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# Cavendish experiment

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### Cavendish experiment

The Cavendish experiment, performed in 1797–98 by British scientist Henry Cavendish, was the first experiment to measure the force of gravity between masses in the laboratory[1] and the first to yield accurate values for the gravitational constant.[2][3] Because of the unit conventions then in use, the gravitational constant does not appear explicitly in Cavendish's work. Instead, the result was originally expressed as the specific gravity of the Earth,[4] or equivalently the mass of the Earth. His experiment gave the first accurate values for these geophysical constants. The experiment was devised sometime before 1783[5] by geologist John Michell,[6] who constructed a torsion balance apparatus for it. However, Michell died in 1793 without completing the work. After his death the apparatus passed to Francis John Hyde Wollaston and then to Henry Cavendish, who rebuilt the apparatus but kept close to Michell's original plan. Cavendish then carried out a series of measurements with the equipment and reported his results in the Philosophical Transactions of the Royal Society in 1798.[7]

## Contents

• The experiment 1
• Whether Cavendish determined G 2
• Derivation of G and the Earth's mass 3
• Notes 5
• References 6

## The experiment

The apparatus constructed by Cavendish was a torsion balance made of a six-foot (1.8 m) wooden rod suspended from a wire, with a 2-inch (51 mm) diameter 1.61-pound (0.73 kg) lead sphere attached to each end. Two 12-inch (300 mm) 348-pound (158 kg) lead balls were located near the smaller balls, about 9 inches (230 mm) away, and held in place with a separate suspension system.[8] The experiment measured the faint gravitational attraction between the small balls and the larger ones.

Vertical section drawing of Cavendish's torsion balance instrument including the building in which it was housed. The large balls were hung from a frame so they could be rotated into position next to the small balls by a pulley from outside. Figure 1 of Cavendish's paper.
Detail showing torsion balance arm (m), large ball (W), small ball (x), and isolating box (ABCDE).

The two large balls were positioned on alternate sides of the horizontal wooden arm of the balance. Their mutual attraction to the small balls caused the arm to rotate, twisting the wire supporting the arm. The arm stopped rotating when it reached an angle where the twisting force of the wire balanced the combined gravitational force of attraction between the large and small lead spheres. By measuring the angle of the rod and knowing the twisting force (torque) of the wire for a given angle, Cavendish was able to determine the force between the pairs of masses. Since the gravitational force of the Earth on the small ball could be measured directly by weighing it, the ratio of the two forces allowed the density of the earth to be calculated, using Newton's law of gravitation.

Cavendish found that the Earth's density was 5.448 ± 0.033 times that of water (due to a simple arithmetic error, found in 1821 by Francis Baily, the erroneous value 5.48 ± 0.038 appears in his paper).[9]

To find the wire's torsion coefficient, the torque exerted by the wire for a given angle of twist, Cavendish timed the natural oscillation period of the balance rod as it rotated slowly clockwise and counterclockwise against the twisting of the wire. The period was about 20 minutes. The torsion coefficient could be calculated from this and the mass and dimensions of the balance. Actually, the rod was never at rest; Cavendish had to measure the deflection angle of the rod while it was oscillating.[10]

Cavendish's equipment was remarkably sensitive for its time.[9] The force involved in twisting the torsion balance was very small, 1.74 x 10–7 N,[11] about 1/50,000,000 of the weight of the small balls[12] or roughly the weight of a large grain of sand.[13] To prevent air currents and temperature changes from interfering with the measurements, Cavendish placed the entire apparatus in a wooden box about 2 feet (0.61 m) thick, 10 feet (3.0 m) tall, and 10 feet (3.0 m) wide, all in a closed shed on his estate. Through two holes in the walls of the shed, Cavendish used telescopes to observe the movement of the torsion balance's horizontal rod. The motion of the rod was only about 0.16 inches (4.1 mm).[14] Cavendish was able to measure this small deflection to an accuracy of better than one hundredth of an inch using vernier scales on the ends of the rod.[15] Cavendish's accuracy was not exceeded until C. V. Boys' experiment in 1895. In time, Michell's torsion balance became the dominant technique for measuring the gravitational constant (G) and most contemporary measurements still use variations of it. This is why Cavendish's experiment became the Cavendish experiment.[16]

## Whether Cavendish determined G

The formulation of Newtonian gravity in terms of a gravitational constant did not become standard until long after Cavendish's time. Indeed, one of the first references to G is in 1873, 75 years after Cavendish's work.[17]

Cavendish expressed his result in terms of the density of the Earth; he referred to his experiment in correspondence as 'weighing the world'. Later authors reformulated his results in modern terms.[18][19][20]

G = g\frac{R_\text{earth}^2}{M_\text{earth}} = \frac{3g}{4\pi R_\text{earth}\rho_\text{earth}}\,

After converting to SI units, Cavendish's value for the Earth's density, 5.448 g cm−3, gives

G = 6.74 × 10−11 m3 kg−1 s−2,

which differs by only 1% from the currently accepted value: 6.67384 × 10−11 m3 kg−1 s−2.

