# Seconds pendulum

Longcase clock with a seconds pendulum.
The seconds pendulum, with a period of two seconds so each swing takes one second

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 1/2 Hz. A pendulum is a weight suspended from a pivot so that it can swing freely.[1] 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.

In 1670 the seconds pendulum was employed by William Clement in his improved version of the original pendulum clock by Christiaan Huygens, creating the longcase clock which could tick seconds.[2]

The length of a seconds pendulum was determined (in toises) by Marin Mersenne in 1644. In 1660, the Royal Society proposed that it be the standard unit of length. In 1671 Jean Picard measured this length at the Paris observatory. He found the value of 440.5 lines of the Toise of Châtelet which had been recently renewed. He proposed an universal toise (French: Toise universelle) which was twice the length of the seconds pendulum.[3][4] However, it was soon discovered that the length of a seconds pendulum varies from place to place: French astronomer Jean Richer had measured the 0.3% difference in length between Cayenne (in French Guiana) and Paris.[5]

Jean Richer and Giovanni Domenico Cassini measured the parallax of Mars between Paris and Cayenne in French Guiana when Mars was at its closest to Earth in 1672. They arrived at a figure for the solar parallax of 91/2 inches, equivalent to an Earth–Sun distance of about 22000 Earth radii. They were also the first astronomers to have access to an accurate and reliable value for the radius of Earth, which had been measured by their colleague Jean Picard in 1669 as 3269 thousand toises. Picard's geodetic observations had been confined to the determination of the magnitude of the earth considered as a sphere, but the discovery made by Jean Richer turned the attention of mathematicians to its deviation from a spherical form. The determination of the figure of the earth became a problem of the highest importance in astronomy, inasmuch as the diameter of the earth was the unit to which all celestial distances had to be referred.

In 1790, one year before the metre was ultimately based on a quadrant of the Earth, Talleyrand proposed that the metre be the length of the seconds pendulum at a latitude of 45°.[1] This option, with one-third of this length defining the foot, was also considered by Thomas Jefferson and others for redefining the yard in the United States shortly after gaining independence from the British Crown.[6]

In the beginning of the 19th century François Arago and Jean-Baptiste Biot determined the length of the seconds pendulum along the Paris meridian arc (French: Méridienne de France) which extended from Balearic Islands to Shetland.[7]

Standard gravity was established in 1901 according to the value of the gravitational strength found by Gilbert Étienne Defforges at the International Bureau of Weights and Measures in 1888. According to his results the length of a seconds pendulum at this place would be about 0.994 m (39.1 in). The standard gravity (g0) defined by standard as 9.806 65 m/s2 is a nominal midrange value on Earth, originally based on the acceleration of a body in free fall at sea level at a geodetic latitude of 45°. Indeed the acceleration of a body near the surface of the Earth is due to the combined effects of gravity and centrifugal acceleration. The resulting acceleration towards the ground is about 0.5% greater at the poles than at the Equator.

## Length of the pendulum

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

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

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

Being ${\displaystyle T_{1/2}=1\ \mathrm {s} }$ and ${\displaystyle g=9.81\ \mathrm {\frac {m}{s^{2}}} }$ therefore ${\displaystyle \ell =0.994\ \mathrm {m} }$.