Seconds pendulum

Riefler precision pendulum clocks with seconds pendulum and compensation

A second pendulum is a pendulum that needs exactly one second for a half oscillation ( called “strike” in watchmaking ) . When accurate clocks were not yet available, it was used to measure short periods of time and for physical experiments. From the 17th century it was used for precise pendulum clocks , especially in observatories to determine the time and for the precise measurement of star words .

In 1790 the Paris Academy discussed whether it was suitable for defining a new measure of length - the meter ; in fact, its length is about a meter. However, the period of oscillation is independent of the mass of the pendulum body, but is influenced by the prevailing force of gravity . Therefore, the seconds pendulum is between 99.1 and 99.6 cm long, depending on the geographical latitude of the location.

Mathematical pendulum

The theoretically ideal pendulum would be a point mass at the end of a massless rod that swings with an infinitely small amplitude around a frictionless axis. At the 45th parallel , this mathematical second pendulum has a length of 99.4 cm. This length results from the fact that the period of oscillation of an ideal pendulum depends only on its length and the acceleration due to gravity ${\displaystyle T}$ ${\displaystyle l}$ ${\displaystyle g}$

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

The required pendulum length depends on the duration of a half oscillation ${\displaystyle T_{1/2}}$

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

So with and you get . ${\displaystyle T_{1/2}=1\ \mathrm {s} }$ ${\displaystyle g=9{,}806\ \mathrm {\frac {m}{s^{2}}} }$ ${\displaystyle l=0{,}9936\ \mathrm {m} }$

The value for g = 9.806 m / s² only applies to sea ​​level and medium geographical latitudes . At the earth's equator it is 9.7803 and at the poles 9.8322 m / s².

The given formula for the period of oscillation is a linearization of the equation of motion. It only applies to a simplified mathematical pendulum with an infinitely small oscillation amplitude, which is strictly not possible physically. Neither mass distribution ( center of gravity of the pendulum body) nor amplitude errors are taken into account. However, the formula is useful for a rough estimate of the length of a clock pendulum.

Influence of the amplitude

The linearized formula (without the higher order terms) simulates an isochronism to the user , as if the same period would apply to different oscillation amplitudes. The calculation error of this small-angle approximation is 0.02% at an operating amplitude of 120 arc minutes (2 °), but can be in the percentage range for pendulums that swing far outwards. A clock pendulum with an oscillation amplitude of 31 ° would lose 100 seconds a day compared to a similar clock that oscillates with 30 ° amplitude. At the amplitudes of 11 ° and 10 ° this value is 35 s. In the case of small oscillation amplitudes, the amplitude error is therefore negligible for domestic use.

The seconds pendulum has generally established itself as the frequency standard in precision pendulum clocks . When the more precise mechanism of the watch escapement was able to reduce friction and keep the amplitude constant, the accuracy increased to a few millionths. The second hand was connected to the shaft of the escape wheel so that it jumps on exactly with the pendulum swing and significantly refines the eye-ear method of measuring time.

Special constructive tricks such as compensation of thermal effects, evacuation and systematic suppression of external disturbance variables made it possible to achieve accuracies better than a tenth of a second per day as early as the 18th century , which was only exceeded by the first quartz watches around 1930 . At this time, the almost frictionless Shortt clocks were already reaching 0.01 s / day.

In the course of the introduction of the meter, it was originally planned to define this length measurement using a seconds pendulum at 45 ° N; Instead, when the meter was introduced in 1793, a more precise, geodetic definition (1 m = 1 / 10,000,000 of the length of the meridian quadrant running through Paris ) was used, as an ellipsoid was already assumed for the earth figure .

history

Pendulum device with one-second reversion pendulum by Adolf Repsold from 1869, GeoForschungsZentrum , Potsdam

Marin Mersenne (1588–1648), Jean Richer (1630–1696), Jean-Charles de Borda (1733–1799), Jean-Baptiste Biot (1774–1862) and François Arago (1786–1853 ) undertook experimental studies to determine the length of the pendulum ), Henry Kater (1777-1835). Friedrich Wilhelm Bessel (1784–1846) carried out extensive investigations into the length of the pendulum and the factors influencing it with a pendulum apparatus designed by him and manufactured by Johann Georg Repsold . Heinrich Christian Schumacher also carried out corresponding investigations at Gut Güldenstein in Holstein in 1829/30 .

See also

• Riefler pendulum

Individual evidence

1. ↑ on the original basis of one second as part of a day divided into 24 hours with 60 minutes of 60 seconds each
2. ↑ Virtual Museum of TU Graz: Second pendulum ( Memento from December 11, 2013 in the Internet Archive )
3. ^ Zdeněk Martínek and Jaroslav Řehoř: Mechanical watches. VEB Verlag Technik, Berlin 1988; ISBN 3-341-00022-4 , p. 15 f.
4. ^ Günter Krug: Mechanical clocks, VEB Verlag Technik, Berlin 1987; ISBN 3-341-00356-8 , p. 183 f.
5. ↑ Based on an older proposal by Pierre Bouguer , which Talleyrand introduced to the French National Assembly, where it was adopted by decree on May 8, 1790 .