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Pendulum (mathematics)

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Title: Pendulum (mathematics)  
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Subject: Classical mechanics, Horology, Euler's laws of motion, Frame of reference, Harmonic oscillator
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Pendulum (mathematics)

The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allows the equations of motion to be solved analytically for small-angle oscillations.

Simple gravity pendulum

Animation of a pendulum showing the velocity and acceleration vectors.

A so-called "simple pendulum" is an idealization of a "real pendulum" but in an isolated system using the following 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 differential equation which represents the motion of a simple pendulum is

{d^2\theta\over dt^2}+{g\over \ell} \sin\theta=0

 

 

 

 

(Eq. 1)

where g is acceleration due to gravity, \ell is the length of the pendulum, and \theta is the angular displacement.



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