Kater's Reversible Pendulum: Measuring Gravity by Flipping a Bar

A pendulum is a clock that gravity winds. Time its swing, measure its length, and you can calculate the local acceleration of gravity g. The catch is that “length” is easy to define only for an idealised point mass on a weightless string. In 1817 the British physicist and army captain Henry Kater found a way around the problem with a bar that swings from either end Kater's pendulum, Wikipedia^[1] Kater 1818^[2]. That instrument is the subject of our Kater's Reversible Pendulum poster.

Why is a real pendulum hard to measure?

For small swings, a simple pendulum of length L has period Kater's pendulum, Wikipedia^[1]

T = 2π√(L/g)

where T is the period in seconds, L the length in metres and g the acceleration of gravity in m s⁻². A real pendulum is a rigid body, called a compound or physical pendulum. Its small-swing period is Pendulum (mechanics), Wikipedia^[3]

T = 2π√(I_O/(mgh))

where m is the mass in kilograms, h the distance in metres from the pivot to the centre of mass, and I_O the moment of inertia about the pivot in kg m². It swings like a simple pendulum of “equivalent length” L_eq Pendulum (mechanics), Wikipedia^[3]:

L_eq = I_O/(mh) = h + k²/h

The second form follows from the parallel-axis theorem, I_O = m(k² + h²), where k is the radius of gyration about the centre of mass. Before Kater, nobody could locate the point a distance L_eq below the pivot, called the centre of oscillation, accurately enough. It depends on how mass is spread through the metal, and the metallurgy and mathematics of the time could not pin it down Kater's pendulum, Wikipedia^[1]. Experimenters used a metal ball on a fine wire instead, but the ball wobbled and the wire stretched Kater's pendulum, Wikipedia^[1].

What was Kater’s trick?

In 1673 Christiaan Huygens proved, in his Horologium Oscillatorium, that the pivot and the centre of oscillation are interchangeable. Hang a pendulum upside down from its centre of oscillation and it swings with the same period, with the old pivot as the new centre Kater's pendulum, Wikipedia^[1]. You can see this in the formula above: replace h with k²/h and L_eq is unchanged. Huygens is also on our Christiaan Huygens sticker.

Kater turned the theorem into an instrument. His pendulum was a brass rod about 2 m long, carrying two knife-edge pivots facing each other about a metre apart, plus a heavy bob and a small sliding weight Kater's pendulum, Wikipedia^[1]. He timed it from one knife edge, flipped it and timed it from the other, and moved the sliding weight until the two periods matched. At that point the distance between the knife edges is exactly the equivalent length L, a distance that can be measured very precisely Kater's pendulum, Wikipedia^[1]. Then

g = 4π²L/T²

The graphic below shows the idea on a computed model of a rigid bar. The periods from the two pivots cross where the sliding weight puts each pivot at the other’s centre of oscillation, and at the crossing the period equals that of a simple pendulum as long as the knife-edge separation.

How accurate was Kater’s pendulum?

Kater measured the period against a precision clock by the method of coincidences, and the knife-edge separation with a microscope to about 2.5 μm Kater's pendulum, Wikipedia^[1]. He corrected for swing amplitude, temperature, air buoyancy and altitude, and reported the length of the seconds pendulum at London: 39.1386 inches at sea level, in vacuum Kater's pendulum, Wikipedia^[1] Kater 1818^[2]. A seconds pendulum has a two-second period, so one swing takes one second, and its length is simply g/π² Seconds pendulum, Wikipedia^[4]. Kater’s length corresponds to g ≈ 9.81158 m s⁻², with results spread by only about 7 parts per million Kater's pendulum, Wikipedia^[1].

For comparison, the standard acceleration of gravity is defined as exactly 9.80665 m s⁻² NIST CODATA: standard acceleration of gravity^[5], which gives a seconds pendulum about 993.6 mm long Seconds pendulum, Wikipedia^[4]. The real value of g varies from place to place Gravimetry, Wikipedia^[6], which is exactly what made pendulum surveys worth doing. In 1824 Parliament made Kater’s seconds-pendulum length the official backup for defining the yard Kater's pendulum, Wikipedia^[1].

What happened to reversible pendulums?

Kater’s accuracy made gravity measurement a regular part of geodesy. His pendulums travelled on the great 19th-century surveys, including the Great Trigonometric Survey of India Kater's pendulum, Wikipedia^[1]. In 1826 Friedrich Bessel showed that the tedious adjustment could be skipped: if the two periods are close, a correction formula gives the equivalent period directly Kater's pendulum, Wikipedia^[1]. Reversible pendulums were eventually replaced by free-fall gravimeters, which drop a mirror inside a vacuum and time its fall with a laser interferometer Kater's pendulum, Wikipedia^[1] Gravimetry, Wikipedia^[6].

Quick FAQ

Does the swing size matter?

Yes. T = 2π√(L/g) holds only for small swings. Larger swings take longer, and the exact period involves an elliptic integral Pendulum (mechanics), Wikipedia^[3]. The graphic above shows how fast the error grows, which is why Kater corrected for amplitude Kater's pendulum, Wikipedia^[1].

Is this the same as a Foucault pendulum?

No. A Foucault pendulum is a long pendulum whose swing plane slowly turns as the Earth rotates Foucault pendulum, Wikipedia^[7]. See our Foucault pendulum guide, and for how Newton linked pendulums, falling apples and orbits, read our guide to the laws of motion and orbits. More prints are in our physics posters.

References

1. Wikipedia, “Kater's pendulum”. https://en.wikipedia.org/wiki/Kater%27s_pendulum
2. Henry Kater (1818), “An account of experiments for determining the length of the pendulum vibrating seconds in the latitude of London,” Philosophical Transactions of the Royal Society of London 108, 33–102 (DOI 10.1098/rstl.1818.0006; metadata verified via Crossref). https://doi.org/10.1098/rstl.1818.0006
3. Wikipedia, “Pendulum (mechanics)”. https://en.wikipedia.org/wiki/Pendulum_(mechanics)
4. Wikipedia, “Seconds pendulum”. https://en.wikipedia.org/wiki/Seconds_pendulum
5. NIST Reference on Constants, Units and Uncertainty, “standard acceleration of gravity g_n” (exact, 9.80665 m s⁻²). https://physics.nist.gov/cgi-bin/cuu/Value?gn
6. Wikipedia, “Gravimetry”. https://en.wikipedia.org/wiki/Gravimetry
7. Wikipedia, “Foucault pendulum”. https://en.wikipedia.org/wiki/Foucault_pendulum

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.