Pulley Calculator

What the pulley calculator does

Two pulleys joined by a belt are one of the oldest ways to send power from one shaft to another, and to change the speed along the way. This calculator analyses such a belt drive. From the two pulley diameters, their speeds, the distance between their centres, and the power being sent through, it works out the length of belt you need, how fast the belt runs, and the tension it carries.

Below is how a belt drive links two pulleys, the equations behind the results, and a worked example.

How to use it

1. Enter the power being transmitted, in watts, and the centre-to-centre distance between the pulley shafts.
2. Enter the driver pulley's diameter and speed, and the driven pulley's diameter and speed. The speeds must match the diameters, since the belt links them.
3. Press Calculate for the belt length, belt velocity, and belt tension, or Reset to clear it.

How a belt drive links two pulleys

A belt wraps around both pulleys and grips them, so the belt and the rims of both pulleys all move at the same speed along the belt. That single fact sets how the two pulleys relate. A point on the rim of a pulley moves faster the larger the pulley is, for a given rate of spin, so to keep the rim speeds equal, the larger pulley must spin slower and the smaller one faster.

This gives the belt drive its speed-changing power. Pair a small driver pulley with a large driven one and the driven shaft turns slower, just as gears would do, but over a distance the belt can span. The ratio of the speeds is the inverse of the ratio of the diameters, so a driven pulley twice the diameter of the driver turns at half the speed.

The equations it uses

The link between the two pulleys is that their rim speeds match, which ties each diameter d to its speed n:

d₁ n₁ = d₂ n₂

The belt's own speed is the rim speed of a pulley, its circumference times how many turns per second, which for a diameter d at n RPM is:

belt velocity = π d n ÷ 60

And the length of belt needed to wrap both pulleys and span the gap between them, with C for the centre distance, is the standard belt-drive length:

L = (π/2)(d₁ + d₂) + 2C + (d₁ − d₂)² ÷ (4C)

The first part wraps the two pulleys, the second crosses and returns along the gap, and the last is a small correction for the pulleys being different sizes.

Belt speed, power, and tension

The belt carries power by being pulled, and the three are tied together simply: power is the pulling force times the speed the belt moves at. So once the belt's speed is known, the effective tension needed to carry a given power follows by dividing the power by that speed.

belt tension = power ÷ belt velocity

This has a practical lesson built in. A belt running fast can carry the same power at a lower tension, while a slow belt needs to be pulled harder to send the same power, which is why belt drives are usually run at a healthy speed. Lower tension means less load on the bearings and a longer life for the belt, so a faster, lighter-pulling belt is often the kinder design.

Units and precision

The calculator works in SI units: diameters and centre distance in metres, speeds in RPM, power in watts. It returns the belt length in metres, the belt velocity in metres per second, and the tension in newtons. Because the belt physically links the two pulleys, the calculator checks that the speeds and diameters you enter are consistent with equal rim speeds, and asks you to correct them if they are not. The belt length here is the geometric requirement; in practice you would round up to the nearest standard belt size.

A worked example

Take a driver pulley 0.1 m across spinning at 1,000 RPM, a driven pulley 0.2 m across, set 0.5 m apart, carrying 1,000 watts.

Because the driven pulley is twice the diameter, it turns at half the speed, 500 RPM. The belt velocity is π × 0.1 × 1,000 ÷ 60 ≈ 5.24 m/s. The belt length is (π/2)(0.3) + 2(0.5) + (0.1)² ÷ 2 ≈ 1.48 m. And the effective tension to carry 1,000 watts is 1,000 ÷ 5.24 ≈ 191 N.

Questions people ask

How does a belt drive change speed?

Through the pulley sizes. The belt keeps both rims moving at the same speed, so a larger pulley turns slower. The speed ratio is the inverse of the diameter ratio.

What is belt velocity?

It is how fast the belt itself travels, equal to a pulley's rim speed, its circumference times its turns per second. It sets the link between the power carried and the tension needed.

Why does a faster belt need less tension?

Because power is tension times belt speed. For a fixed power, a higher speed means a lower tension carries it, which eases the load on the bearings and the belt.

Is this the same as a simple-machine pulley?

No. This is a belt drive, two pulleys linked by a belt to transmit power and change speed. A simple-machine pulley uses a rope through one or more pulleys to multiply lifting force.

References

A quick note on where this comes from. The equal-rim-speed relationship of a belt drive, the belt-length geometry, and the link between power, belt speed, and tension are standard mechanical engineering, set out by Engineers Edge and in references such as Marks' Standard Handbook for Mechanical Engineers. The SI units follow the US National Institute of Standards and Technology.

1. Engineers Edge, Flat Belt Length and Pulley Center Distance Calculation. https://www.engineersedge.com/belt_design/belt_length_pulley_center_dist.htm
2. Marks' Standard Handbook for Mechanical Engineers, belt drive design (power transmission by belts).
3. National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811