Motor Power⚠ unverified
Electrical / Motors · Mechanical power from torque and angular speed
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| T | T | N*m | 10.0 | Torque |
| omega | ω | rad/s | 100.0 | Angular speed |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| power | P | W | Power |
The science & history
Understanding the Parameters
- Torque $T$ — the twisting effort at the shaft, in newton‑metres.
-
Angular speed $\omega$ — in radians per second (not rpm); convert an rpm figure with $\omega = 2\pi\,n/60$.
-
$P$ — mechanical power in watts. The same torque delivers more power the faster the shaft turns.
Derivation (Approaching a Proof)
Power is the rate of doing work. For rotation, the work done by a torque $T$ turning through a small angle $d\theta$ is $dW = T\,d\theta$ (the rotational analogue of $dW = F\,dx$). The power is therefore
$$P = \frac{dW}{dt} = T\,\frac{d\theta}{dt} = T\omega,$$
since $\omega = d\theta/dt$. This is exact for a rigid shaft and holds instantaneously; over a cycle the average mechanical power is $\bar T\,\omega$. It mirrors the electrical Watt's Law $P = VI$ — each is (an effort) × (a flow).
History
The identity $P = T\omega$ follows directly from the 18th–19th‑century mechanics of work and power (the same framework in which James Watt defined horsepower — see Watt's Law). It became the everyday bridge between a motor's torque–speed curve and its power rating as electric motors industrialised in the late 19th century.
Related Concepts: Motor Torque, Watt's Law, Motor Efficiency
Notes: Registry calculator motor-power (unverified). $\omega$ must be in rad/s; the mechanical
(shaft) power is the electrical input power minus the machine's losses (see Motor Efficiency).