
A Kater's pendulum is a reversible free swinging
pendulum
A pendulum is a device made of 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 i ...
invented by British physicist and army captain
Henry Kater in 1817 (made public on 29 January 1818),
for use as a
gravimeter instrument to measure the local
acceleration of gravity. Its advantage is that, unlike previous pendulum gravimeters, the pendulum's
centre of gravity and
center of oscillation do not have to be determined, allowing a greater accuracy. For about a century, until the 1930s, Kater's pendulum and its various refinements remained the standard method for measuring the strength of the Earth's gravity during
geodetic surveys. It is now used only for demonstrating pendulum principles.
Description
A pendulum can be used to measure the
acceleration of gravity ''g'' because for narrow swings (when the
small angle approximation is valid) its
period of swing ''T'' depends only on ''g'' and its length ''L'':
:
So by measuring the length ''L'' and period ''T'' of a pendulum, ''g'' can be calculated.
The Kater's pendulum consists of a rigid metal bar with two pivot points, one near each end of the bar. It can be suspended from either pivot and swung. It also has either an adjustable weight that can be moved up and down the bar, or one adjustable pivot, to adjust the periods of swing. In use, it is swung from one pivot, and the
period timed, and then turned upside down and swung from the other pivot, and the period timed. The movable weight (or pivot) is adjusted until the two periods are equal. At this point the period ''T'' is equal to the period of an 'ideal' simple pendulum of length equal to the distance between the pivots. From the period and the measured distance ''L'' between the pivots, the acceleration of gravity can be calculated with great precision from the equation (1) above.
The acceleration due to gravity by Kater's pendulum is given by:
:
where ''T''
1 and ''T''
2 are the time periods of oscillations when it is suspended from ''K''
1 and ''K''
2 respectively and ''ℓ''
1 and ''ℓ''
2 are the distances of knife edges ''K''
1 and ''K''
2 from the center of gravity respectively.
History
Gravity measurement with pendulums

The first person to discover that gravity varied over the Earth's surface was French scientist
Jean Richer, who in 1671 was sent on an expedition to
Cayenne,
French Guiana
French Guiana, or Guyane in French, is an Overseas departments and regions of France, overseas department and region of France located on the northern coast of South America in the Guianas and the West Indies. Bordered by Suriname to the west ...
, by the French
Académie des Sciences, assigned the task of making measurements with a
pendulum clock. Through the observations he made in the following year, Richer determined that the clock was minutes per day slower than at Paris, or equivalently the length of a pendulum with a swing of one second there was Paris ''lines'', or 2.6 mm, shorter than at Paris. It was realized by the scientists of the day, and proven by
Isaac Newton
Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
in 1687, that this was due to the fact that the Earth was not a perfect sphere but slightly
oblate; it was thicker at the equator because of the Earth's rotation. Since the surface was farther from the Earth's center at Cayenne than at Paris, gravity was weaker there. After that discovery was made, freeswinging pendulums started to be used as precision
gravimeters, taken on voyages to different parts of the world to measure the local gravitational acceleration. The accumulation of geographical gravity data resulted in more and more accurate models of the overall shape of the Earth.
Pendulums were so universally used to measure gravity that, in Kater's time, the local strength of gravity was usually expressed not by the value of the acceleration ''g'' now used, but by the length at that location of the ''
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 tha ...
'', a pendulum with a period of two seconds, so each swing takes one second. It can be seen from equation (1) that for a seconds pendulum, the length is simply proportional to ''g'':
:
Inaccuracy of gravimeter pendulums
In Kater's time, the period ''T'' of pendulums could be measured very precisely by timing them with precision clocks set by the passage of stars overhead. Prior to Kater's discovery, the accuracy of ''g'' measurements was limited by the difficulty of measuring the other factor ''L'', the length of the pendulum, accurately. ''L'' in equation (1) above was the length of an ideal mathematical 'simple pendulum' consisting of a point mass swinging on the end of a massless cord. However the 'length' of a real pendulum, a swinging rigid body, known in mechanics as a
compound pendulum, is more difficult to define. In 1673 Dutch scientist
Christiaan Huygens
Christiaan Huygens, Halen, Lord of Zeelhem, ( , ; ; also spelled Huyghens; ; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor who is regarded as a key figure in the Scientific Revolution ...
in his mathematical analysis of pendulums, ''Horologium Oscillatorium'', showed that a real pendulum had the same period as a simple pendulum with a length equal to the distance between the pivot point and a point called the ''
center of oscillation'', which is located under the pendulum's
center of gravity and depends on the mass distribution along the length of the pendulum. The problem was there was no way to find the location of the center of oscillation in a real pendulum accurately. It could theoretically be calculated from the shape of the pendulum if the metal parts had uniform density, but the metallurgical quality and mathematical abilities of the time didn't allow the calculation to be made accurately.
