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### The Gravitational Constant

July 31, 2023 Gravitation is an important physical principle, but its*gravity*shouldn't prevent us from making a few jokes.

Why did the female physicist suspend her gravitation research for a short time? She was gravid.The first joke could have been based on a true story, but the concept of anti-gravity is fiction. Gravitational force is always attractive, and the equivalence of the gravitational properties of antimatter and normal matter has been confirmed to a high precision.[1] As I wrote in a previous article (Fictional Materials, April 21, 2016), H. G. Wells (1866-1946) introduced the fictional material, Cavorite, in his 1901 novel, The First Men in the Moon. Cavorite has an anti-gravity property that enables a voyage to the Moon in a spherical spaceship having movable sheets of Cavorite to allow steered propulsion.

I've been reading this book about anti-gravity and I can't seem to put it down.

*While H. G. Wells was a successful author with an estimated net worth in today's money of about $5 million, the concept of the starving author is truer today than in his time.There are so many books available, that any one author is drowning in a sea of many authors; and, things will only get worse. This blog, less the personal anecdotes from my research career, can be written nearly as well by an artificial intelligence agent. This is true for magazine articles and novels as well, and it's happening all the time.(Illustration from the 1901 first edition of The First Men in the Moon, via Wikimedia Commons, showing weightlessness. Click for larger image.)*

The launching point for gravitation in modern physics was the 1687 publication of Philosophiae Naturalis Principia Mathematica, commonly called

*The Principia*, by Isaac Newton (1642-1727). Newton's law of universal gravitation states that there's an attractive force between every bit of matter in the universe and every other, and that the force is proportional to the product of their masses divided by the square of the distance between their centers (an inverse square law). This force

**F**is mathematically written as

,n which

**m**and

_{1}**m**are the masses of the objects,

_{2}**r**is the distance between their centers, and

**G**is the gravitational constant. The commonly accepted value of

**G**is now 6.67430 x 10

^{-11}N·m

^{2}/kg

^{2}.[2] The first measurement of the gravitational constant was done a century after Newton's Principia by English chemist and physicist, Henry Cavendish (1731-1810). This Cavendish experiment, done in 1797-1798, was actually designed to measure the density of the Earth, but

**G**pops out with a simple calculation. Since the gravitational force is so weak, Cavendish used a torsion balance with two 2-inch-diameter 1.61-pound (0.73 kilogram) lead spheres at the ends of a six foot (1.8 meter) wooden rod acting against two massive twelve inch, 348-pound (158 kilogram) lead spheres (see figure). The gravitational attraction between the small and large spheres caused just a 0.16" deflection of the balance.

*Diagram of the torsion balance used in the Cavendish experiment.In this figure, *

*Θ*is the deflection caused by the gravitational attraction of the two smaller lead spheres to the two larger ones.Thin wires, with small torsion coefficients

**k**, give more deflection; but, any wire needs to hold the weight of the two 1.61 pound (0.73 kg) lead spheres and their supporting wooden rod.(Wikimedia Commons image by Chris Burks. Click for larger image.)

Cavendish performed many experiments on electricity, and James Clerk Maxwell (1831-1879) was interested enough to edit his research notes a century later.[3]

*(Wellcome Trust image M0014171 of Henry Cavendish, from a drawing in the British Museum, via Wikimedia Commons. Click for larger image.)*

Henry Cavendish was an interesting man. He was extremely shy, he dressed in clothes of an earlier period, and he was not able to converse with women, communicating with his female servants only in writing, and having a back staircase to avoid meeting any of them. In 2001, neurologist, Oliver Sacks (1933-2015), speculated that Cavendish had Asperger syndrome (AS), an opinion embraced by other professionals.[4-5] There's a present realization that AS people make good scientists, engineers, mathematicians and computer programmers, but not much research has been done on this topic. A century after Cavendish, in 1894, Charles Vernon Boys (1855-1944) determined

**G**to the unprecedented precision of five significant digits.[6] Boys created his torsion balance with fine fused quartz threads that he created using a crossbow to shoot quartz rods with molten quartz at their tip.[6] As I wrote in an earlier article (Strength of Materials, May 11, 2020), materials free of surface cracks, such as freshly-drawn glass fibers, have high strength. Gravity was a puzzle, also, to the ancients, and a timeline of gravity appears in a 2020 arXiv paper.[7] Always notable is Aristotle's conception of gravity in which the

*four elements*seek their natural place, the natural place of matter (Earth) being at the center of the universe; which, at his time, was the center of the Earth. Fire's place is, of course, in the heavens. Gravitational force is small; and, as a consequence, the gravitational constant is known to a very small precision, six significant digits, as compared with the twelve significant digits of the fine structure constant. That's one reason why there are frequent experiments to get better values. In 2018, Katelyn Horstman, now at the California Institute of Technology (Pasadena, California), and Virginia Trimble of the University of California Irvine (Irvine, California) did an analysis of papers on the gravitational constant with the results shown in the figure.[8]

*Papers on the gravitational constant published before and after 1938. The United States is seen to be a world leader in such research. (Created using Gnumeric from data in ref. 8.)[8]*

### References:

- M. J. Borchert, J. A. Devlin, S. R. Erlewein, M. Fleck, J. A. Harrington, B. M. Latacz, F. Voelksen, E. J. Wursten, F. Abbass, M. A. Bohman, A. H. Mooser, M. Wiesinger, C. Will, K. Blaum, Y. Matsuda, C. Ospelkaus, W. Quint, J. Walz, Y. Yamazaki, C. Smorra, and S. Ulmer, "A 16-parts-per-trillion measurement of the antiproton-to-proton charge–mass ratio," Nature, vol. 601 (January 5, 2022), pp. 53-57, https://doi.org/10.1038/s41586-021-04203-w.
- Newtonian constant of gravitation, CODATA Internationally recommended 2018 values of the Fundamental Physical Constants, NIST website.
- Isobel Falconer, "Editing Cavendish: Maxwell and The Electrical Researches of Henry Cavendish," arXiv, April 28, 2015, https://doi.org/10.48550/arXiv.1504.07437.
- Oliver Sacks, "Henry Cavendish: An early case of Asperger’s syndrome?." Neurology, vol. 57, no. 7 (October 9, 2001), pp. 1347ff., DOI: https://doi.org/10.1212/WNL.57.7.1347.
- Hugo Lidbetter, "Henry Cavendish and Asperger’s syndrome: A new understanding of the scientist," Personality and Individual Differences, vol. 46, no. 8 (June, 2008), pp. 784-793, https://doi.org/10.1016/j.paid.2009.01.032.
- Isobel Falconer, "Historical Notes: The Gravitational Constant," Mathematics Today, vol. 58, no. 4 (In press, 2022), pp. 126-127. Also at arXiv, https://doi.org/10.48550/arXiv.2306.06411.
- Arshia Anjum, Sriman Srisa, and Saran Mishra, "The Timeline Of Gravity," arXiv, November 20, 2020, https://doi.org/10.48550/arXiv.2011.14014.
- Katelyn Horstman and Virginia Trimble, "A citation history of measurements of Newtons constant of Gravity," arXiv, November 26, 2018, https://doi.org/10.48550/arXiv.1811.10556.

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