Seconds pendulum

A seconds pendulum is a pendulum whose period is precisely two seconds; one second for a swing in one direction and one second for the return swing, a frequency of 0.5 Hz.

Principles A pendulum is a weight suspended from a pivot so that it can swing freely. 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 combined with 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, is called the period. The period depends on the length of the pendulum, and also to a slight degree on its weight distribution (the moment of inertia about its own center of mass) and the amplitude (width) of the pendulum's swing. For a simple gravity pendulum — a point mass on a weightless string of length ℓ {\displaystyle \ell } swinging with an infinitesimally small amplitude, without resistance — the period of the pendulum will be:

T = 2 π ℓ g . {\displaystyle T=2\pi {\sqrt {\frac {\ell }{g}}}.}

The length of the pendulum is a function of the time lapse of half a cycle T 1 / 2 {\displaystyle T_{1/2}}

ℓ = g ( T 1 / 2 π ) 2 . {\displaystyle \ell =g\left({\frac {T_{1/2}}{\pi }}\right)^{2}.}

With T 1 / 2 = 1 s {\displaystyle T_{1/2}=1\ \mathrm {s} } , gives g = ℓ ⋅ π 2 {\displaystyle g={\ell \cdot \pi ^{2}}}

where g is the acceleration due to gravity, with quantity dimension of length per time squared. Using the standard acceleration of gravity g0 = 9.80665 m/s2, the length of the string will be approximately 993.6 millimetres, i.e. less than a centimetre short of one metre everywhere on Earth. The arc of a simple gravity pendulum is not isochronous motion: larger amplitude swings take slightly longer. To obtain motion independent of amplitude, the pendulum needs to move along a cycloidal path rather than a circle.

Defining the second

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