THE MICHELL-CAVENDISH EXPERIMENT

Laurent Hodges, Department of Physics and Astronomy, Iowa State University, Ames, IA

Copyright 1998 by Laurent Hodges

 

The "Cavendish apparatus" for determining the gravitational constant has always seemed to me – and, surely, to many other physics teachers – to be one of the jewels of physics, a marvel of simplicity and beauty. I never fail to show it to my students when we are learning about gravitation. Our textbooks generally credit Henry Cavendish (1731-1810) with having been the first scientist to determine the value of the G. However, his experiment is often referred to as having "weighed the earth" because from the knowledge of R, the earth's radius, and the gravitational field strength g at the surface of the earth, it is possible to determine the mass M of the earth.

Recently, as I was preparing notes on the gravitational interaction for an introductory calculus-level physics course at Iowa State University, I became curious as to how Cavendish expressed his results. Did he give a numerical value for G or for M? How accurate was his value, and in what units did he express his result? Not finding an answer to these questions in any of the textbooks that I checked, I decided to read Cavendish's original paper in the Philosophical Transactions for 1798, a copy of which resides in our library.

I recommend reading famous original papers because they always seem to contain unexpected surprises. It is a pleasure to read Balmer's paper on his formula for the spectral lines of hydrogen and encounter his glowing encomium for Avogadro's experimental techniques: the fact that a simple formula fit the data to better than 1 part in 40,000 suggested to Balmer that Avogadro's precision must have been better than 1 part in 40,000, and this impressed him greatly. On the other hand, Thomson's 1897 paper on the discovery of the electron contains dozens and dozens of numbers but no units, something most of us would never let our students get away with.

Cavendish's paper did not disappoint me in this respect. There were three major surprises awaiting me:

(1) First, Cavendish did not determine either G or M: he determined the average density of the earth, which he expressed as a specific gravity, or a ratio of the earth's density to the density of water.

(2) Second, while he presents his raw data very completely and carefully, Cavendish made an error in his calculations and gave the average density as 5.48 times that of water (the accepted value today is 5.518) when his data clearly give the slightly smaller value 5.45.

(3) Third, and most surprisingly, the famous "Cavendish experiment" and "Cavendish apparatus" were not designed by Cavendish, but by a geologist named John Michell.

 

MICHELL'S APPARATUS

Cavendish gives Michell full credit in a long introduction describing Michell's experiment and apparatus. In fact, the very first paragraph of Cavendish's paper to the Royal Society summarizes the circumstances:

"Many years ago, the late Rev. John Michell, of this Society, contrived a method of determining the density of the earth, by rendering sensible the attraction of small quantities of matter; but, as he was engaged in other pursuits, he did not complete the apparatus till a short time before his death, and did not live to make any experiments with it. After his death, the apparatus came to the Rev. Francis John Hyde Wollaston, Jacksonian professor at Cambridge, who, not having conveniences for making experiments with it, in the manner he could wish, was so good as to give it to me."

Those of us familiar with the modern tabletop "Cavendish apparatus" sold today may be surprised to learn that Michell's apparatus was much larger. The wire of the torsional pendulum was 40 inches long and supported a wooden arm 6 feet long from whose ends hung lead balls 2 inches in diameter, this part all enclosed in a wooden case "to defend it from the wind." Michell's lead balls on the outside were 8 inches in diameter.

By the time Michell's apparatus came into Cavendish's possession, the wooden arm was warped, and needed to be replaced. Cavendish also decided to place the whole apparatus in a room which remained constantly shut, moving the weights from the outside and observing the motion of the arm from the outside using a telescope, and this required some further changes in Michell's apparatus. Consequently, the apparatus on which his experiments were carried out was mainly rebuilt.

 

CAVENDISH'S RESULTS

Cavendish's paper contains a long and impressive discussion of how he carried out his experiments and how he investigated and sought to reduce sources of errors.

The table in which Cavendish summarized his experiments lists 29 different determinations of the earth's density (relative to that of water), ranging from 4.88 to 5.79. Their average is 5.45 with a standard deviation of 0.22, although Cavendish incorrectly reports the average as 5.48.(1) Cavendish expresses the opinion that "it seems very unlikely that the density of the earth should differ from 5.48 by so much as 1/14 of the whole." Actually, his (corrected) average of 5.45 is only 1.3 % lower than the currently accepted value of the density of the earth.

Nowhere in Cavendish's paper does he make any reference to either the gravitational constant G or the earth's mass. Assuming that he had very accurate values for the radius of the earth and the value of g, his (corrected) result, which is 1.3 % low, would correspond to a 1.3 % underestimate of M or a 1.3 % overestimate in the value of G, corresponding to the values 5.90 x 1024 kg and 6.76 x 10-11 N · m/kg in SI units.

 

JOHN MICHELL

The name of John Michell is not well known to most physicists, but he merits entries in most encyclopedias and scientific biographical dictionaries, where he is generally referred to as a geologist.

Michell, who lived from 1724 to 1793, was elected to the Royal Society in 1760 – the same year as Henry Cavendish – and became the Woodwardian professor of geology at Cambridge two years later. He is best known for having invented the torsion balance – a key device in his gravitational apparatus – and for having been "the father of seismology" (Isaac Asimov’s term) as a result of his study of the earthquake. He also wrote a treatise on "artificial magnets" which explained how to make magnets that were stronger than any found in nature, something that was known to magnet-makers but apparently not previously published.

