We describe a Bessel pendulum for use in the teaching laboratory, and measurements of the local acceleration of gravity made with it to an accuracy of better than one part in The Bessel pendulum is a reversible pendulum that eliminates atmospheric corrections that apply to the more familiar Kater pendulum. The physical principles underlying the Kater pendulum as well as Bessel’s refinement are reviewed, and construction details are given for a realization of the pendulum.
REFERENCES
1.
D. Kleppner and R. J. Kolenkow, An Introduction to Mechanics (McGraw–Hill, New York, 1973), pp. 257–258.
2.
J. B. Marion, Classical Dynamics of Particles and Systems (Sanders College Publishing, Fort Worth, 1995), 4th ed., p. 455.
3.
V. F. Lenzen and R. P. Multhauf, Development of Gravity Pendulums in the 19th Century, Contributions from the Museum of History and Technology Paper 44 (Smithsonian Institution, Washington, 1965), pp. 303–352.
4.
F. W. Bessel, “Untersuchungen über die Länge des einfachen Secundenpendels,” Abhandlungen der Königliche Akademie der Wissenshaften zu Berlin (1826);
also published as Untersuchungen über die Länge des einfachen Secundenpendels (Bruns, Leipzig, 1889).
5.
Precision pendulums based on Bessel’s design were first constructed in 1861 (after Bessel’s death) by the firm A. Repsold and Sons, hence the Bessel pendulum is often called the Repsold–Bessel or Bessel–Repsold reversible pendulum.
6.
A. H.
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The frequency shift due to viscous damping by the air remains present in a Bessel pendulum, and must be corrected for if significant.
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K. E.
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The finite-amplitude correction is the same for a physical pendulum as it is for a simple pendulum. As the fractional change in period is simply a function of the angular oscillation amplitude the Kater and Bessel pendulums can be adjusted for reversibility at fixed, finite Then the finite-amplitude correction for this is applied before using the Kater relation to determine g.
14.
Internet address: bgi.cnes.fr
15.
P. Vanı́ček and E. Krakiwsky, Geodesy: The Concepts (North-Holland, Amsterdam, 1986), 2nd ed., p. 79.
16.
The height correction for locally flat topography is where is the height difference and and are the mean density and radius of the Earth, respectively. The mean density of the local crust material beneath the higher of the two points being compared is The first term expresses the dependence of gravity in free space, while the second accounts for the gravitational attraction of the local material. The second term (the so-called Bouguer correction) is comparable in size to the first and cannot be omitted. For example, the free-air gravity gradient is while for typical we have The second term is easily derived by modeling the local material as a disk much wider than it is thick, but much smaller than the Earth. For a more detailed discussion including the effects of topography, see Ref. 15, Chap. 21.
17.
Mathematical and Physical Papers (Cambridge U.P., London, 1922), Vol. 3, p. 1 (1901).
18.
L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1987), 2nd ed., p. 83ff.
19.
The pendulum is a simple harmonic oscillator with potential energy and kinetic energy Using energy conservation gives hence For the pendulum A is g times the numerator of Eq. (3) or (7), while B is the denominator.
20.
In writing Eq. (B2) we assume that the resonant frequency of the support for horizontal vibrations is much greater than the frequency of the pendulum. In this limit the mass of the support can be ignored.
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© 2001 American Association of Physics Teachers.
2001
American Association of Physics Teachers
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