# Pulley Calculator

By Bogna Szyk
Last updated: Aug 23, 2021

This pulley calculator analyzes a system of two pulleys joined by a conveyor belt (also called a belt drive). You can use it to calculate the pulley RPM (revolutions per minute), but also its diameter, and some properties of the whole system (such as the pulley speed, belt tension or torque). You can use this tool right away or continue reading to learn more about the logic behind the pulley formulas.

## Pulley system A pulley system consists of two pulleys (usually of different diameters) and a belt loop to link the pulleys. In the figure above, the belt is marked with a red color.

One of the two pulleys is called the driver pulley - it means that transmitting power is applied to it, causing it to rotate. The other pulley is called the driven pulley. It rotates because of the force transmitted through the belt.

There are two main parameters associated with each of the pulleys. The first one is the diameter (twice the radius), and the second one is its angular velocity measured in revolutions per minute. (For more information about the angular velocity check out our centrifugal force calculator!)

## Some pulley formulas

Once you have created your pulley system, you can start determining various parameters with this pulley calculators. The values that you can find are:

1. Diameter and RPM of each pulley

For a pulley system like this, the product of pulley diameter `d` and RPM `n` is the same for both driver and driven pulley. It means that

`d₁ * n₁ = d₂ * n₂`

You can use this formula to find any of these four values: driver pulley diameter `d₁`, it's angular velocity `n₁`, the driven pulley diameter `d₂` or it's angular velocity `n₂`.

2. Belt velocity

The speed of the belt can be calculated according to the formula

`v = π * d₁ * n₁ / 60`

where the angular frequency is expressed in RPM and the belt velocity in meters per second.

3. Belt length

The length of the belt is dependent on the diameters of both pulleys and the distance between their centers `D`:

`L = (d₁ * π / 2) + (d₂ * π / 2) + 2D + ((d₁ - d₂)² / 4D)`

You can also reverse this formula to calculate the distance between the pulleys for a known belt length.

4. Belt tension

The tension in the belt is dependent on the belt velocity and the transmitting power `P`:

`F = P / v`

Naturally, you can use the pulley speed calculator to find the power as well - simply input the values of belt tension and velocity.

5. Torque

The last values that can be found with this pulley calculator are the drive torque (torque of the driver pulley) and the driven torque (of the driven pulley). Use the following equation:

`T = P /(2 * π * n / 60)`

where the angular velocity `n` of each pulley is expressed in revolutions per minute.

## Calculating pulley RPM and speed: an example

1. Start with writing down the known values. Let's say that you know the diameter and RPM of the driver pulley (`d₁ = 0.4 m` and `n₁ = 1000 RPM`), the diameter of the driven pulley (`d₂ = 0.1 m`) and the transmitting power (`P = 1500 W`). You have also measured the distance between the pulley centers to be equal to `D = 1 m`.

2. Determine the angular velocity of the driven pulley, using the formula 1:

`d₁ * n₁ = d₂ * n₂`

`n₂ = d₁ * n₁ / d₂ = 0.4 * 1000 / 0.1 = 4000 RPM`

1. Calculate the pulley speed:

`v = π * d₁ * n₁ / 60 = π * 0.4 * 1000 / 60 = 20.944 m/s`

1. You can also use the next formula for the belt length:

`L = (d₁ * π / 2) + (d₂ * π / 2) + 2D + ((d₁ - d₂)² / 4D)`

`L = (0.4 * π / 2) + (0.1 * π / 2) + 2 * 1 + ((0.4 - 0.1)² / (4 * 1) ) = 2.808 m`

1. Finally, use the formulas for belt tension and torque to find the remaining parameters:

`F = P / v = 1500 / 20.944 = 71.62 N`

`T₁ = P /(2 * π * n₁ / 60) = 1500 / (2 * π * 1000 / 60) = 14.324 Nm`

`T₂ = P /(2 * π * n₂ / 60) = 1500 / (2 * π * 4000 / 60) = 3.581 Nm`

Bogna Szyk
Transmitting power
W
Pulley centers
ft
Driver pulley
Diameter
ft
Angular velocity
RPM
Drive torque
Nm
Driven pulley
Diameter
ft
Angular velocity
RPM
Driven torque
Nm
Belt
Belt length
ft
Belt velocity
ft/s
Belt tension
N
We made a video explaining how pulleys work! Watch it here:
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