per hour. In the direction of the equator the hourly deviation decreases (here in Dresden — approximately on the 51st latitude 11.7 degrees per hour). At the equator the oscillation axis of the Foucault pendulum does not turn at all!

Following the technical data and details of the Foucault pendulum in the MATHEMATICS ADVENTURE LAND:

  • Geographic Breadth: 51^\circ 2' 31,5240'' (51,04209^\circ)
  • Placed are 71 stones to illustrate the rotation.
  • Pendulum bob: ball bearing sphere 100\mathrm{Cr}6, diameter d=15\mathrm{cm}, mass about 13.8\mathrm{kg}.
  • Pendulum rope: dyneemata 4\mathrm{mm}.
  • Rope length l\approx 18\mathrm m.
  • Period of oscillation of the pendulum: T=2\pi\sqrt{\frac{l}{g}}=2\pi\sqrt{\frac{18\mathrm m}{9,81\mathrm m/\mathrm s^2}}=8,51\mathrm s, with acceleration due to gravity g=9,81\mathrm m/\mathrm s^2.
  • Lagerung: Nadellager, kardanisch.

The Foucault pendulum was set up in MATHEMATICS ADVENTURE LAND with the kind support of the Dresden Max Planck Institute for Chemical Physics of Solids in an old elevator shaft from the 3rd floor of the Technical Collections to the 9th floor in the 48\mathrm high Ernemannturm.


And now … the mathematics of it:

Only at the geographic poles — through which the axis of rotation of the earth passes — the plane of oscillation of the pendulum turns exactly by 360^\circ in 24 hours. Strictly speaking, it is 23 hours 56 minutes and 4 seconds (23.93\mathrm h). This is, in fact, the length of a sidereal day of the Earth. The angular velocity of the rotation of the pendulum plane decreases now to zero up to the equator. The mathematical relation is described with the sine of the latitude:

    \[\omega=\omega_0\sin(\varphi)\]

with the angular velocity \omega (in ^\circ/\mathrm h), the angular velocity at the pole \omega_0 of 360^\circ/23,93\mathrm h\approx 15^\circ/\mathrm h, and the latitude \varphi.

Now what does this look like at a pole? The latitude of the poles is 90^\circ. Thus \omega=\omega_0\sin(90^\circ)=\omega_0\approx 15^\circ/\mathrm h.

The equator has zero latitude. Thus we likewise get \omega=\omega_0\sin(0^\circ)=0, so there is no rotation of the plane of oscillation at the equator.

The location of Foucault’s pendulum in the MATHEMATICS ADVENTURE LAND of the Dresden Technical Collections in Junghansstraße is at the position 51.04209^\circ north latitude. How fast does our pendulum turn here in one hour? We get:

    \[\omega=\omega_0\sin(51,04209^\circ)\approx 11,7^\circ/\mathrm h.\]

The 71 stones in the full circle of our experiment with Foucault’s pendulum are placed at a distance of about 5.1^\circ from each other (see the figure 1 below) and are knocked over by the swinging pendulum.

But how long must one wait in each case until the next of the stones is knocked over by the pendulum?

In one hour the pendulum turns by 11.7^\circ. So in 26 minutes the next stone should fall. — But this is not true, because the stones are on gap. If the pendulum knocks over a stone on one side, the next one falls on the opposite side. So the pendulum plane has to turn only about 2.55^\circ to hit the next stone. This takes about 13 minutes.

Figure 1: Foucault’s pendulum in the MATHEMATICS ADVENTURE LAND