For this reason, historians of science have argued that Cavendish did not measure the gravitational constant.[21][22][23][24]

Physicists, however, often use units where the gravitational constant takes a different form. The Gaussian gravitational constant used in space dynamics is a defined constant and the Cavendish experiment can be considered as a measurement of the astronomical unit. In Cavendish's time, physicists used the same units for mass and weight, in effect taking g as a standard acceleration. Then, since R_\text{earth} was known, \rho_\text{earth} played the role of an inverse gravitational constant. The density of the Earth was hence a much sought-after quantity at the time, and there had been earlier attempts to measure it, such as the Schiehallion experiment in 1774.

For these reasons, physicists generally do credit Cavendish with the first measurement of the gravitational constant.[25][26][27][28][29]

## Derivation of G and the Earth's mass

For the definitions of terms, see the drawing below and the table at the end of this section.

The following is not the method Cavendish used, but shows how modern physicists would calculate the results from his experiment.[28][30][31] From Hooke's law, the torque on the torsion wire is proportional to the deflection angle \theta of the balance. The torque is \kappa\theta where \kappa is the torsion coefficient of the wire. However, the torque can also be written as a product of the attractive forces between the balls and the distance to the suspension wire. Since there are two pairs of balls, each experiencing force F at a distance L / 2 from the axis of the balance, the torque is LF. Equating the two formulas for torque gives the following:

\kappa\theta\ = LF \,

For F, Newton's law of universal gravitation is used to express the attractive force between the large and small balls:

Diagram of torsion balance
F = \frac{G m M}{r^2}\,

Substituting F into the first equation above gives

To find the torsion coefficient (\kappa\,) of the wire, Cavendish measured the natural resonant oscillation period T of the torsion balance:

T = 2\pi\sqrt{I/\kappa}

Assuming the mass of the torsion beam itself is negligible, the moment of inertia of the balance is just due to the small balls:

I = m(L/2)^2 + m(L/2)^2 = 2m(L/2)^2 = mL^2/2\,,

and so:

T = 2\pi\sqrt{\frac{mL^2}{2\kappa}}\,

Solving this for \kappa, substituting into (1), and rearranging for G, the result is:

G = \frac{2 \pi^2 L r^2}{M T^2} \theta\,

Once G has been found, the attraction of an object at the Earth's surface to the Earth itself can be used to calculate the Earth's mass and density:

mg = \frac{GmM_{earth}}{R_{earth}^2}\,
M_{earth} = \frac{gR_{earth}^2}{G}\,
\rho_{earth} = \frac{M_{earth}}{4 \pi R_{earth}^3/3} = \frac{3g}{4 \pi R_{earth} G}\,
 Definition of terms SYMBOL UNITS DEFINITION \theta\, \mbox{radians}\, Deflection of torsion balance beam from its rest position F\, \mbox{N}\, Gravitational force between masses M and m G\, \mbox{m}^3 {\mbox{kg}}^{-1} \mbox{s}^{-2}\, Gravitational constant m\, \mbox{kg}\, Mass of small lead ball M\, \mbox{kg}\, Mass of large lead ball r\, \mbox{m}\, Distance between centers of large and small balls when balance is deflected L\, \mbox{m}\, Length of torsion balance beam between centers of small balls \kappa\, \mbox{N}\,\mbox{m}\,\mbox{radian}^{-1}\, Torsion coefficient of suspending wire I\, \mbox{kg}\,\mbox{m}^2\, Moment of inertia of torsion balance beam T\, \mbox{s}\, Period of oscillation of torsion balance g\, \mbox{m}\,\mbox{s}^{-2}\, Acceleration of gravity at the surface of the Earth M_{earth}\, \mbox{kg}\, Mass of the Earth R_{earth}\, \mbox{m}\, Radius of the Earth \rho_{earth}\, \mbox{kg}\,\mbox{m}^{-3}\, Density of the Earth

