Cavendish experiment

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The Cavendish experiment, performed in 1797-1798 by British scientist Henry Cavendish, was the first experiment to measure the force of gravity between masses in the laboratory and the first to yield accurate values for the gravitational constant. 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, 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 by geologist John Michell, 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.

Video Cavendish experiment

The experiment

The apparatus constructed by Cavendish was a torsion balance made of a six-foot (1.8 m) wooden rod horizontally suspended from a wire, with a two 2-inch (51 mm) diameter 1.61-pound (0.73 kg) lead spheres, each 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. The experiment measured the faint gravitational attraction between the small balls and the larger ones.

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 5.480±0.038 appears in his paper).

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.

Cavendish's equipment was remarkably sensitive for its time. The force involved in twisting the torsion balance was very small, 1.74×10^-7 N, about ¹/_50,000,000 of the weight of the small balls. 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). 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. Cavendish's accuracy was not exceeded until C. V. Boys's 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.

Cavendish's result was also the first evidence for a planetary core made of metal. The result of 5.4 g·cm^-3 is close to 80% of the density of liquid iron, and 80% higher than the density of the Earth's outer crust, suggesting the existence of a dense iron core.

Maps Cavendish experiment

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.

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.

${\displaystyle G=g{\frac {R_{\text{earth}}^{2}}{M_{\text{earth}}}}={\frac {3g}{4\pi R_{\text{earth}}\rho _{\text{earth}}}}\,}$

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 2014 CODATA value of 6.67408×10^-11 m³ kg^-1 s^-2.

For this reason, historians of science have argued that Cavendish did not measure the gravitational constant.

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 this constant. In Cavendish's time, physicists used the same units for mass and weight, in effect taking g as a standard acceleration. Then, since R_earth was known, ?_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.

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Derivation of G and the Earth's mass

The following is not the method Cavendish used, but shows how modern physicists would calculate the results from his experiment. From Hooke's law, the torque on the torsion wire is proportional to the deflection angle ? of the balance. The torque is ?? where ? is the torsion coefficient of the wire. However, a torque in opposite direction is also generated by the gravitationnal pull of the masses. It can 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. At equilibrium (when the balance has been stabilized at an angle ?), the total amount of torque must be zero, as the these two sources of torque cancel out. Thus, we can equate their intensities given by the formulas above, which gives the following:

${\displaystyle \kappa \theta \ =LF\,}$

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

${\displaystyle F={\frac {GmM}{r^{2}}}\,}$

Substituting F into the first equation above gives

${\displaystyle \kappa \theta \ =L{\frac {GmM}{r^{2}}}\qquad \qquad \qquad (1)\,}$

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

${\displaystyle T=2\pi {\sqrt {\frac {I}{\kappa }}}}$

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

${\displaystyle I=m\left({\frac {L}{2}}\right)^{2}+m\left({\frac {L}{2}}\right)^{2}=2m\left({\frac {L}{2}}\right)^{2}={\frac {mL^{2}}{2}}\,}$,

and so:

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

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

${\displaystyle G={\frac {2\pi ^{2}Lr^{2}\theta }{MT^{2}}}\,}$

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:

${\displaystyle mg={\frac {GmM_{earth}}{R_{earth}^{2}}}\,}$

${\displaystyle M_{earth}={\frac {gR_{earth}^{2}}{G}}\,}$

${\displaystyle \rho _{earth}={\frac {M_{earth}}{{\tfrac {4}{3}}\pi R_{earth}^{3}}}={\frac {3g}{4\pi R_{earth}G}}\,}$

Definitions of terms

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See also

• Schiehallion experiment

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Notes

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References

• Boys, C. Vernon (1894). "On the Newtonian constant of gravitation". Nature. 50 (1292): 330-4. Bibcode:1894Natur..50..330.. doi:10.1038/050330a0. Retrieved 2013-12-30. 
• 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. Bibcode:1987AmJPh..55..210C. doi:10.1119/1.15214.  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. Bibcode:1999MeScT..10..470F. doi:10.1088/0957-0233/10/6/310. 
• "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. Bibcode:1999PhTea..37...34L. doi:10.1119/1.880145. 
• McCormmach, Russell; Jungnickel, Christa (1996). Cavendish. Philadelphia, Pennsylvania: American Philosophical Society. ISBN 978-0-87169-220-7. Retrieved 2013-12-30. 
• 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.

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External links

• Cavendish's experiment in the Feynman Lectures on Physics
• Sideways Gravity in the Basement, The Citizen Scientist, July 1, 2005. Homebrew Cavendish experiment, showing calculation of results and precautions necessary to eliminate wind and electrostatic errors.
• "Big 'G'", Physics Central, retrieved Dec. 8, 2013. Experiment at Univ. of Washington to measure the gravitational constant using variation of Cavendish method.
• Eöt-Wash Group, Univ. of Washington. "The Controversy over Newton's Gravitational Constant". Archived from the original on 2016-03-04. Retrieved December 8, 2013. . Discusses current state of measurements of G.
• Model of Cavendish's torsion balance, retrieved Aug. 28, 2007, at Science Museum, London.
• Weighing the Earth - background and experiment

Source of article : Wikipedia