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The animations below depict the motion of a simple (frictionless) pendulum with increasing amounts of initial displacement of the bob, or equivalently increasing initial velocity. The small graph above each pendulum is the corresponding phase plane diagram; the horizontal axis is displacement and the vertical axis is velocity. With a large ...
"Simple gravity pendulum" model assumes no friction or air resistance. A pendulum is a device made of a weight suspended from a pivot so that it can swing freely. [1] 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 toward the equilibrium position.
A simple pendulum exhibits approximately simple harmonic motion under the conditions of no damping and small amplitude. Assuming no damping, the differential equation governing a simple pendulum of length l {\displaystyle l} , where g {\displaystyle g} is the local acceleration of gravity , is d 2 θ d t 2 + g l sin θ = 0. {\displaystyle ...
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 ...
Rayleigh–Lorentz pendulum (or Lorentz pendulum) is a simple pendulum, but subjected to a slowly varying frequency due to an external action (frequency is varied by varying the pendulum length), named after Lord Rayleigh and Hendrik Lorentz. [1] This problem formed the basis for the concept of adiabatic invariants in mechanics. On account of ...
Illustration of how a phase portrait would be constructed for the motion of a simple pendulum Time-series flow in phase space specified by the differential equation of a pendulum. The X axis corresponds to the pendulum's position, and the Y axis its speed.
Simple pendulum, see picture (right). Simple harmonic oscillator where the phase portrait is made up of ellipses centred at the origin, which is a fixed point. Damped harmonic motion, see animation (right). Van der Pol oscillator see picture (bottom right).
The relationship between the use of generalized coordinates and Cartesian coordinates to characterize the movement of a mechanical system can be illustrated by considering the constrained dynamics of a simple pendulum. [12] [13] A simple pendulum consists of a mass M hanging from a pivot point so that it is constrained to move on a circle of ...