Enter a beat interval and wheel tooth count to compare ideal pendulum length, beat rate and wheel speed. The calculation assumes one tooth per beat; the hooked-wheel illustration does not solve contact, impulse, recoil or clock accuracy.
Hook-tooth Escapement Interactive Calculator
Convert an entered beat interval to the small-angle simple-pendulum equivalent length, beat rate, ideal escape-wheel speed and angular tooth pitch.
Equation Used
- Small-angle ideal simple pendulum using rounded gravitational acceleration g = 9.81 m/s².
- Entered beat interval is one half of the full pendulum period.
- Exactly one ideal escape-wheel tooth is released per beat.
- No tooth/pallet geometry, locking, drop, recoil, impulse, amplitude, circular error, friction, air drag, train torque, wear or timekeeping-rate model.
Pendulum length uses g = 9.81 m/s² and the small-angle simple-pendulum relation. Wheel speed assumes exactly one tooth release per beat. The tooth and pallet animation is qualitative, not a contact solution.
Timing relationships only
A pendulum crossing from one side to the other takes one beat interval, so its full ideal period is twice that interval. If an escapement releases one wheel tooth at each beat, tooth count converts beats per minute directly to wheel revolutions per minute.
The visual uses a stylized hooked wheel and pallet pair to illustrate timing. It does not solve their contact, locking, impulse or recoil. Displayed pendulum length is a numerical result; the drawn rod has a fixed display length. Playback may be slowed for viewing.
Use as a timing conversion
The calculator can check ideal period, pitch and release-count arithmetic. It cannot design, restore, adjust, authenticate or rate an escapement or clock.
Simple-pendulum and release-count identities
With beat interval tb, full period is T=2tb. Rearranging T=2π√(L/g) gives L=g(T/2π)². Beat rate is 60/tb. Under the explicit one-tooth-per-beat assumption, wheel speed is beat rate divided by N and angular pitch is 360°/N.
Worked example
At a 1.0 s beat, the ideal full period is 2.0 s and the equivalent simple-pendulum length is about 994.0 mm using g = 9.81 m/s². A 30-tooth wheel releasing one tooth per beat turns at 2.0 rpm and has 12° pitch.
Scope and limitations
Real timing depends on physical-pendulum mass distribution, local gravity, amplitude and circular error, suspension and temperature, escapement impulse and recoil, train torque, friction, air drag, tooth/pallet geometry, level, wear and adjustment.
Questions
Is the displayed length the rod length?
Not necessarily. It is the equivalent simple-pendulum distance from pivot to centre of oscillation.
Does the calculator predict seconds per day?
No. It omits the effects that determine real clock rate.
Does the animation prove the hooked tooth will lock and release?
No. Exact tooth and pallet geometry has not been solved.
Calculation basis
OpenStax University Physics: Pendulums explains the small-angle simple-pendulum relation and the distinction from a physical pendulum. The calculator uses rounded g = 9.81 m/s².
Wheel speed and pitch follow the stated one-tooth-per-beat assumption. These timing identities do not determine the geometry or performance of a particular hook-tooth escapement.
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