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Measuring the Gravitational Constant

November 14, 2014

The geek motif is not a recent phenomenon. Henry Cavendish (1731-1810), the prominent British scientist who lived more than two
Henry Cavendish centuries ago, could rightly be counted among the geeks. Technically competent, as are geeks of the present day, he was also exceptionally shy. He was especially shy around women, so much so that he communicated with his female servants only by notes; and, according to Wikipedia, he is said to have had a back door added to his house so that he could leave without seeing the housekeeper.

(Illustration by George Wilson, from "The Life of the Hon. Henry Cavendish," 1851, via Wikimedia Commons.)

Cavendish is credited with the discovery of hydrogen, which he made by dissolving metals in acid; for example, Fe + 2HClFeCl2 + H2. He also reacted carbonates with acid to get carbon dioxide, which was called "fixed air" at that time. He concluded that air was a 1:4 mixture of oxygen ("dephlogisticated air") and nitrogen ("phlogisticated air"). In his experiments, Cavendish found a gas that wasn't oxygen or nitrogen existing at about the 1% level in air. This was later found by William Ramsay and Lord Rayleigh to be argon.

Jumping from chemistry to physics, Cavendish did experiments on electricity, discovering electrical principles that were later credited to others since he rarely published. Our knowledge of these experiments have come through James Clerk Maxwell, who was named the first Cavendish Professor of Physics at Cambridge University in 1871. Maxwell collected, edited, and published the research papers of Cavendish in 1879.

Of special interest to physics is the Cavendish experiment in which he determined the density of the Earth; and, indirectly, the value of the gravitational constant G in Newton's law of universal gravitation,

 Newton's Law of Universal Gravitation

where F is the gravitational force between two masses, m1 and m1, and r is the distance between them. Although Cavendish calculated only the density of the Earth, and not G, the constant could be calculated from his data. The present CODATA value of G is 6.67384 x 10-11 m3 kg-1 s-2.

Cavendish obtained the apparatus to do this experiment through a serendipitous chain of events, as he detailed in the first paragraph of his paper on the measurement (see figure). Cavendish improved the apparatus to lessen the affects of temperature and air currents.

First paragraph of the  1798 paper by Henry Cavendish on the density of the Earth
(Henry Cavendish, Esq. F. R. S. and A. S., "Experiments to Determine the Density of the Earth," Phil. Trans. R. Soc. Lond., vol. 88 (January 1, 1798), pp. 469-526, via Wikimedia Commons.)

I wrote about the Cavendish experiment in an earlier article (Big G, October 12, 2010). A schematic diagram of the experiment is shown in the figure. The larger spheres, cast from lead, were each 350 pounds, while the smaller spheres, also of lead, were each 1.61 pounds. Such large masses notwithstanding, the force acting on the torsion wire was just a ten millionth of a newton. The torsion constant was obtained by measuring the resonant frequency of the torsion assembly. The Cavendish value of G was within a percent of the presently accepted value.

Schematic of the torsion balance used in the Cavendish experimentSchematic of the torsion balance used in the 1798 Cavendish experiment.

The larger masses (M) are stationary lead balls, and the smaller balls (m) are also made of lead.

Gravitational force causes the masses to twist the wire with torsion coefficient κ, which allows a force measurement.

(Diagram by Chris Burks, via Wikimedia Commons.)

Since gravitational force is so weak, the gravitational constant is the fundamental constant known to the least precision. There have been many attempts to improve on the Cavendish experiment, one recent example of which is given in the references.[1-2] As the figure shows, the experimental values for G are consistent to just a few hundred parts per million.

Figure captionMeasurement of G using torsion balances (circles) and other methods (squares). The shaded area is the one-sigma confidence interval of the 2010 CODATA value. Marked is the value of the University of Zürich experiment described below.

(Fig. 5 of ref. 3, modified, via arXiv.[3])

A G experiment designed without a torsion balance was done by a team of seven scientists at the University of Zürich. This experiment, concluded in 2006, was a decade in the making, and a summary of the experiment has been posted on arXiv.[3] The original motivation for the experiment was the detection of a possible "fifth force."[4] A fifth force has not been found, but the apparatus, shown conceptually in the figure, produced a new measurement of the gravitational constant.

Schematic diagram of the Zurich gravitational constant apparatusSchematic diagram of the University of Zürich gravitational constant measurement apparatus.

In configuration A, the weight of the top mass is increased, and the weight of the bottom mass is reduced, by the gravitational attraction of the large masses, while the reverse happens in B.

(Simplified version of fig. 5 of ref. 3, via arXiv.[3])

Attention to detail is important in a measurement like this. The large masses were 500 liter stainless steel vessels each filled with 6,760 kilograms of liquid mercury. Liquid mercury was used, since homogeneity of the masses could be assured, and mercury has a relatively high density (13.54 g/cc). The smaller masses were gold-plated oxygen-free, high-conductivity (OFHC) copper about 1100 grams in mass. Copper was chosen to avoid any magnetic forces.[3]

The gravitational signal in this experiment was large, with the differential force being about 7.7 μN. The value of G obtained was 6.674 252 x 10−11 m3 kg−1 s−2 with a relative standard uncertainty of 18 x 10−6.[3]

References:

  1. Harold V. Parks, James E. Faller, "A Simple Pendulum Determination of the Gravitational Constant," arXiv Preprint (September 7, 2010).
  2. Eugenie Samuel Reich, "G-whizzes disagree over gravity," Nature, vol. 466, no. 7310 (August 26, 2010), p.1030.
  3. S. Schlamminger, R.E. Pixley, F. Nolting, J. Schurr, and U. Straumann, "Reflections on a Measurement of the Gravitational Constant Using a Beam Balance and 13 Tons of Mercury," arXiv Preprint Server, July, 19, 2014
  4. Ephraim Fischbach, Daniel Sudarsky, Aaron Szafer, Carrick Talmadge, and S. H. Aronson, "Reanalysis of the Eötvös experiment," Phys. Rev. Lett., vol. 56, no. no. 1 (January 6, 1986), pp.3-6.

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