Torque, Power, and Gears

How twisting force becomes useful work over time.

Force becomes torque

A force pushes. When that push is applied away from a pivot — not through it — it creates rotation. The farther from the axis, the more effectively the same force twists. We call this rotational effect torque.

\tau = r F \sin\theta

$\tau$ is maximized when force is perpendicular to the lever arm ( $\theta = 90°$ ). Pushing along the lever arm ( $\theta = 0$ ) produces no torque.

Torque through angle becomes work

Torque alone stores no energy — it is a force waiting to act. Torque becomes work only when rotation actually happens. The more angle swept, the more work done. Radians make the math clean: one radian of rotation at unit torque is exactly one joule.

W = \tau\,\theta

Compare to linear work $W = F d$ : torque replaces force, angle replaces displacement.

Work over time becomes power

Power is how quickly work happens. Two engines that both do 100 J of work are equal in energy — but the one that does it in one second is ten times as powerful as the one that takes ten seconds. Power is work per unit time.

P = \frac{dW}{dt}

Rotational power

Combining the previous two steps: a rotating object delivers power equal to its torque times its angular velocity. High torque at low RPM equals the same power as low torque at high RPM — if the product is the same.

P = \tau\,\omega

$\omega$ is in rad/s. To convert: $\omega = 2\pi \cdot \text{RPM} / 60$ .

RPM and horsepower

Engineers measure engine speed in RPM and power in horsepower. Horsepower is simply power in Imperial units (1 HP ≈ 746 W). The constant 5252 in the formula below comes from converting RPM to rad/s and watts to horsepower.

\text{HP} = \frac{\tau_{\text{lb·ft}} \times \text{RPM}}{5252}

A key consequence: torque and horsepower curves always intersect at exactly 5252 RPM when both are plotted together.