# Pendulum

A pendulum. Yellow arrow down shows the gravity force on the bob. Yellow arrow up shows the tension force towards suspension point. This force is stronger when the bob is moving faster as the centrifugal force adds to the pendulum weight. The red arrow shows the pendulum speed. You can adjust the pendulum length with the control at the top.
A pendulum is a weight suspended from a pivot that swings freely, converting between potential (displacement) and kinetic (speed) energy in a closed loop.

When a pendulum is displaced from its resting equilibrium position and then released, the gravity force accelerates it back toward the equilibrium position. However when the accelerating pendulum reaches this position, it does not stop because of the kinetic energy that bob accumulates while moving downward. The pendulum passes the equilibrium point and then gravity force starts decelerating it. Finally the pendulum stops after reaching more or less the same angle from equilibrium as it was initially displaced, but now in the opposite side.

The pendulum converts between kinetic (bob speed) and potential (bob height above the ground) energy in a closed loop that only terminates because the efficiency of conversion is finite and with every period, part of the energy is lost.

The restoring force combined with the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time required for one complete cycle, a left swing and a right swing, is called the oscillation period of the pendulum. A period of the same pendulum is relatively constant and was the world's most accurate timekeeping technology until the 1930s.[1]. It depends mainly on its length. The word 'pendulum' is new Latin, from pendulus, meaning 'hanging'.[2]

The simple gravity pendulum[3][4] [5] [6] is a weight (or bob) on the end of a massless cord suspended from a pivot, without friction. Once pushed, it swings with the constant amplitude forever. Real pendulums slow down and stop because of the friction and air drag. A simple gravity pendulum only exists as a convenient abstraction that may help to solve real examples with sufficient accuracy.

## Period of oscillation

The period of swing of a simple gravity pendulum depends on its length, the acceleration of gravity, and to a small extent on the maximum angle that the pendulum swings away from vertical, [[Math:c|\theta_0]], called the amplitude.[7] It is independent of the mass of the bob. If the amplitude is limited to small swings, the period T of a simple pendulum, the time taken for a complete cycle, is:[8]

[[Math:c|T \approx 2\pi \sqrt {\frac L g} ]]

where

[[Math:c|\theta_0 \lt\lt 1]]

where L is the length of the pendulum and g is the local acceleration of gravity.

For small swings, the period is independent on the oscillation amplitude. This is the reason why pendulums are so useful for timekeeping.[9]

For larger amplitudes, the period increases gradually with amplitude so it is longer than given by equation (1). For example, at an amplitude of [[Math:c|\theta_0=23^o]] it is 1% larger than given by (1). The true period cannot be represented by a closed formula but is given by an infinite series:[10]

[[Math:c| T = 2\pi \sqrt{L\over g} \left( 1\plus \frac{1}{16}\theta_0^2 \plus \frac{11}{3072}\theta_0^4 \plus \frac{173}{737280}\theta_0^6 \plus \frac{22931}{1321205760}\theta_0^8 \plus . . . \right) ]]

The difference between this true period and the period of "small swings" is called the circular error.

A pendulum that oscillates with relatively small amplitude is governed by the interesting "universal" laws that describe very different processes with the same equations. In the case of pendulum, the same laws also cover the spring oscillations, electric resistor–inductor–capacitor circuit and may even cover size oscillation in some biological populations. For small swings the pendulum oscillates as a harmonic oscillator:

[[Math:c|\theta (t) = \theta_0 \cos{\Big ( 2 \pi t \over T \Big )}\,]]

where [[Math:c|\theta]] can be either deviation from the equilibrium position (potential energy) or the bob speed (kinetic energy).

Unless attachment point restricts movement, pendulum tends to keep its oscillation plane regardless of factors like Earth rotation. From the perspective of the ground-based observer, the pendulum plane slowly rotates, reflecting the rotation of the Earth. However pendulum must be very long and protected from wind to observe this reliably.

## References

1. 1 Warren Marrison (1948). The Evolution of the Quartz Crystal Clock. Bell System Technical Journal 27 510–588.
2. 2 Morris William, Ed. The American Heritage Dictionary, New College Ed.
3. 3 defined by Christiaan Huygens: Horologium Oscillatorium, Part 4, Definition 3, translated July 2007 by Ian Bruce
4. 4 Simple pendulum
5. 5 Pendulum Systems
6. 6 Simple Pendulum
7. 7 Willis Milham. Time and Timekeepers, p.188-194
8. 8 David Halliday, Robert Resnick, Jearl Walker. Fundamentals of Physics, 5th Ed.
9. 9 Herbert J. Cooper. Scientific Instruments
10. 10 R Nelson, M. G. Olsson (February 1986). The pendulum - Rich physics from a simple system. American Journal of Physics 2 112–121.

Acknowledgements

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