To get around this problem, most early gravity researchers, such as
Jean Picard (1669),
Charles Marie de la Condamine (1735), and
Jean-Charles de Borda (1792) approximated a simple pendulum by using a metal sphere suspended by a light wire. If the wire had negligible mass, the center of oscillation was close to the center of gravity of the sphere. But even finding the center of gravity of the sphere accurately was difficult. In addition, this type of pendulum inherently wasn't very accurate. The sphere and wire didn't swing back and forth as a rigid unit, because the sphere acquired a slight
angular momentum
Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
during each swing. Also the wire stretched elastically during the pendulum's swing, changing ''L'' slightly during the cycle.
Kater's solution
However, in ''Horologium Oscillatorium'', Huygens had also proved that the pivot point and the center of oscillation were interchangeable. That is, if any pendulum is suspended upside down from its center of oscillation, it has the same period of swing, and the new center of oscillation is the old pivot point. The distance between these two conjugate points was equal to the length of a simple pendulum with the same period.
As part of a committee appointed by the
Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
in 1816 to reform British measures, Kater had been contracted by the House of Commons to determine accurately the length of the seconds pendulum in London. He realized Huygens' principle could be used to find the center of oscillation, and so the length ''L'', of a rigid (compound) pendulum. If a pendulum were hung upside down from a second pivot point that could be adjusted up and down on the pendulum's rod, and the second pivot were adjusted until the pendulum had the same period as it did when swinging right side up from the first pivot, the second pivot would be at the center of oscillation, and the distance between the two pivot points would be ''L''.
Kater was not the first to have this idea. French mathematician
Gaspard de Prony first proposed a reversible pendulum in 1800, but his work was not published until 1889. In 1811
Friedrich Bohnenberger again discovered it, but Kater independently invented it and was first to put it in practice.
The pendulum
Kater built a pendulum consisting of a brass rod about 2 meters long, inches wide and one-eighth inch thick, with a weight ''
(d)'' on one end.
For a low friction pivot he used a pair of short triangular 'knife' blades attached to the rod. In use the pendulum was hung from a bracket on the wall, supported by the edges of the knife blades resting on flat agate plates. The pendulum had two of these knife blade pivots ''
(a)'', facing one another, about a meter (40 in) apart, so that a swing of the pendulum took approximately one second when hung from each pivot.
Kater found that making one of the pivots adjustable caused inaccuracies, making it hard to keep the axis of both pivots precisely parallel. Instead he permanently attached the knife blades to the rod, and adjusted the periods of the pendulum by a small movable weight ''
(b,c)'' on the pendulum shaft. Since gravity only varies by a maximum of 0.5% over the Earth, and in most locations much less than that, the weight had to be adjusted only slightly. Moving the weight toward one of the pivots decreased the period when hung from that pivot, and increased the period when hung from the other pivot. This also had the advantage that the precision measurement of the separation between the pivots had to be made only once.
Experimental procedure
To use, the pendulum was hung from a bracket on a wall, with the knife blade pivots supported on two small horizontal agate plates, in front of a precision pendulum clock to time the period. It was swung first from one pivot, and the oscillations timed, then turned upside down and swung from the other pivot, and the oscillations timed again. The small weight ''
(b)'' was adjusted with the adjusting screw, and the process repeated until the pendulum had the same period when swung from each pivot. By putting the measured period ''T'', and the measured distance between the pivot blades ''L'', into the period equation (1), ''g'' could be calculated very accurately.
Kater performed 12 trials.
He measured the period of his pendulum very accurately using the clock pendulum by the ''method of coincidences''; timing the interval between the ''coincidences'' when the two pendulums were swinging in synchronism. He measured the distance between the pivot blades with a microscope comparator, to an accuracy of 10
−4 in. (2.5 μm). As with other pendulum gravity measurements, he had to apply small corrections to the result for a number of variable factors:
*the non-zero width of the pendulum's swing, which increased the period
*temperature, which caused the length of the rod to vary due to
thermal expansion
Thermal expansion is the tendency of matter to increase in length, area, or volume, changing its size and density, in response to an increase in temperature (usually excluding phase transitions).
