Kater's Pendulum: Weighing Gravity With a Swinging Bar

The problem: you can't see the center of oscillation

A real pendulum is never a point mass on a string. Swing any rigid body about a horizontal axis and it obeys the physical (compound) pendulum equation of motion. Newton's rotational form, τ = Iα, gives −mgh·sinθ = I·(d²θ/dt²), where m is the mass, h the distance from the pivot to the center of gravity, I the moment of inertia about the pivot, and θ the swing angle. For small θ, sinθ ≈ θ and the motion is simple harmonic with period:

• T = 2π√(I / (mgh)) — the master equation for any swinging rigid body.

Define an equivalent simple pendulum length L_eq = I/(mh). Then T = 2π√(L_eq/g), exactly like a mass on a massless string of length L_eq. To extract g you would need to know T (easy — just time it) and L_eq (hard). The trouble is that L_eq depends on I and on h, and both of those depend on knowing precisely where the center of gravity sits and how the mass is distributed. For a heavy machined bar those are the two things you can measure worst. That single blind spot capped 18th-century gravimetry at roughly a part in a thousand.

Kater's insight: reverse the bar and cancel the unknowns

Use the parallel-axis theorem, I = I_cm + mh² = m(k² + h²), where k is the radius of gyration about the center of mass (I_cm = mk²). The period becomes:

• T = 2π√((k² + h²) / (gh)), so the equivalent length is L_eq = (k² + h²)/h = h + k²/h.

Here is the beautiful symmetry Kater exploited. For a given L_eq, the equation h + k²/h = L_eq is a quadratic in h with two roots, h₁ and h₂, that satisfy h₁·h₂ = k² and h₁ + h₂ = L_eq. In plain terms: there are two distinct pivot distances on opposite sides of the center of mass that give the exact same period. One of them is the center of suspension, the other the center of oscillation — and they are interchangeable (a fact first proven by Huygens in 1673).

Kater's move: build a bar with two fixed knife edges at distances h₁ and h₂ from the (unknown) center of gravity, and add a sliding weight to tune the mass distribution until the period timed from edge 1 equals the period timed from edge 2. When T₁ = T₂ and h₁ ≠ h₂, those two edges are the conjugate points, so L_eq = h₁ + h₂ = the physical distance between the knife edges. You measure that distance directly with a micrometer. Everything else cancels.

The payoff equation: g = 4π²L / T²

Once the two periods are matched, the physical pendulum collapses onto an ideal simple pendulum of length L equal to the knife-edge spacing:

• Set T = 2π√(L/g) and solve → g = 4π²L / T².

That is the entire experiment in one line. Note what is absent: no mass, no center-of-gravity location, no moment of inertia, no radius of gyration k. They vanished because the reversal condition forced h₁·h₂ = k² and h₁ + h₂ = L without ever solving for k individually. You need only two measurable quantities:

• T — the period, measured by timing many hundreds of swings against a precise clock (Kater used the coincidence method, watching his pendulum beat against a standard clock pendulum and timing when they drifted in and out of phase).
• L — the knife-edge separation, measured mechanically.

Because g scales as L/T², and because L can be measured to microns and T to microseconds over a long swing count, the accuracy is extraordinary. Kater reported the length of the seconds pendulum in London — the pendulum whose half-period is exactly 1 s — as 39.1386 inches, corresponding to g = 9.81158 m/s².

What actually limits the precision

The elegance hides real work. In practice T₁ and T₂ can never be tuned to be perfectly equal, so Bessel's correction (Friedrich Bessel, 1826) handles the small residual. Writing the two exact physical-pendulum relations and eliminating k gives:

• 8π²/g = (T₁² + T₂²)/(h₁ + h₂) + (T₁² − T₂²)/(h₁ − h₂).

The first term contains only the well-measured sum (h₁ + h₂) = L, the knife-edge distance. The second term contains the poorly known difference (h₁ − h₂), but it is multiplied by (T₁² − T₂²), which is driven nearly to zero by matching the periods. So the CM location only ever appears in a term that has been made vanishingly small — the cancellation is robust, not just exact at one point. That is why the design tolerates imperfect tuning. Real error budget:

• Finite-amplitude error: the small-angle formula is only the leading term; the true period grows as T ≈ T₀(1 + θ₀²/16 + …), so a 3° swing already shifts T by ~1.7 × 10⁻⁴. Keep θ₀ small, or correct for it.
• Air buoyancy and added mass: air reduces the effective restoring weight and drags along a co-moving mass; Kater's best numbers were reduced to a vacuum swing.
• Knife-edge flex and finite radius: a real edge is not a perfect line; its rounding shifts the effective pivot by microns.
• Thermal expansion: brass expands, so L drifts with temperature — hence Kater quoting his value at 62 °F (17 °C).

