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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 ...
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 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
When calculating the period of a simple pendulum, the small-angle approximation for sine is used to allow the resulting differential equation to be solved easily by comparison with the differential equation describing simple harmonic motion.
Other phenomena can be modeled by simple harmonic motion, including the motion of a simple pendulum, although for it to be an accurate model, the net force on the object at the end of the pendulum must be proportional to the displacement (and even so, it is only a good approximation when the angle of the swing is small; see small-angle ...
The Foucault pendulum or Foucault's pendulum is a simple device named after French physicist Léon Foucault, conceived as an experiment to demonstrate the Earth's rotation. If a long and heavy pendulum suspended from the high roof above a circular area is monitored over an extended period of time, its plane of oscillation appears to change ...
For a simple harmonic oscillator released from rest, regardless of its initial displacement, the time it takes to reach the lowest potential energy point is always a quarter of its period, which is independent of its amplitude. Therefore, the Lagrangian of a simple harmonic oscillator is isochronous.
Derivation of the frequency response Using the method of harmonic balance, an approximate solution to the Duffing equation is sought of the form: [ 9 ] x = a cos ( ω t ) + b sin ( ω t ) = z cos ( ω t − ϕ ) , {\displaystyle x=a\,\cos(\omega t)+b\,\sin(\omega t)=z\,\cos(\omega t-\phi ),} with z 2 = a 2 + b 2 {\displaystyle z^{2 ...