Substances usually contract with decreasing temp ...
*atmospheric pressure, which reduced the effective mass of the pendulum by the buoyancy of the displaced air, increasing the period
*altitude, which reduced the gravitational force with distance from the center of the Earth. Gravity measurements are always referenced to
sea level
Mean sea level (MSL, often shortened to sea level) is an mean, average surface level of one or more among Earth's coastal Body of water, bodies of water from which heights such as elevation may be measured. The global MSL is a type of vertical ...
.
He gave his result as the length of the
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 tha ...
. After corrections, he found that the mean length of the solar seconds pendulum at London, at sea level, at , swinging in vacuum, was 39.1386 inches. This is equivalent to a gravitational acceleration of 9.81158 m/s
2. The largest variation of his results from the mean was . This represented a precision of gravity measurement of 0.7×10
−5 (7
milligals).
In 1824, the British Parliament made Kater's measurement of the seconds pendulum the official backup standard of length for defining the
yard
The yard (symbol: yd) is an English units, English unit of length in both the British imperial units, imperial and US United States customary units, customary systems of measurement equalling 3 foot (unit), feet or 36 inches. Sinc ...
if the yard prototype was destroyed.
[''An Act for ascertaining and establishing Uniformity of Weights and Measures'', British Parliament, 17 June 1824, reprinted in The wording of the Act indicates that the pendulum definition is to be used to restore the yard if the prototype is destroyed.]
Use

The large increase in gravity measurement accuracy made possible by Kater's pendulum established
gravimetry as a regular part of
geodesy
Geodesy or geodetics is the science of measuring and representing the Figure of the Earth, geometry, Gravity of Earth, gravity, and Earth's rotation, spatial orientation of the Earth in Relative change, temporally varying Three-dimensional spac ...
. To be useful, it was necessary to find the exact location (latitude and longitude) of the 'station' where a gravity measurement was taken, so pendulum measurements became part of
surveying
Surveying or land surveying is the technique, profession, art, and science of determining the land, terrestrial Plane (mathematics), two-dimensional or Three-dimensional space#In Euclidean geometry, three-dimensional positions of Point (geom ...
. Kater's pendulums were taken on the great historic
geodetic surveys of much of the world that were being done during the 19th century. In particular, Kater's pendulums were used in the
Great Trigonometric Survey
The Great Trigonometrical Survey of India was a project that aimed to carry out a wikt:Special:Search/survey, survey across the Indian subcontinent with scientific precision. It was begun in 1802 by the British infantry officer William Lambton ...
of India.
Reversible pendulums remained the standard method used for absolute gravity measurements until they were superseded by free-fall
gravimeters in the 1950s.
Repsold–Bessel pendulum

Repeatedly timing each period of a Kater pendulum, and adjusting the weights until they were equal, was time-consuming and error-prone.
Friedrich Bessel
Friedrich Wilhelm Bessel (; 22 July 1784 – 17 March 1846) was a German astronomer, mathematician, physicist, and geodesy, geodesist. He was the first astronomer who determined reliable values for the distance from the Sun to another star by th ...
showed in 1826 that this was unnecessary. As long as the periods measured from each pivot, ''T''
1 and ''T''
2, are close in value, the period ''T'' of the equivalent simple pendulum can be calculated from them:
:
Here
and
are the distances of the two pivots from the pendulum's center of gravity. The distance between the pivots,
, can be measured with great accuracy.
and
, and thus their difference
, cannot be measured with comparable accuracy. They are found by balancing the pendulum on a knife edge to find its center of gravity, and measuring the distances of each of the pivots from the center of gravity. However, because
is so much smaller than
, the second term on the right in the above equation is small compared to the first, so
doesn't have to be determined with high accuracy, and the balancing procedure described above is sufficient to give accurate results.
Therefore, the pendulum doesn't have to be adjustable at all, it can simply be a rod with two pivots. As long as each pivot is close to the
center of oscillation of the other, so the two periods are close, the period ''T'' of the equivalent simple pendulum can be calculated with equation (2), and the gravity can be calculated from ''T'' and ''L'' with (1).
In addition, Bessel showed that if the pendulum was made with a symmetrical shape, but internally weighted on one end, the error caused by effects of air resistance would cancel out. Also, another error caused by the non-zero radius of the pivot knife edges could be made to cancel out by interchanging the knife edges.
