ThatsMaths

That’s Maths in The Irish Times this week (TM050, or Search for “thatsmaths” at irishtimes.com) is about how a simple pendulum can demonstrate the rotation of the Earth.

Spectators gathered in Paris in March 1851 were astonished to witness visible evidence of the Earth’s rotation. With a simple apparatus comprising a heavy ball swinging on a wire, Léon Foucault showed how the Earth rotates on its axis. His demonstration caused a sensation, and Foucault achieved instant and lasting fame.

The announcement of the experiment read “You are invited to see the Earth spinning … ”. A bob of mass 28 kg was suspended from the dome of the Panthéon by a 67 metre wire. The pendulum swung through a diameter of about six metres, and the position was indicated on a large circular scale. Demonstrations were held daily, and attracted large crowds.

Following Foucault’s demonstration, pendulum mania raged across Europe and the United States, and the experiment was repeated hundreds of times. Many of these attempts were done without due care. The London Literary Gazette reported on several cases in which “to the horror of the spectators, the Earth has been shown to turn the wrong way”. These errors probably resulted from elliptical bob trajectories due to incorrect starting conditions or to stray air currents disturbing the movement of the bob.

The observed change in the swing plane of the pendulum is often stated to be due to the Earth turning beneath it. This is roughly correct, but it is an over-simplification. The turning rate for a pendulum swinging at the North Pole is one revolution per day. At other locations the rate depends on the latitude and at the Equator there is no turning at all. Thus, after one day, the swing plane does not return to its original position. At Paris the turning period is about 31 hours. The mathematical term for this phenomenon is “anholonomy”, and it has been a source of confusion ever since Foucault’s demonstration.

Foucault Pendulum in Dublin

In September 1851, the American Journal of Science surveyed several pendulum demonstrations in Europe and America. It included details of the experiments carried out in Dublin by two reverends, Joseph Galbraith and Samuel Haughton. They were close contemporaries and life-long collaborators, both Fellows of Trinity College. They were well-known for their many mathematical textbooks, which were widely used and which earned them handsome royalties.

Galbraith and Haughton replicated the pendulum experiment shortly after Foucault had reported his findings. Their experiment was done at the engine-factory of the Dublin and Kingstown Railway, where Samuel’s cousin Wilfred Haughton was chief engineer. They also analysed the effects of ellipticity of the trajectory and derived a mathematical expression for the precession due to this effect.

The pendulum length for the experiments of Galbraith and Haughton was 35 feet and its swing length 4 feet. The bob was an iron sphere, 30 lbs in weight, with a downward-pointing spike to indicate its position on a scale. The theoretical precession rate at the latitude of Dublin is 12.07 degrees per hour. The mean rate observed in the experiments was 11.9 degrees per hour, which is surprisingly accurate considering the many possible sources of error. According to an article in the Philosophical Magazine, “Messrs. Galbraith and Haughton … have pursued their research with all imaginable precautions”. Their impressive results confirm this view.

Sources

[1] Tobin, William, 2003: The Life and Science of Leon Foucault. Camb. Univ. Press, 338pp. ISBN: 9-780-521-80855-2.

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