Michell also did important work in astronomy. He was the first to point out that double stars were much more common that one would expect from a random distribution of stars in the sky, so that many of them were probably binary system. He also made the first reasonable estimate of the distance to a star outside the solar system; his estimate of the distance to Vega was off by only a factor of 4.

Finally Michell was apparently the first person to describe, in a 1784 paper, the possibility of black holes (the modern term, not his). He argued, using the concept of light as particles, that light could not escape from a sufficiently massive star, but pointed out that the existence of such a star could still be inferred from its gravitational attraction on nearby objects. Laplace is usually given credit as the first to describe the concept of a black hole, but this was in his 1796 book on celestial mechanics, 12 years after Michell’s ideas were published.

It is not clear exactly when Michell devised his gravitational experiment, but perhaps it was in the early 1780s. Cavendish, writing in 1798, says it was "many years ago" although Michell did not complete his apparatus until "a short time before his death," which was in 1793. Coulomb investigated the electric force using a torsion balance, and is usually credited with having invented it, but Cavendish states in a footnote that Michell had described his torsion balance to Cavendish before the 1785 publication of Coulomb's results. Most references give Michell and Coulomb equal credit for having separately invented this balance.

 

WHO DESERVES THE CREDIT?

The evidence is clear that Michell and Cavendish both deserve credit for the "Cavendish experiment," which should probably be referred to as the "Michell-Cavendish experiment." It was Michell who thought the experiment up, including inventing the torsion balance that is at the heart of the experiment. It was Michell who built the original apparatus for the experiment, an apparatus clearly capable of producing the results desired. The fact that the purpose of the experiment was to determine the specific gravity (density) of the earth suggests that geology rather than theoretical physics was the motivation.

On the other hand, Cavendish recognized the value of the experiment, rebuilt the apparatus and made minor improvements, and carried out very precise measurements that yielded an excellent value of the density of the earth. To appreciate Cavendish's accuracy it is only necessary to compare his results with those of later scientists, to see how slowly his results were improved upon:

1798 5.45 1.3 % low H. Cavendish

1838 5.49 0.5 % low F. Reich

1842 5.67 2.7 % high F. Baily

1852 5.58 1.1 % high F. Reich

1883 5.56 0.8 % high A. Cornu and J. Baille

1895 5.53 0.2 % high C. V. Boys

It would have been interesting to see how accurate a measurement Michell could have made with his apparatus, but it would be surprising had he done as well as Cavendish.

 

REFERENCE

 

  1. I thought perhaps no one had ever checked Cavendish's numbers and discovered this error, until I read the following in the famous 11th edition of the Encyclopedia Britannica (1911, volume 5, page 581): "The figure he gives for the specific gravity of the earth is 5.48, water being 1, but in fact the mean of the 29 results he records works out to 5.448." Perhaps Cavendish originally calculated the average as 5.448 and subsequently accidentally dropped one of the 4s. His figure of 5.48 is repeated several times in his paper, so it is not a typographical error.

 

CHRONOLOGY OF JOHN MICHELL

• Lived from 1724 to 1793.

• Usually described as a geologist ("the father of seismology") or as an astronomer

• Studied at Cambridge University and became a fellow.

• 1750: Published a book on how to make artificial magnets, previously a secret known to only a few magnet-makers. Michell pointed out that they had many advantages over lodestones: cheapness, abundance, strength, shape, and restorability.

• 1760: Published a fundamental paper, Conjectures Concerning the Cause, and Observations upon the Phenomena of Earthquakes for which he became known as "the father of seismology." In this paper, which followed the great Lisbon earthquake (1755) by a few years, Michell assigned the cause of earthquakes to the force generated when water suddenly met subterranean fires. He appreciated that such a force would generate waves in the Earth's crust and tried to estimate the velocity of these, giving a not unreasonable figure of 1200 miles per hour. Finally, Michell showed various means to determine the point of origin of the earthquake, the concept of the epicenter.

• 1762: Appointed Woodward Professor of Geology

• 1764: left academic life to take up a post as rector at Thornhill, Yorkshire, in 1764.

• 1767: Published a paper on double stars, pointing out with originality and insight that there are far too many of them to result from a random scattering and therefore they must in many cases constitute a genuine binary system. He also devised a method for calculating the distance of the stars, giving the first reasonable estimate of the distance to a star, Vega, based on its apparent brightness. He estimated it to be 460,000 astronomical units away, about 1/4 the actual distance.

• 1784: Argued in a paper in Philosophical Transactions that sufficiently massive stars (400-500 times the mass of the sun) would not allow light to leave, but that these dark stars might nevertheless be detected by their gravitational effect on nearby stars.

• 1790: Constructed a torsion balance (he is considered the co-inventor, with Coulomb, of this balance) and related apparatus to measure gravitational attraction and thus the mean density of the Earth.

• 1793: Michell died on April 9 at Thornhill.

• 1798: Five years after Michell's death, Henry Cavendish determined the mean density of the earth using Michell's apparatus, which he had to rebuild.