# Pulley Calculator

Created by Bogna Szyk
Reviewed by Steven Wooding
Last updated: Jun 05, 2023

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. If you're into the real-life pulleys application, make sure to check our winch size calculator.

💡 You might also be interested in our torque calculator and tension calculator.

## 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 the transmitting power is applied to it, causing it to rotate. The other pulley is called the driven pulley. It turns 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!)

Pulleys are simple machines. We analyzed another one at our lever calculator: discover how mechanics makes our life easier!

## Some pulley formulas

Once you have created your pulley system, you can start determining various parameters with this pulley calculator. 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, its angular velocity n₁, the driven pulley diameter d₂, or its angular velocity n₂.

2. Belt velocity

We can calculate the speed of the belt 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 you can find 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

3. Calculate the pulley speed:

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

4. You can also use the following 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

5. 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 N·m

T₂ = P /(2 × π × n₂ / 60) = 1500 / (2 × π × 4000 / 60) = 3.581 N·m

## FAQ

### What is the belt length for two fixed pulleys?

You can use Omni Calculator's pulley calculator or do as follows:

1. Define the distance between pulleys D.

2. Obtain the diameter of the driver pulley d1 and the driven pulley d2.

3. Use the following equation:

(d1 × pi/ 2) + (d2 × π / 2) + (2 × D) + ((d1 - d2)² / (4 × D)).

### Why we use big chainrings and a small sprockets for riding fast?

The reason is, in that configuration, as per the pulley formula, the sprocket (driven pulley) has higher angular velocity. Try using the Omni Calculator's pulley calculator, and you will verify that the smaller the driven pulley, the bigger will be its angular velocity for the same transmitting power.

### How to lift myself (75 kg) with a pulley?

Considering the pulley formula, the more pulleys you use, the more mechanical advantage you get. In that case, if you use six pulleys, you get a mechanical advantage of 12, meaning the force you will need to apply is 75 kg × 9.81 m/s² / 12 = 61.31 N.

### Why bike shifts are so useful on an uphill road?

According to the pulley formula, it is because they are designed to interchange bike chain speed by force but generate the same power: Power = Force × Chain velocity. Remember that riding a bike requires less effort to pedal (less force) when using higher shifts, but you pedal faster (bike chain velocity higher).

Bogna Szyk
Transmitting power
W
Pulley centers
ft
Driver pulley
Diameter
ft
Angular velocity
RPM
Drive torque
N·m
Driven pulley
Diameter
ft
Angular velocity
RPM
Driven torque
N·m
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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