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 104. 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.

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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.
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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 θ0, the Kater and Bessel pendulums can be adjusted for reversibility at fixed, finite θ0. Then the finite-amplitude correction for this θ0 is applied before using the Kater relation to determine g.
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Internet address: bgi.cnes.fr
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The height correction for locally flat topography is Δg/Δh=[−2+(3/2)(ρce)]g/Re where Δh is the height difference and ρe and Re 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 ρc. The first term expresses the 1/R2 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 c=0) gravity gradient is Δg/Δh=−3.08 mGal/m while for typical ρce≈0.5 we have Δg/Δh≈−2 mGal/m. 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.
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19.
The pendulum is a simple harmonic oscillator with potential energy Ep=Aθ2/2 and kinetic energy Ek=Bθ̇2/2. Using energy conservation Ėp=−Ėk gives θ̈=−(A/B)θ, hence ω2=A/B. 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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