Hiscox figure 157 shows a ball on an arm pivoted to a clamp on a revolving vertical shaft. Its position is read against a curved index. The arm swings outward as speed increases; the scale and its support remain stationary. This differs from a two-ball governor linked to a spring-loaded sleeve and dial.
Centrifugal Speed Indicator Interactive Calculator
Adjust shaft speed, arm length, pivot offset and ball mass. The model calculates equilibrium arm angle, orbit radius and inward force, while the animation shows the pointer passing a stationary curved scale.
Equation Used
- Point-mass ball on a massless rigid arm; positive pivot offset from the shaft.
- Vertical shaft, uniform rotation and gravity 9.80665 m/s².
- No pivot friction, arm weight, spring, linkage to a sleeve, vibration or transient settling.
- The pointer is compared with the stationary scale only when crossing its plane.
- No instrument accuracy, calibration error, balancing tolerance or manufacturing specification is predicted.
This is the single-ball, curved-index indicator shown in Hiscox figure 157. The calculator uses ideal gravity and centrifugal moment balance, not a calibration-error prediction. Shaft bearings, pointer thickness and scale supports are illustrative; the ball symbol is not a sized component.
A ball, a hinged arm and a curved index
Hiscox figure 157 describes an arm and ball pivoted to a clamp on a revolving vertical shaft. The ball’s position indicates rotational speed against a curved index bar. The drawing includes a fixed support for that index.
The animation preserves that arrangement. The shaft and clamp rotate, carrying the arm around the axis. The ball follows a horizontal orbit at the calculated steady angle. The curved index stays in a fixed plane, so the pointer is read when it crosses that plane; a projected pointer position elsewhere is not a scale reading.
The illustrative scale is recalculated for the chosen arm length and pivot offset. Those dimensions affect equilibrium, whereas ideal ball mass affects force but cancels from the angle equation. The ball marker is symbolic and does not prescribe diameter or balance.
Mechanical speed indication
The reference places this mechanism among devices for measuring speed and power. It uses the position of a rotating pendulous mass as a speed-dependent indication.
The simple open mechanism should not be confused with every centrifugal tachometer or engine governor. A spring-loaded sleeve, geared pointer or enclosed dial needs its own geometry and calibration model.
Moment balance about the arm pivot
Let L be pivot-to-ball-center length, e the pivot’s radial offset from the shaft, and θ the arm angle from downward vertical. The ball’s orbit radius is r = e + Lsinθ.
In the rotating frame, the outward centrifugal term is mω²r. Its moment arm about the hinge is Lcosθ. Weight mg acts with moment arm Lsinθ. Equilibrium is therefore mω²r Lcosθ = mg Lsinθ, or ω²(e + Lsinθ)cosθ = g sinθ.
The calculator solves that equation for 0 ≤ θ < 90°. Inward force in the inertial description is F = mω²r. Vertical drop below the hinge is h = Lcosθ. The ideal scale relationship is N = 60/(2π)√[g tanθ/(e + Lsinθ)].
At a fixed radius, inward force varies with the square of speed. An instrument’s reading error is a different quantity: it depends on its mechanical response, scale and calibration. This equilibrium calculation does not predict that error.
Checks on the ideal model
At zero speed, the arm hangs vertically: θ = 0°, orbit radius equals the pivot offset e, vertical drop equals L and inward force is zero.
With a 250 mm arm and 50 mm pivot offset, a 60° arm position corresponds to about 76.24 rpm. A 100 g ball then requires about 1.699 N inward force, and its center sits 125 mm below its hinge.
Doubling ball mass doubles inward force at the same speed, but does not change the ideal angle. Changing arm length or pivot offset changes both the orbit geometry and the scale calibration.
Model limits
A real arm has weight, friction and finite motion limits. Its steady position can also be affected by vibration and drive variation. The massless-arm model is intended to explain the mechanism, not establish accuracy or service condition.
The fixed index becomes harder to resolve as angle approaches horizontal. This geometric limitation does not supply a universal usable RPM range or an acceptable calibration tolerance.
Calculator questions
Why does the pointer not stay on the scale?
The ball and arm rotate around the shaft, while the curved index occupies one fixed plane. The pointer passes that plane once per revolution.
Why does ball mass not change the angle?
In this ideal model the gravitational and centrifugal moments both contain mass, which cancels from equilibrium. Mass still scales the inward force.
Is there a spring or sliding sleeve?
Neither is part of the selected single-arm model. Those belong to other centrifugal indicator and governor arrangements.
Does doubling speed quadruple a tachometer’s RPM error?
Not in general. Force at fixed radius scales with speed squared, but reading error depends on the device’s mechanical response, scale and calibration. The equilibrium model calculates arm position, not instrument error.
Does the animation show the transient swing outward?
No. It shows steady rotation at the calculated equilibrium. Changing an input updates that equilibrium immediately; settling dynamics are not modeled.
References & Further Reading
- Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances (1901), printed page 52, figure 157: pivoted arm and ball, revolving shaft and curved index.
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