REVERSIBLE PENDULUM
Match the periods from two pivots and g = 4π²L/T² — the moment of inertia cancels completely. Kater weighed gravity without knowing I.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
You want g to five digits with a swinging pendulum. The trouble: a real pendulum is a rigid body, and its period depends on its moment of inertia — which you cannot measure precisely. Captain Kater, in 1817, found a way to make that unknown vanish.
Two pivots
His pendulum has a knife edge near each end, a fixed distance L apart, and a weight you can slide. Swung from either edge it is a physical pendulum, its period set by the moment of inertia through k, the radius of gyration — the very thing you do not know.
Match them
The trick: slide the weight until the two periods, from the top edge and the bottom edge, are EQUAL. That single condition forces the pendulum to behave exactly like a simple one of length L, so g = 4π²L/T² — with no k anywhere in sight.
Nothing left
Both surviving quantities are things you can measure beautifully: the distance between two knife edges, and a period averaged over hundreds of swings. Everything about the mass — how much, how spread out, where centred — has cancelled clean away.
Audit
Audited: at the match, g = 4π²L/T² returns 9.80665 for any radius of gyration k, checked across values; the reversibility condition is k² = d₁d₂, which makes the equivalent length exactly L. The simple pendulum needs a point mass to be honest — this one needs nothing at all.