Editing Pendulum (section)

=== Kater's pendulum ===
{{Main|Kater's pendulum}}
{| style="float:right;"
|- 
|[[File:PenduloCaminos.jpg|thumb|115px|Kater's pendulum and stand]]
||[[File:Kater pendulum use.png|thumb|200px|Measuring gravity with Kater's reversible pendulum, from Kater's 1818 paper]]
|}
[[File:Kater pendulum vertical.png|thumb|upright=0.4|A Kater's pendulum]]
{{anchor|huygens-law}}<!-- anchor used by (at least) [[Huygens' law of the pendulum]] -->
The precision of the early gravity measurements above was limited by the difficulty of measuring the length of the pendulum, ''L'' .  ''L'' was the length of an idealized simple gravity pendulum (described at top), which has all its mass concentrated in a point at the end of the cord. In 1673 Huygens had shown that the period of a rigid bar pendulum (called a ''compound pendulum'') was equal to the period of a simple pendulum with a length equal to the distance between the [[wikt:pivot|pivot]] point and a point called the [[Center of percussion|center of oscillation]], located under the [[center of gravity]], that depends on the mass distribution along the pendulum. But there was no accurate way of determining the center of oscillation in a real pendulum. Huygens' discovery is sometimes referred to as ''Huygens' law of the (cycloidal) pendulum''.<ref>{{cite book |doi=10.1093/acprof:oso/9780199570409.003.0005 |title=Isaac Newton's Scientific Method: Turning Data into Evidence about Gravity and Cosmology |chapter=Christiaan Huygens: A Great Natural Philosopher Who Measured Gravity and an Illuminating Foil for Newton on Method |last=Harper |first=William L. |date=Dec 2011|pages=194–219 |isbn=978-0-19-957040-9 }}</ref>

To get around this problem, the early researchers above approximated an ideal simple pendulum as closely as possible by using a metal sphere suspended by a light wire or cord. If the wire was light enough, the center of oscillation was close to the center of gravity of the ball, at its geometric center. This "ball and wire" type of pendulum wasn't very accurate, because it didn't swing as a rigid body, and the elasticity of the wire caused its length to change slightly as the pendulum swung.

However Huygens had also proved that in any pendulum, the pivot point and the center of oscillation were interchangeable.<ref name="HuygensReciprocity" />  That is, if a pendulum were turned upside down and hung from its center of oscillation, it would have the same period as it did in the previous position, and the old pivot point would be the new center of oscillation.

British physicist and army captain [[Henry Kater]] in 1817 realized that Huygens' principle could be used to find the length of a simple pendulum with the same period as a real pendulum.<ref name="Kater1818" />  If a pendulum was built with a second adjustable pivot point near the bottom so it could be hung upside down, and the second pivot was adjusted until the periods when hung from both pivots were the same, the second pivot would be at the center of oscillation, and the distance between the two pivots would be the length ''L'' of a simple pendulum with the same period.

Kater built a reversible pendulum (''see drawing'') consisting of a brass bar with two opposing pivots made of short triangular "knife" blades ''<span style="color:red;">(a)</span>'' near either end. It could be swung from either pivot, with the knife blades supported on agate plates. Rather than make one pivot adjustable, he attached the pivots a meter apart and instead adjusted the periods with a moveable weight on the pendulum rod ''<span style="color:red;">(b,c)</span>''. In operation, the pendulum is hung in front of a precision clock, and the period timed, then turned upside down and the period timed again. The weight is adjusted with the adjustment screw until the periods are equal. Then putting this period and the distance between the pivots into equation (1) gives the gravitational acceleration ''g'' very accurately.

Kater timed the swing of his pendulum using the "''method of coincidences''" and measured the distance between the two pivots with a micrometer. After applying corrections for the finite amplitude of swing, the buoyancy of the bob, the barometric pressure and altitude, and temperature, he obtained a value of 39.13929&nbsp;inches for the seconds pendulum at London, in vacuum, at sea level, at 62&nbsp;°F. The largest variation from the mean of his 12 observations was 0.00028 in.<ref>{{cite book
  | last = Cox
  | first = John
  | title = Mechanics
  | publisher = Cambridge Univ. Press
  | year = 1904
  | location = Cambridge, UK
  | pages = [https://archive.org/details/mechanics00coxgoog/page/n341 311]–312
  | url = https://archive.org/details/mechanics00coxgoog
  }}</ref> representing a precision of gravity measurement of 7×10<sup>−6</sup> (7 [[mGal]] or 70 [[Metre per second squared|μm/s<sup>2</sup>]]). Kater's measurement was used as Britain's official standard of length (see [[#Britain and Denmark|below]]) from 1824 to 1855.

Reversible pendulums (known technically as "convertible" pendulums) employing Kater's principle were used for absolute gravity measurements into the 1930s.