Section 7–6 Cavendish experiment

(Cavendish’s apparatus / Weighing the earth / Gravitational constant)

In this section, the three interesting points discussed are Cavendish’s apparatus, weighing the earth, and gravitational constant.

1. Cavendish’s apparatus:

“…It was first measured by Cavendish with an apparatus which is schematically indicated in Fig. 7–13 (Feynman et al., 1963, section 7.6 Cavendish experiment).”

The heart of Cavendish’s apparatus is a torsion balance that is made of a very fine fiber. The strength of the gravitational force is provided by two large, fixed balls of lead and two smaller balls of lead on the ends of an arm. By measuring the extent the fiber is twisted, one can measure the strength of the force that it is inversely proportional to the square of the distance between the two objects. One difficulty of this experiment is the weakness of the force that may be measured inaccurately if there are disturbances such as electrical forces or a draught. Thus, there is a need to cover the apparatus to keep the air out and make sure it is not electrically charged. Cavendish placed the whole apparatus in a room which remained shut and he observed the motion of the arm from the outside using a telescope.

The apparatus was originally designed by Rev. John Michell. To recognize Michell’s design and work, it should be renamed as Michell-Cavendish apparatus. In his seminal paper, Cavendish (1798) writes that “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 (p. 469).”

2. Weighing the Earth:

“This experiment has been called ‘weighing the earth’. Cavendish claimed he was weighing the earth… (Feynman et al., 1963, section 7.6 Cavendish experiment).”

Feynman mentions that Cavendish’s experiment has been called “weighing the earth.” In his first Messenger lecture, he explains that “[w]ith pedantic and careful teaching today we would not let our students say that; we would have to say ‘measuring the mass of the earth’ (p. 28).” Essentially, we can calculate the mass of the earth by determining the Newtonian gravitational constant (G), the attractive gravitational force, and the distances between the two objects from this experiment. However, Cavendish did not determine either G through this experiment. In fact, he determined the average density of the earth, which he expressed as a ratio of the earth’s density to the density of water.

The title of Cavendish’s (1798) paper is “Experiments to determine the density of the Earth.” Cavendish’s main purpose in this paper was to solve an eighteenth-century dispute: what is the mean density of the Earth? In other words, Cavendish’s aim was not to determine the gravitational constant, but rather to deduce the Earth’s density. This experiment was motivated by criticisms of astronomical investigations which ignored the gravitational attraction of mountains.

Note: In the Feynman lectures websites, it is recently changed to “This experiment has been called ‘weighing the earth’ by some people.”

3. Gravitational constant:

“This is the only way in which the mass of the earth can be determined. G turns out to be 6.670×10⁻¹¹ newton⋅m²/kg² (Feynman et al., 1963, section 7.6 Cavendish experiment).”

Feynman explains that Cavendish’s apparatus can be used to determine the coefficient G of the gravity law instead of only weighing the Earth. Interestingly, in his first Messenger lecture, he adds that Cavendish “was weighing the sun and everything else at the same time, because the pull of the sun is known in the same manner (Feynman, 1965, p. 29).” Simply put, one may accurately determine the gravitational constant from the formula, F = Gmm′/r². According to Feynman, Newtonian gravitational constant turns out to be 6.670 × 10⁻¹¹ newton⋅m²/kg². However, the 2014 Committee on Data for Science and Technology (CODATA) recommended the value of Newtonian gravitational constant to be 6.67408(31) × 10⁻¹¹ m³ kg^-1 s^-2.

Newtonian gravitational constant (G) is extremely difficult to measure precisely and it has no definitive relationship with other fundamental constants. Furthermore, variations of gravitational constant measured by different laboratories suggest that the uncertainty in G could be much larger. Recently, Kuroda (1995) suggested that internal friction in the torsion fiber may have caused errors in the previous measurements. One way to resolve this problem is to include a turntable and adjust its rotation rate such the torsion fiber is not twisted anymore. Alternatively, physicists have determined G by using an atomic fountain in which atoms are allowed to rise and fall (Peters, Chung, & Chu, 1999).

Note: One may argue that Feynman has goofed again when he says that “[t]his is the only way in which the mass of the earth can be determined.” This is because there are many different ways to determine the mass of earth, for example, by using a turntable or atomic fountain to deduce the gravitational constant.

Questions for discussion:

1. What are the main features of Cavendish’s apparatus?

2. Should Cavendish’s experiment be known as “weighing the Earth”?

3. What are the other ways of determining Newtonian gravitational constant?

The moral of the lesson: Newtonian gravitational constant can be determined by many ways such as the use of Cavendish’s apparatus.

References:

1. Cavendish, H. (1798). Experiments to determine the density of the Earth. By Henry Cavendish, Esq. FRS and AS. Philosophical Transactions of the Royal Society of London, 88, 469-52.

2. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Kuroda, K. (1995). Does the time-of-swing method give a correct value of the newtonian gravitational constant?. Physical Review Letters, 75(15), 2796-2798.

5. Peters, A., Chung, K. Y., & Chu, S. (1999). Measurement of gravitational acceleration by dropping atoms. Nature, 400(6747), 849-852.