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The period increases asymptotically (to infinity) as θ 0 approaches π radians (180°), because the value θ 0 = π is an unstable equilibrium point for the pendulum. The true period of an ideal simple gravity pendulum can be written in several different forms (see pendulum (mechanics)), one example being the infinite series: [11] [12
The real period is, of course, the time it takes the pendulum to go through one full cycle. Paul Appell pointed out a physical interpretation of the imaginary period: [ 16 ] if θ 0 is the maximum angle of one pendulum and 180° − θ 0 is the maximum angle of another, then the real period of each is the magnitude of the imaginary period of ...
A simple pendulum exhibits approximately simple harmonic motion under the conditions ... is the amplitude of the pendulum). The period, the time for one complete ...
The period depends on the length of the pendulum, and also to a slight degree on its weight distribution (the moment of inertia about its own center of mass) and the amplitude (width) of the pendulum's swing. For a simple gravity pendulum-- a point mass on a weightless string of length L swinging with an infinitesimally small amplitude, without ...
The time period is able to be calculated by = ... In the small-angle approximation, the motion of a simple pendulum is approximated by simple harmonic motion.
The seconds pendulum (also called the Royal pendulum), 0.994 m (39.1 in) long, in which the time period is two seconds, became widely used in quality clocks. The long narrow clocks built around these pendulums, first made by William Clement around 1680, who also claimed invention of the anchor escapement, [ 4 ] became known as grandfather clocks .
Repeatedly timing each period of a Kater pendulum, and adjusting the weights until they were equal, was time-consuming and error-prone. Friedrich Bessel showed in 1826 that this was unnecessary. As long as the periods measured from each pivot, T 1 and T 2, are close in value, the period T of the equivalent simple pendulum can be calculated from ...
The equation of the simple harmonic motion with frequency for the displacement () is given by ¨ + =. If the frequency is constant, the solution is simply given by = (+).But if the frequency is allowed to vary slowly with time = (), or precisely, if the characteristic time scale for the frequency variation is much smaller than the time period of oscillation, i.e., | |, then it can be shown ...