Belt Power from Tight and Slack Tensions

Power out of a belt is the difference between the two tensions, multiplied by the speed:

\[ P = (F_1 - F_2)\,v \]

The quantity $F_1 - F_2$ has a name — the effective pull, or net tension — and it is worth understanding why the difference is what matters. On the tight side the belt is pulling forward on the pulley; on the slack side it is pulling backward. The net moment about the shaft is $(F_1 - F_2)r$, so the torque, and therefore the power, comes from the difference alone. Both tensions individually contribute to the load on the shaft and the bearings, and neither contributes anything to the work done.

The field mistake that this equation explains

A drive is slipping, so somebody tightens it. Adding the same amount to both tensions changes $F_1 - F_2$ by nothing at all, so the drive transmits exactly the power it did before — and now the shaft load $F_1 + F_2$ has gone up, the bearings are working harder, the belt is running hotter, and something is going to fail early. Over-tensioning is comfortably the most common way a correctly specified belt drive is ruined in service.

What tightening the belt can do is raise the ceiling on the ratio $F_1/F_2$ before slip. The capstan equation says slip begins when $F_1/F_2$ reaches $e^{\mu\theta}$, and since $F_2$ sets that ceiling, more $F_2$ does permit more net pull. But there is a correct amount, it is the least that will carry the peak load without slipping, and it is measured — with a belt-deflection gauge, or with a frequency meter that reads the span's natural frequency and reports tension directly — not judged by feel.

Where the friction limit lands

Combine the two equations. With slip impending, $F_1 = F_2 e^{\mu\theta}$, so

\[ P_{max} = F_1\left(1 - e^{-\mu\theta}\right) v \]

and the whole design problem is visible: capacity rises with the maximum tension the belt can stand, with the wrap angle, with the friction, and with speed. That is why V-belts exist. Wedging the belt into a groove multiplies the effective friction by $1/\sin(\beta/2)$, roughly three times the flat-belt value for a 38° groove, and the exponential turns that threefold gain in $\mu$ into a very large gain in the ratio the drive can hold.

Subtracting the centrifugal share

At speed, part of the tension in the belt is doing nothing but holding the belt in its own circular path. The more careful form subtracts it before applying the friction limit: $P_{max} = (F_1 - F_c)(1 - e^{-\mu\theta})v$, with $F_c = \rho_L v^2$. This is what produces the peak in a belt's power-versus-speed curve, and why running a belt faster stops helping somewhere around 25–30 m/s.