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The Pendulum

Swinging physics

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Introduction | Approximate | Exact
Frequency | Maximum Velocity

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Introduction

There are some parallels between a pendulum and an elliptical orbit:

  • As the pendulum swings, it exchanges kinetic and potential energy.
  • The highest velocity, and highest kinetic energy, coincides with the pendulum's closest approach to the center of the earth.
  • The lowest velocity, always zero, coincides with the pendulum's farthest distance from the center of the earth.
  • If we disregard frictional losses, the pendulum's total energy is a constant, just as with an elliptical orbit.

As simple as they seem, gravity pendulums cannot be described analytically (i.e. by means of closed-form equations), as a result of which, when high precision is needed, they are modeled numerically.

Approximate

This is only an approximate solution for a pendulum's period t, suitable only for small swings:

(1) $ \displaystyle t \approx 2 \pi \sqrt{\frac{L}{g}} $

Where:

  • t = Pendulum period, seconds
  • L = Pendulum arm length, meters
  • g = Little-g, described above

Some explanation:

  • The assumption is that the pendulum consists of a weight attached to the end of a massless arm of length L, that there is no friction, and there is no air resistance.
  • Equation (1) is only accurate for infinitesimally small swings. As soon as the pendulum's weight swings through more than a tiny angle, the accuracy of equation (1) declines. Strictly speaking, the equation is only valid if the pendulum is swung through an angle φ so small that:
    (2) $ \displaystyle \sin \phi = \phi$ (radians)

    But a better equation appears next.

Exact

Here is an exact pendulum equation for period t:

(3) $ \displaystyle t = 2\pi \sqrt{\frac{L}{g}}\ \sum_{n=0}^{\infty}\frac{\sin\left(\frac{1}{2} \phi \right)^{{\left(2 \, n\right)}} \left(2 \, n\right)!^{2}}{2^{{\left(4 \, n\right)}} n!^{4}} $

Where:

  • t = Pendulum period, seconds
  • L = Pendulum arm length, meters
  • g = Little-g, described above
  • φ = Pendulum initial angle, radians

In practice, equation (3) will provide any desired level of accuracy, by writing an algorithm to solve it that replaces ∞ with a numerical value consistent with the needed accuracy.

Frequency

The frequency f of a pendulum in Hertz is trivially derived from the period t acquired with the above methods:

(4) $ \displaystyle f = \frac{1}{t} $
Maximum Velocity

In keeping with the earlier discussion of kinetic and potential energy, the maximum velocity of a pendulum bob occurs at the moment it is pointing straght down, when it has the most kinetic energy. The equation for maximum velocity:

(5) $ \displaystyle v_m = \sqrt{2gL(1-\cos(\phi))}$

Where:
  • vm = Maximum velocity, m/s
  • g = Little-g, described above
  • L = Pendulum arm length, meters
  • φ = Pendulum release angle, radians
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