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Concept
Pendulum (mechanics)
Summary
A pendulum is a body suspended from a fixed support so that it swings freely 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 toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging it back and forth. The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allow the equations of motion to be solved analytically for small-angle oscillations.
A simple gravity pendulum is an idealized mathematical model of a real pendulum. This is a weight (or bob) on the end of a massless cord suspended from a pivot, without friction. Since in this model there is no frictional energy loss, when given an initial displacement it will swing back and forth at a constant amplitude. The model is based on these assumptions:
The rod or cord on which the bob swings is massless, inextensible and always remains taut.
The bob is a point mass.
Motion occurs only in two dimensions, i.e. the bob does not trace an ellipse but an arc.
The motion does not lose energy to friction or air resistance.
The gravitational field is uniform.
The support does not move.
The differential equation which represents the motion of a simple pendulum is
where g is the magnitude of the gravitational field, l is the length of the rod or cord, and θ is the angle from the vertical to the pendulum.
The differential equation given above is not easily solved, and there is no solution that can be written in terms of elementary functions. However, adding a restriction to the size of the oscillation's amplitude gives a form whose solution can be easily obtained. If it is assumed that the angle is much less than 1 radian (often cited as less than 0.
Official source
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