Pulley Calculator

How the driver and driven pulley talk to each other

Two pulleys joined by a belt agree on one thing: belt speed. Because the belt doesn't stretch or slip in the ideal case, every point on its surface moves at the same linear speed, whether it's wrapped around the small pulley or the big one. A point further from the axle covers more distance per revolution, so that pulley has to spin slower to keep surface speed constant, and that's the entire mechanism behind a belt drive.

Call the pulley attached to the motor or engine the driver (diameter D1, speed N1), and the one doing the work the driven pulley (diameter D2, speed N2). Matching belt speeds gives the equation this calculator is built around: N2 = N1 × D1 / D2. Flip it around for the speed ratio, i = D2 / D1, written "i : 1" — a 2:1 ratio means the driven pulley turns at half the driver's speed. Make D2 bigger than D1 and you slow things down; smaller, and you speed them up. It's the same logic a gear train uses, which is why a gear ratio calculator gives near-identical numbers for a gearbox doing the same job; the choice is mostly cost and noise, not kinematics.

Pitch diameter is the number that actually matters

Here's where back-of-envelope pulley math goes wrong: the diameter that belongs in the ratio formula isn't what you'd measure with calipers around the outside of the pulley. A V-belt doesn't ride on the rim — it wedges into the groove and runs on an effective line partway down the sheave, called the pitch diameter. A synchronous belt is similar: its pitch line sits within the belt, not at the tooth tips. Sheave and sprocket catalogs list pitch diameter precisely because that's the figure that predicts speed and ratio correctly.

The gap between pitch diameter and outside diameter is small on a large pulley and barely matters. On a small one it can be enough to throw your calculated speed off by a few percent. Pulling numbers off a drawing or catalog, use the published pitch diameter. With no documentation, measure the groove's effective running diameter, not the outer edge, and use that for D1 and D2.

What a bigger pulley does to torque and power

Speed and torque trade off because power, ignoring losses, stays constant crossing the belt. Power is torque times angular speed, so if the driven pulley spins at half the driver's speed, its torque doubles to make up the difference — the same way a long lever multiplies force at the cost of travel distance. That's why a slow, large driven pulley on a hoist or mixer feels so much stronger than the motor shaft it's bolted to.

Real drives lose a little, which is where the efficiency factor η comes in. Friction, belt flexing and slip mean output torque and power are always a bit less than the bare ratio predicts: T2 = η × T1 × D2 / D1. According to the U.S. Department of Energy, a V-belt drive can reach a peak efficiency of 95% or more when it is installed, but can lose as much as 5% over time if the belt slips because nobody retensions it. Notched V-belts run cooler and more efficiently than standard ones, and synchronous (toothed) belts hold about 98% across a wide load range because they grip by tooth engagement and run slip-free.

Here's what that looks like across a set of driven-pulley sizes, holding the driver at 80 mm and 1450 rpm with 8 N·m on the motor shaft and a 95% efficient belt:

Double the driven pulley and the speed halves while the torque doubles. The 95% efficiency trims every row by the same 5%, which is why even the 1:1 row delivers 7.6 N·m rather than the full 8.

A motor, two pulleys, and a conveyor: full worked example

Take a 4-pole induction motor on a 50 Hz supply. Its synchronous speed is 120 × 50 / 4 = 1500 rpm, but an induction motor always runs a little slower under load because of slip, so use the full-load speed printed on the nameplate — say 1450 rpm. It drives an 80 mm pitch-diameter sheave, delivering 8 N·m at the shaft, into a 200 mm driven sheave through a V-belt running at 95% efficiency.

Ratio first: i = D2/D1 = 200/80 = 2.5, a 2.5:1 reduction drive. Driven speed: N2 = N1 × D1/D2 = 1450 × 80/200 = 580 rpm. Belt speed, the same anywhere on the loop: v = π × D1 × N1/60 = π × 0.08 × 1450/60 ≈ 6.07 m/s. Output torque: T2 = η × T1 × D2/D1 = 0.95 × 8 × 2.5 = 19.0 N·m. Output power: P2 = η × T1 × 2πN1/60 ≈ 0.95 × 8 × 151.8 ≈ 1154 W, against roughly 1215 W entering the belt from the motor side — the missing 61 W, about 5% of the input, is exactly what the 95% efficiency figure says should be lost.

Change one input and the rest cascades: drop D2 to 160 mm and N2 jumps to 725 rpm while T2 falls to 15.2 N·m; swap the V-belt for a synchronous belt at 98% efficiency and P2 climbs to roughly 1190 W for the same motor. That's the point of running the numbers before you cut a belt or order a sheave.

Running the numbers backwards: picking a pulley for a target speed

Sometimes the question isn't "what speed do I get," it's "what pulley gets me the speed I need." Knowing your driver pulley and its speed, plus a target N2 — a fan that has to hit a given airflow, a spindle rated for a maximum rpm — the second mode solves the ratio formula for the unknown diameter: D2 = D1 × N1/N2. Plug in the numbers and it hands back the driven diameter to shop for, plus the resulting ratio and belt speed.

Both modes assume no belt slip for speed — efficiency only scales torque and power, never the rpm numbers, which stay true to the geometric ratio as long as the belt doesn't skip. Once you've landed on diameters, the next question is how long a belt to buy for that center distance, a separate calculation handled on the belt length calculator. If what you're sizing is a block-and-tackle or rope-and-pulley lifting rig rather than a rotating belt drive, that's a different kind of mechanical advantage altogether — see the pulley system calculator for that. And if torque itself is what you're after, the torque calculator and motor torque calculator dig deeper into shaft torque, with more tools on the physics hub.