Bessel didn't construct such a pendulum, but in 1864 Adolf Repsold, under contract to the Swiss Geodetic Commission, developed a symmetric pendulum 56 cm long with interchangeable pivot blades, with a period of about second. The Repsold pendulum was used extensively by the Swiss and Russian Geodetic agencies, and in the
Survey of India. Other widely used pendulums of this design were made by
Charles Peirce and C. Defforges.
International Association of Geodesy
The 1875 Conference of the
European Arc Measurement dealt with the best instrument to be used for the determination of gravity. The association decided in favor of the reversion pendulum and it was resolved to redo in Berlin, in the station where
Friedrich Wilhelm Bessel made his famous measurements, the determination of gravity by means of devices of various kinds employed in different countries, in order to compare them and thus to have the equation of their scales, after an in-depth discussion in which an American scholar,
Charles Sanders Peirce
Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American scientist, mathematician, logician, and philosopher who is sometimes known as "the father of pragmatism". According to philosopher Paul Weiss (philosopher), Paul ...
, took part. Indeed, as the
figure of the Earth
In geodesy, the figure of the Earth is the size and shape used to model planet Earth. The kind of figure depends on application, including the precision needed for the model. A spherical Earth is a well-known historical approximation that is ...
could be inferred from variations of the
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 tha ...
length, the
United States Coast Survey's direction instructed
Charles Sanders Peirce
Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American scientist, mathematician, logician, and philosopher who is sometimes known as "the father of pragmatism". According to philosopher Paul Weiss (philosopher), Paul ...
in the spring of 1875 to proceed to Europe for the purpose of making pendulum experiments to chief initial stations for operations of this sort, in order to bring the determinations of the forces of gravity in America into communication with those of other parts of the world; and also for the purpose of making a careful study of the methods of pursuing these researches in the different countries of Europe.
The determination of gravity by the reversible pendulum was subject to two types of error. On the one hand the resistance of the air and on the other hand the movements that the oscillations of the pendulum imparted to its plane of suspension. These movements were particularly important with the apparatus designed by the Repsold brothers on the indications of Bessel, because the pendulum had a large mass in order to counteract the effect of the viscosity of the air. While
Emile Plantamour was carrying out a series of experiments with this device,
Adolph Hirsch found a way to demonstrate the movements of the pendulum's suspension plane by an ingenious process of optical amplification. Isaac-Charles Élisée Cellérier, a mathematician from Geneva and Charles Sanders Peirce would independently develop a correction formula that allowed the use of the observations made with this type of gravimeter.
President of the Permanent Commission of the European Arc Measurement from 1874 to 1886,
Carlos Ibáñez Ibáñez de Ibero became the first president of the
International Geodetic Association (1887–1891) after the death of
Johann Jacob Baeyer. Under Ibáñez's presidency, the International Geodetic Association acquired a global dimension with the accession of the
United States
The United States of America (USA), also known as the United States (U.S.) or America, is a country primarily located in North America. It is a federal republic of 50 U.S. state, states and a federal capital district, Washington, D.C. The 48 ...
,
Mexico
Mexico, officially the United Mexican States, is a country in North America. It is the northernmost country in Latin America, and borders the United States to the north, and Guatemala and Belize to the southeast; while having maritime boundar ...
,
Chile
Chile, officially the Republic of Chile, is a country in western South America. It is the southernmost country in the world and the closest to Antarctica, stretching along a narrow strip of land between the Andes, Andes Mountains and the Paci ...
,
Argentina
Argentina, officially the Argentine Republic, is a country in the southern half of South America. It covers an area of , making it the List of South American countries by area, second-largest country in South America after Brazil, the fourt ...
and
Japan
Japan is an island country in East Asia. Located in the Pacific Ocean off the northeast coast of the Asia, Asian mainland, it is bordered on the west by the Sea of Japan and extends from the Sea of Okhotsk in the north to the East China Sea ...
. As a result of the work of the International Geodetic Association, in 1901,
Friedrich Robert Helmert found, mainly by gravimetry, parameters of the
ellipsoid
An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
remarkably close to reality.
References
External links
The Accurate Measurement of ''g'' using Kater's pendulum, U. of Sheffield Has derivation of equations
Kater, Henry (June 1818) An Account of the Experiments for determining the length of the pendulum vibrating seconds in the latitude of London, The Edinburgh Review, Vol. 30, p.407Has detailed account of experiment, description of pendulum, value determined, interest of French scientists
{{DEFAULTSORT:Kater's Pendulum
Pendulums