A Kater's pendulum is a reversible free swinging pendulum 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:
T = 2 π L g ( 1 ) {\displaystyle T=2\pi {\sqrt {\frac {L}{g}}}\qquad \qquad \qquad (1)\,}
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:
g = 8 π 2 T 1 2 + T 2 2 ℓ 1 + ℓ 2 + T 1 2 − T 2 2 ℓ 1 − ℓ 2 {\displaystyle g={\frac {8\pi ^{2}}{{\dfrac {T_{1}^{2}+T_{2}^{2}}{\ell _{1}+\ell _{2}}}+{\dfrac {T_{1}^{2}-T_{2}^{2}}{\ell _{1}-\ell _{2}}}}}}
where T1 and T2 are the time periods of oscillations when it is suspended from K1 and K2 respectively and ℓ1 and ℓ2 are the distances of knife edges K1 and K2 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, 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 2+1⁄2 minutes per day slower than at Paris, or equivalently the length of a pendulum with a swing of one second there was 1+1⁄4 Paris lines, or 2.6 mm, shorter than at Paris. It was realized by the scientists of the day, and proven by Isaac Newton 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 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:
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