Rotational Power (P = τω)

This is $P = Fv$ rewritten for a rotating shaft: power equals torque times angular velocity, $P = \tau\omega$. Torque alone tells you how hard something is being twisted and says nothing about how fast work is being done; multiply by the rotation rate and you have the rate of energy delivery. It is the single most useful equation in drivetrain work, because torque is what a shaft has to be built to survive and power is what it actually delivers.

An electric motor producing 200 N·m at 3000 rpm: convert first, $\omega = 3000 \times 0.10472 = 314$ rad/s, so $P = 200 \times 314 = 62\,800$ W, about 63 kW or 84 hp. Run the same motor at 1500 rpm at the same torque and it delivers half the power, having done nothing different at the shaft except turn more slowly.

This is what a dyno chart is plotting, and it explains the shape everyone recognises. An engine's torque curve peaks somewhere in the mid range and falls away, yet its power keeps climbing past that point, because $\omega$ is still rising faster than $\tau$ is falling. Power finally peaks where the two rates of change balance. It is also why gearing works: a gearbox trades $\tau$ against $\omega$ at constant power, so a low gear multiplies torque and divides speed, and the product — the useful output — is unchanged apart from losses.

Units are the whole difficulty here, and North American practice hides one conversion inside a magic number. The shop formula $\text{hp} = \tau_{\text{lb·ft}} \times \text{rpm}/5252$ is exactly this equation with the pound-foot, the revolution and the horsepower folded into a single constant. That is why every horsepower and pound-foot curve ever plotted on shared axes crosses at 5252 rpm — not a property of engines, a property of the unit system. In SI the equation needs no constant at all, but it does need $\omega$ in rad/s: feed rpm in directly and the power comes out 9.55 times too high, which is the difference between an 84 hp motor and an 800 hp one. Two further cautions. Torque and power do not peak at the same speed, so a machine specified by its peak torque and a machine specified by its peak power are being described at different operating points, and quoting one at the other's rpm is meaningless. And this is power at the shaft; a motor's electrical input is larger by whatever its efficiency costs, and the output at the far end of a gearbox is smaller again.