Belt Power⚠ unverified
Mechanical / Flexible Elements · Power transmitted by a belt drive
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| F1 | F1 | N | 600.0 | Tight-side tension |
| F2 | F2 | N | 200.0 | Slack-side tension |
| v | v | m/s | 10.0 | Belt speed |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| P | P | W | Power |
The science & history
Understanding the Parameters
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Net driving tension $F_1 - F_2$ — the effective force. A belt does not push; it transmits torque by the difference between its two spans. Increasing $F_1$ (tighter belt) or decreasing $F_2$ raises the driving force — but $F_1/F_2$ is capped at $e^{\mu\theta}$ (Belt Tension Ratio) before slip.
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Belt speed $v$ — power scales linearly with speed, so belts are run fast to transmit more power for a given tension. But too fast and centrifugal tension (Centrifugal Tension) eats into the usable tension, capping power (Maximum Belt Power).
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Torque link — the same power at the pulley is $P = T\omega$, and the belt force $(F_1-F_2)$ acts at the pulley radius $r$, so torque $T = (F_1-F_2)r$ — the belt equation and the rotational equation are the same statement.
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Two limits — the transmissible power is bounded both by friction (slip, via the tension ratio) and by belt strength (the tight-side stress, Belt Stress); the smaller governs.
Derivation (Approaching a Proof)
Consider the belt passing over a driving pulley of radius $r$ turning at angular speed $\omega$, with belt speed $v = r\omega$. The tight side pulls forward with $F_1$, the slack side back with $F_2$, so the net tangential force the belt applies to the rim is $F_1 - F_2$. The torque delivered is
$$T = (F_1 - F_2)\,r.$$
Power is torque times angular speed:
$$P = T\omega = (F_1 - F_2)\,r\,\omega = (F_1 - F_2)\,v,$$
using $v = r\omega$. Equivalently, power is force times the velocity at which that force acts — the belt surface moving at $v$. The result is independent of pulley radius because a larger radius gives more torque but proportionally lower angular speed.
Dimensional check. $P = (F_1 - F_2)\,v = \text{N}\cdot(\text{m}/\text{s}) = \text{N}\cdot\text{m}/\text{s} = \text{W}$ — power, as required.
History and Development
$P = (F_1 - F_2)v$ has been the basis of belt-drive sizing since the industrial revolution, when line shafts and flat belts distributed power throughout factories. Though electric motors and gearboxes displaced line shafting, belt drives remain ubiquitous (HVAC blowers, pumps, conveyors, automotive accessory drives) because they are cheap, quiet, tolerant of misalignment, and forgiving of shock. The equation, paired with the capstan tension ratio (Belt Tension Ratio) and V-belt factors, underlies every manufacturer's selection chart.
Related Concepts: Belt Tension Ratio, Maximum Belt Power, Centrifugal Tension, Belt Stress, V Belt Speed, V-Belt Design
Notes: $F_1 - F_2$ is the effective driving force; torque $= (F_1-F_2)r$. Power $\propto v$ — run fast, but centrifugal tension caps it (Maximum Belt Power). Bounded by both friction (slip, Belt Tension Ratio) and belt strength (Belt Stress). Assumes no slip.