# Pulley Calculator

You can find belt and pulley systems in your car's engine, the luggage belt at the airport, and in many other places: learn how to calculate pulleys' speed, diameter, and much more with our comprehensive and helpful tool!

Learn:

- What is a belt and pulley system;
- How to calculate the RPM of a pulley and its diameter;
- How to calculate the pulley ratio;
- How to calculate the torque in a belt and pulley system;
- The formulas to calculate the belt speed, tension, and length.

## What is a pulley?

A pulley is a **simple machine**: a mechanical device with a basic construction (i.e., no gears, no heat cycles, and so on) that changes the direction and/or the magnitude of a force.

In the most basic configuration, a single pulley is a **wheel** rotating around an **axle**, only able to **change the direction of a force** (without giving you any mechanical advantage). This is pretty straightforward to understand — the story changes when the number of pulleys increases.

## Adding pulleys: what is a belt and pulley system

When you add a pulley, your simple machine can change the magnitude of the applied force too. You can see a similar example of this in our gear ratio calculator. Pulleys connected by a rope (an **open** stretch of material in which you can identify a beginning and an end) allow for the multiplication and rotation of the **tension** of the rope, with a final mechanical advantage ideally not dependent on the size of the pulleys. If you connect the set of pulleys through a **closed loop of material**, a **belt**, however, the size of the pulleys matter.

But what is a belt-pulley system? We are talking of a system of two to more pulleys connected by a flat, smooth belt. The system is not dissimilar from the chain transmission of bikes, and it's actually more widespread than you think: joined to the astounding math of **Möbius strips** it allows your car to move (and to require less maintenance).

When you build a simple belt-pulley system, you have to consider only a few important parameters:

- The radiuses of the pulleys;
- The rotational speeds of the pulleys; and
- The distance between the pulleys.

Many other parameters and quantities can be calculated starting from these.

If you want to move your belt and pulley system, you need to apply a driving force to it: it may be an engine, your legs (in a bike), the wind, or anything else. The pulley that is moved in the first place is called the **driving pulley**; the other ones are, regardless of their position and role, **driven pulleys**.

## Pulley calculations: pulley RPM and diameter calcualtions

We begin our analysis of the belt and pulley calculator with quantities independent from the belt. The diameters and rotations per minute of two pulleys connected by a belt are **related** by a simple formula:

Here you can find:

- The
**diameters**of the pulleys, $d_1$ and $d_2$; - The
**RPM**of both pulleys: $n_1$ and $n_2$.

We can also calculate the **pulley ratio**, though it has not much importance as for gears:

The diameters of the pulleys in a belt and pulley system are the quantities defining the **mechanical advantage** of your machine.

#### Pulley torque calculations

When you know a pulley's diameter and RPM, you can also define its **torque**. The torque depends, obviously, on the **input power** on the **driving pulley**. In **rotational systems**. the power is equal to the **product of torque and rotational speed**:

Where:

- $\tau$ is the
**torque**on the pulley; and - $\omega$ its rotational speed.

The rotational speed, or angular velocity, is usually defined in **degrees per second** or **radians per second**. To convert it to **rotations per minute**, use this formula:

If you know the input power in **watts**, the formula for a pulley's torque is:

Where $n_1$ and $\tau_1$ are quantities related to the driving pulley.

The same formula applies when you calculate the torque of the driven pulley. Since, ideally, the power is transmitted without dissipation, and you can calculate the pulley rotational speed, we can calculate the torque in a belt and pulley system with a generic equality:

## Belt and pulley calculator: belt speed, length and tension formulas

Let's finally introduce the belt in our pulley calculator. Some of the calculations require the knowledge of an additional parameter: the **distance between the pulley's centers**, $D$.

#### Belt speed calculator

To calculate the speed of a belt in a belt and pulley system, just... think! In an ideal case, there is no **slippage** (which means that the points of contact between the belt and pulleys don't move). The speed of the belt, then, is equal to the tangential speed of **both pulleys**, which is (unsurprisingly) the same:

This equality is easily understandable if you return to the previous section, where you'd see that the product of diameter and RPM is a constant in a belt and pulley system.

#### Calculation for the tension in a pulley and belt system

To calculate the tension in a pulley system driven by a belt, we use the following formula:

Where $P$ is the power driving the pulleys.

#### Lenght of the belt in a belt and pulley system

The length of the belt is a fundamental quantity: when you translate your machine from the drawing board to the physical world, you want the belt to be as taut as possible. If your belt is too long, the stretch "leaving" the driving pulley would be slack, and the tension on that side would be lost.

To calculate the length of a belt in a machine composed by belts and pulleys, this formula comes in handy:

Mind that this is an approximation of the exact formula!