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Pendulum and environment
Choose the rod material first, then change temperature, gravity or amplitude and use fast-forward to read the drift.
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Model checks
The science
Pendulum and escapement
A pendulum of length L has the exact period T = 2π√(L/g) / AGM(1, cos(θ0/2)), which for small swings is 2π√(L/g): 0.9936 m gives a 2 s "seconds pendulum". A larger swing lengthens the period by about θ02/16 (circular error). Friction is modelled as damping θ'' = -(g/L) sinθ - (ω0/Q) θ'; the escapement gives the bob a small impulse Δω = J each time it passes the centre. The swing settles where the energy of the impulses equals the friction loss, J = ω0θ0(√(1 + π/Q) - 1).
Gear train
A gear pair with z1 and z2 teeth turns the driven shaft by -z1/z2 of the driver's angle. The escape wheel (30 teeth) advances one beat of 6 degrees per second, so it turns once a minute. The train 10:60, 8:80, 10:30 and 8:32 slows it by 60 to 1 (minute wheel) and by another 12 to 1 (hour wheel).
Rate
A steel rod lengthens by αΔT, so the clock loses about ½αΔT × 86400 s per day: 0.5 s per kelvin for steel, 0.05 s per kelvin for Invar.
Honest simplifications: the escapement is an ideal impulse at the centre (real anchor escapements add recoil, lock and a small arc-dependent rate error); the gear drawing uses straight-sided teeth, not involute profiles; hands are driven by the beat count through the train ratios; no friction in the train, no suspension spring.
What to try
- Seconds pendulum preset at 30x fast-forward: the clock error graph stays flat.
- Hot summer day with a steel rod, then the Invar rod: compare seconds lost per day.
- Big swing preset: the circular error makes the clock run slow.
- Lower Q: the amplitude settles lower and takes less time to build up.