Shaft Power ⇄ Torque
P = Tω — convert between power, torque, and rotational speed.
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The engineering
Same energy story, two dialects: torque is what twists the coupling, power is what the electricity bill sees. The 2π/60 bookkeeping is where errors creep in — a motor plate says rpm, every formula wants rad/s, and the factor of ~9.55 between 'kW per N·m at 1 rpm' is the number that bites when someone skips it.
The practical consequence: for a fixed power, torque is inversely proportional to speed. That's why a 5 kW motor at 3000 rpm needs a slim shaft while the same 5 kW after a 100:1 gearbox needs one you can't wrap a hand around — the shaft is sized by torque, never by power alone.
Where this math comes from
James Watt needed to sell steam engines to mine owners who thought in horses, so around 1783 he defined the horsepower — 33,000 ft·lbf per minute — from observations of dray horses turning mill wheels. It was marketing arithmetic that became an SI-adjacent unit and still refuses to die on motor nameplates.
Measuring shaft power honestly took Gaspard de Prony's 1821 friction brake — clamp a lever on the spinning shaft, weigh the reaction, multiply by speed. Every dynamometer since is the Prony brake wearing better instrumentation, and 'brake horsepower' still carries his apparatus in its name.
- 1783James WattDefines the horsepower to price steam engines against horses.
- 1821Gaspard de PronyProny brake — the first practical shaft-power dynamometer.
- 1889British AssociationThe watt adopted as the unit of power, honoring Watt.
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