A worked feel for the numbers

Design a seconds pendulum: you want a full period T = 2 s (half-period beats at 1 s). Invert g = 4π²L/T² for the required edge spacing at g = 9.81 m/s²:

• L = gT²/(4π²) = 9.81 × 4 / 39.48 ≈ 0.994 m.

So the knife edges sit just under a meter apart — Kater's bar was about 1.6 m long overall to hang the adjustable masses beyond the edges. Now see the sensitivity. Differentiate g = 4π²L/T²: dg/g = dL/L − 2·dT/T. To reach 1 part in 10⁶ in g you need L good to 1 µm in a meter (achievable with a comparator) and T good to 5 × 10⁻⁷ — impossible in one swing, but trivial if you time 10,000 swings: a timing error spread over ~5.5 hours of swinging shrinks by the swing count. This is the deep reason pendulum gravimetry beat everything else for a century: time is the most precisely measurable quantity in physics, and the pendulum converts g into a frequency you can average down indefinitely.

The regional variation the instrument was built to chase is real and large by these standards: g runs from ≈ 9.780 m/s² at the equator to ≈ 9.832 m/s² at the poles — a 0.5% swing from the Earth's oblateness plus centrifugal reduction. A Kater's pendulum resolves parts per million, so it maps that difference and even the tiny anomalies over dense ore bodies or ocean deeps.

From geodesy to the meter itself

Kater's pendulum was not a lab curiosity — it was survey hardware. Through the 19th century, expeditions carried reversible pendulums across latitudes to chart the shape and density of the Earth. Because g at a point encodes the mass beneath it, a pendulum gravimeter is a remote-sensing probe of geology: measure g on a grid and you infer buried density contrasts. This is the direct ancestor of the gravity surveys the oil and mining industries still run, now with spring and superconducting gravimeters that push to 10⁻⁹ g.

The pendulum also touched the definition of length. In the era when a 'seconds pendulum' was proposed as a natural length standard, Kater's precise London value fed directly into the metrology of the yard and, on the continent, into pendulum-based checks on the meter. For decades his 39.1386-inch seconds-pendulum length was a reference number in British weights and measures. The same physics — a rigid body's period set by g and its mass distribution — reappears everywhere a swing must be timed precisely: pendulum clocks, seismometer suspensions, and the torsion and mass-on-spring gravimeters that eventually replaced it.

Misconceptions and subtleties

'The two knife edges must be symmetric about the center.' No — the opposite. If the edges were equidistant from the CM they would trivially give equal periods but tell you nothing, because h₁ = h₂ makes the Bessel difference term ill-defined and L = h₁ + h₂ would just be twice a distance you don't know. The design deliberately places the edges asymmetrically (unequal masses on each end) so h₁ ≠ h₂, which is exactly what makes h₁ + h₂ equal the measurable conjugate-point spacing.

'You're tuning it to resonance.' No. Nothing is being driven at resonance; you are matching two free periods to each other by sliding a small mass, which shifts k² = h₁·h₂ until the two L_eq values coincide.

'Period depends on mass, so a heavier bar swings slower.' Mass cancels in T = 2π√(I/(mgh)) because I ∝ m — the pendulum period is famously mass-independent, just as for the simple pendulum. What matters is the distribution of mass, captured by k.

'Small-angle means it barely moves.' The small-angle approximation isn't about amplitude being invisible — it's that the θ³/6 term in sinθ is dropped. At 5° the correction is ~4.8 × 10⁻⁴, tolerable for a demo but not for a ppm measurement, which is why precision runs used amplitudes of a degree or less and applied the amplitude correction anyway.

Simple pendulum vs. Kater's reversible pendulum as gravimeters

Property	Simple pendulum	Kater's reversible pendulum
Period law	T = 2π√(L/g)	T = 2π√((k²+h²)/(gh))
What you must measure	Length L to the bob's center	Only knife-edge spacing L = h₁+h₂
Center of mass needed?	Yes — a major error source	No — canceled by reversal
Moment of inertia needed?	N/A (idealized)	No — canceled by reversal
Typical accuracy in g	~1% (bob size, string mass)	~10⁻⁴ to 10⁻⁶
Limiting error	Locating the effective length	Timing + edge-spacing + buoyancy