## Notes

1. ^ Boys 1894 p. 355
2. ^ Encyclopædia Britannica 1910 p. 385 'The aim [of experiments like Cavendish's] may be regarded either as the determination of the mass of the Earth,...conveniently expressed...as its "mean density", or as the determination of the "gravitation constant", G'. Cavendish's experiment is generally described today as a measurement of G (Clotfelter 1987 p. 210).
3. ^ Many sources incorrectly state that this was the first measurement of G (or the Earth's density); for instance: Feynman, Richard P. (2013) [1963]. "7. The Theory of Gravitation". mainly mechanics, radiation and heat. The Feynman lectures on physics. Volume I. Pasadena, California: California Institute of Technology. 7–6 Cavendish’s experiment. There were previous measurements, chiefly by Bouguer (1740) and Maskelyne (1774), but they were very inaccurate (Poynting 1894)(Encyclopædia Britannica 1910).
4. ^ Clotfelter 1987, p. 210
5. ^ McCormmach & Jungnickel 1996, p.336: A 1783 letter from Cavendish to Michell contains '...the earliest mention of weighing the world'. Not clear whether 'earliest mention' refers to Cavendish or Michell.
6. ^ Cavendish 1798, p. 59 Cavendish gives full credit to Michell for devising the experiment
7. ^ Cavendish, H. 'Experiments to determine the Density of the Earth', Philosophical Transactions of the Royal Society of London, (part II) 88 p.469-526 (21 June 1798), reprinted in Cavendish 1798
8. ^ Cavendish 1798, p.59
9. ^ a b Poynting 1894, p.45
10. ^ Cavendish 1798, p.64
11. ^ Boys 1894 p.357
12. ^ Cavendish 1798 p. 60
13. ^ A 2 mm sand grain weighs ~13 mg. Theodoris, Marina (2003). "Mass of a Grain of Sand". The Physics Factbook. Retrieved 2013-12-30.
14. ^ Cavendish 1798, p. 99, Result table, (scale graduations = 1/20 in ≈ 1.3 mm) The total deflection shown in most trials was twice this since he compared the deflection with large balls on opposite sides of the balance beam.
15. ^ Cavendish 1798, p.63
16. ^ McCormmach & Jungnickel 1996, p.341
17. ^ Cornu, A.; Baille, J. B. (1873). "Détermination nouvelle de la constante de l'attraction et de la densité moyenne de la Terre" [New Determination of the Constant of Attraction and the Average Density of Earth]. C. R. Acad. Sci. (in Français) (Paris) 76: 954–958.
18. ^ Boys 1894, p.330 In this lecture before the Royal Society, Boys introduces G and argues for its acceptance
19. ^ Poynting 1894, p.4
20. ^ MacKenzie 1900, p.vi
21. ^ Clotfelter 1987
22. ^ McCormmach & Jungnickel 1996, p.337
23. ^ Hodges 1999
24. ^ Lally 1999
25. ^ Halliday, David; Resnick, Robert (1993). Fundamentals of Physics. John Wiley & Sons. p. 418. 'The apparatus used in 1798 by Henry Cavendish to measure the gravitational constant'
26. ^ Feynman, Richard P. (1963). "Lectures on Physics, Vol.1". Addison-Wesley. pp. 6–7. 'Cavendish claimed he was weighing the Earth, but what he was measuring was the coefficient G...'
27. ^ Feynman, Richard P. (1967). "The Character of Physical Law". MIT Press. p. 28. 'Cavendish was able to measure the force, the two masses, and the distance, and thus determine the gravitational constant G'
28. ^ a b "Cavendish Experiment, Harvard Lecture Demonstrations, Harvard Univ". Retrieved 2013-12-30.. '[the torsion balance was]...modified by Cavendish to measure G.'
29. ^ Shectman, Jonathan (2003). Groundbreaking Experiments, Inventions, and Discoveries of the 18th Century. Greenwood. pp. xlvii. 'Cavendish calculates the gravitational constant, which in turn gives him the mass of the earth...'
30. ^ Poynting 1894, p.41
31. ^ Clotfelter 1987 p.212 explains Cavendish's original method of calculation

## References

• Boys, C. Vernon (1894). "On the Newtonian constant of gravitation". Nature 50 (1292): 330–4.
• Cavendish, Henry (1798). "Experiments to Determine the Density of the Earth". In MacKenzie, A. S. Scientific Memoirs Vol.9: The Laws of Gravitation. American Book Co. (published 1900). pp. 59–105. Retrieved 2013-12-30. Online copy of Cavendish's 1798 paper, and other early measurements of gravitational constant.
• Clotfelter, B. E. (1987). "The Cavendish experiment as Cavendish knew it". American Journal of Physics 55 (3): 210–213. Establishes that Cavendish didn't determine G.
• Falconer, Isobel (1999). "Henry Cavendish: the man and the measurement". Measurement Science and Technology 10 (6): 470–477.
• "Gravitation Constant and Mean Density of the Earth". Encyclopædia Britannica, 11th Ed. 12. The Encyclopædia Britannica Co. 1910. pp. 385–389. Retrieved 2013-12-30.
• Hodges, Laurent (1999). "The Michell-Cavendish Experiment, faculty website, Iowa State Univ.". Retrieved 2013-12-30. Discusses Michell's contributions, and whether Cavendish determined G.
• Lally, Sean P. (1999). "Henry Cavendish and the Density of the Earth". The Physics Teacher 37 (1): 34–37.
• McCormmach, Russell; Jungnickel, Christa (1996). Cavendish. Philadelphia, Pennsylvania:
• Poynting, John H. (1894). The Mean Density of the Earth: An essay to which the Adams prize was adjudged in 1893. London: C. Griffin & Co. Retrieved 2013-12-30. Review of gravity measurements since 1740.