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In Kater's time, the period T of pendulums could be measured very precisely by timing them with precision clocks set by the passage of stars overhead. Prior to Kater's discovery, the accuracy of g measurements was limited by the difficulty of measuring the other factor L, the length of the pendulum, accurately.
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To illustrate, Figure 1 shows the so-called Normal PDF, which will be assumed to be the distribution of the observed time periods in the pendulum experiment. Ignoring all the biases in the measurements for the moment, then the mean of this PDF will be at the true value of T for the 0.5 meter idealized pendulum, which has an initial angle of 30 ...
His first major contribution to science was the comparison of the merits of the Cassegrainian and Gregorian telescopes; Kater determined the latter to be an inferior design. [ 1 ] His most substantial work was the invention of Kater's pendulum , enabling the strength of gravity to be determined, first at London [ 2 ] and subsequently at various ...
More formulas of this nature can be given, as explained by Ramanujan's theory of elliptic functions to alternative bases. Perhaps the most notable hypergeometric inversions are the following two examples, involving the Ramanujan tau function τ {\displaystyle \tau } and the Fourier coefficients j {\displaystyle \mathrm {j} } of the J-invariant ...
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The fundamental rectangle in the complex plane of . There are twelve Jacobi elliptic functions denoted by (,), where and are any of the letters , , , and . (Functions of the form (,) are trivially set to unity for notational completeness.) is the argument, and is the parameter, both of which may be complex.
A pendulum is a body suspended from a fixed support such that it freely swings back and forth under the influence of gravity. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back towards the equilibrium position.