Q60C - How Cavendish didn’t measure G and Boys did
Newton’s law
Newton stated his law of gravity in terms of proportionalities; i.e. the force is proportional to the mass and inversely proportional to the square of the distance. The ‘gravitational constant’ implicit in this law did not make its appearance in research papers and text-books until about 200 years after the Principia. We do not even know who first denoted this constant by the symbol G, or when. It was not Newton, and it was certainly not Henry Cavendish whose ‘experiment to measure G ’ falsely appears in many physics text-books. What Cavendish set out to measure in 1798 was the mass of Earth in order to estimate its mean density. His paper mentions neither G nor even g. His experiment was important as being the first to depend on an accurate measurement of the small deflection produced by the tiny gravitational force between objects of everyday size.

The gravitational force between two 5 kg spheres whose centres are separated by 10 cm is a fraction of a millionth of a newton. This force, equivalent to less than 100 mg weight would have been barely detectable, let alone accurately measurable, by the best chemical balances of the day. However, the French physicist Coulomb had recently developed a highly sensitive ‘torsion balance’ for his experiments to establish the laws of attraction and repulsion of electric charges. An English investigator, Rev. John Michell (who was the first scientist to speculate about the existence of what we now call ‘black holes’) built a large-scale version of the torsion balance for experiments on the gravitational force. He died before these could be carried out, and his apparatus was passed on to Cavendish, who installed a modified version of it in a specially built sealed room at his London home 

Cavendish paid great attention to environmental conditions – freedom from draughts and temperature stability (to avoid convection currents) was essential. He also made careful estimates of the corrections to be made for the gravitational forces on the test spheres due to all the rest of the apparatus (e.g. the mahogany cabinet which enclosed the spheres).
The ‘Cavendish experiment’ described by most text-books was carried out by Charles Boys in Oxford in or around 1893. Boys’ equipment was similar but far more compact – it could be fitted into a small cupboard rather than a medium-sized room – and he really did want to measure G rather than the mass of Earth.

The force F between known masses m and M at a known distance d apart is deduced from the angle a through which the torsion bar twists in the presence of the large masses M. The only unknown in the gravitational equation is then G:

so

The twisting force in the suspension can be found by measuring the period of twisting of the oscillations of the bar.
Had Boys wished to find the mass of the Earth ME , he could have deduced it from a knowledge of G and g (the local acceleration due to gravity) and the radius of the Earth R using the equation (see Advancing Physics A2 student’s book, chapter 11)

in the form

Cavendish, however, found ME without reference to either G or g or needing to know the value of the small mass m. He did not even work out forces explicitly – there was no agreed unit of force at that time – but argued entirely in terms of ratios. His complete reasoning is ingenious and quite elaborate, but a key step was to relate the ratio ME / M to the ratio of the gravitational force FE exerted on m by the Earth to the gravitational force F exerted on m by M. Newton’s laws of gravity told Cavendish that FE is proportional to ME / R2 and F is proportional to M / d 2 so

F was the force needed to produce the torsional deflection a which Cavendish observed (just over half a degree corresponding to an arc length of about 1 cm). FE was the weight of the small mass. Cavendish was able to deduce the ratio FE / F from the ratio of the period of the torsional pendulum to that of a simple pendulum of the same length oscillating under gravity.
There were several attempts to measure G towards the end of the nineteenth century, but Boys’ experiment was soon accepted as the most accurate, and more recent measurements have essentially served to confirm a value close to 6.67 ´ 10–11 N m2 kg–2.
It is interesting to speculate about the sudden interest in the value of G 200 years after Newton and a century after Cavendish. It would seem that geophysical measurements (e.g. the size, shape and mass of the Earth) were no longer of such great concern. Interest was growing in more fundamental aspects of the forces of nature. Boys’ experiment belonged to an era of classic determinations of ‘fundamental physical constants’ e.g. Michelson’s determination of the speed of light (1881–7) and Millikan’s measurement of the charge on the electron (1907–13). One of the motives for repeated measurements of G was to test whether it was truly a universal constant, hence different experimenters used different materials (e.g. steel, platinum, gold, quartz) and different shapes (e.g. cubes, cylinders).

Questions
1.	The text quotes the ‘gravitational force between two 5 kg spheres whose centres are separated by 10 cm’. 
	Suggest suitable materials that could be used to make these spheres.

2.	Cavendish quotes the length of a ‘seconds-pendulum’ in London in 1798 as 39.14 inches. 
	Use this value to calculate the local value of g at that time.

3.	Cavendish quotes the mean diameter of the Earth as ‘41 800 000 feet’ and derives a mean density of 5.48 relative to the density of water. 
	Calculate the mass in kg which this implies for the Earth. 
	Supposing Cavendish had wished to calculate G from his data, what value would he have obtained?