A pendulum is a mechanical device consisting of a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting mechanical equilibrium position, it is subject to a restoring force due to gravity, which 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 back and forth. The time for one complete cycle, a left swing and a right swing, or frequency, is commonly known as its "period". The period depends both on the length of the pendulum and, somewhat, on the amplitude (the width of the pendulum's swing). The SI unit of the period of a pendulum is the second (s). The regular motion of a pendulum lends itself to timekeeping. The pendulum clock was invented by Christiaan Huygens in 1656. It became the world's standard manner of timekeeping, used in homes and offices for 270 years. The Shortt-Synchronome clock achieved accuracy of about one second per year before it was superseded as a time standard by the quartz clock in the 1930s. Pendulums are also used in scientific instruments such as accelerometers and seismometers. Historically, pendulums were used as gravimeters to measure the acceleration of gravity in geo-physical surveys, and as a standard of length. The word pendulum is Neo-Latin, from the Latin pendulus, meaning 'hanging'.
Mechanics
Simple gravity pendulum The simple gravity pendulum is an idealized mathematical model of a pendulum. This is a weight (or bob) on the end of a massless cord suspended from a pivot, without friction. When given an initial push, it will swing back and forth at a constant amplitude. Real pendulums are subject to friction and air drag, so the amplitude of their swings declines.
Period of oscillation
The period of swing of a simple gravity pendulum depends on its length, the local strength of gravity, and to a small extent on the maximum angle that the pendulum swings away from vertical, θ0, called the amplitude. 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:
where L {\displaystyle L} is the length of the pendulum and g {\displaystyle g} is the local acceleration of gravity. For small swings the period of swing is approximately the same for different size swings: that is, the period is independent of amplitude. This property, called isochronism, is the reason pendulums are so useful for timekeeping. Successive swings of the pendulum, even if changing in amplitude, take the same amount of time. For larger amplitudes, the period increases gradually with amplitude so it is longer than given by equation (1). For example, at an amplitude of θ0 = 0.4 radians (23°) it is 1% larger than given by (1). The period increases asymptotically (to infinity) as θ0 approaches π radians (180°), because the value θ0 = π is an unstable equilibrium point for the pendulum. The true period of an ideal simple gravity pendulum can be written in several different forms (see pendulum (mechanics)), one example being the infinite series:
T = 2 π L g [ ∑ n = 0 ∞ ( ( 2 n ) ! 2 2 n ( n ! ) 2 ) 2 sin 2 n ( θ 0 2 ) ] = 2 π L g ( 1 + 1 16 θ 0 2 + 11 3072 θ 0 4 + ⋯ ) {\displaystyle T=2\pi {\sqrt {\frac {L}{g}}}\left[\sum _{n=0}^{\infty }\left({\frac {\left(2n\right)!}{2^{2n}\left(n!\right)^{2}}}\right)^{2}\sin ^{2n}\left({\frac {\theta _{0}}{2}}\right)\right]=2\pi {\sqrt {\frac {L}{g}}}\left(1+{\frac {1}{16}}\theta _{0}^{2}+{\frac {11}{3072}}\theta _{0}^{4}+\cdots \right)}
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