# Angular Frequency Calculator

With this angular frequency calculator, you can find the **angular frequency** of **rotating** and **oscillating bodies**. If you're interested in how angular frequency *differs* for rotation and oscillation, you've come to the right place! Read the following article for a better understanding of the fundamentals, including:

- What is angular frequency? Units of angular frequency.
- Angular frequency equation for rotating bodies.
- Angular frequency formula of oscillating bodies.
- Examples to understand how to calculate angular frequency.

Understanding frequency is helpful to the discussion below, so we recommend you head first to our frequency calculator.

## What is angular frequency?

Angular frequency is a measure of how fast a body rotates or oscillates. It is a *scalar* quantity and is the *circular counterpart* of frequency. For a **rotating body**, angular frequency is how many *rotations* the body makes in one second. Similarly, for an **oscillating body**, angular frequency measures how many *oscillations* it completes in one second.

The **SI unit** of angular frequency is **radians per second** $(\text{rad/s})$. Although Hertz is also *dimensionally correct*, we refrain from using this to avoid confusion between frequency and angular frequency.

## Angular frequency equation for rotating bodies

As stated earlier, the angular frequency of a rotating body is the number of rotations it completes in one second. The equation for angular frequency of a body whose rotation displaces it by a small angle $\Delta \theta$ is:

Where:

- $\omega$ -
**Angular frequency**of rotation; - $\Delta \theta$ -
**Angular displacement**during rotation; and - $\Delta t$ -
**Time**in which the angular displacement occurs.

Notice that the angular frequency of a rotating object is the same as its angular velocity.

## Angular frequency formula of oscillating bodies

For an oscillating motion, the angular frequency measures the time rate of oscillation. It is given by:

Where:

- $\omega$ -
**Angular frequency**of oscillation; - $T$ -
**Period**of oscillation, or time needed to complete one oscillation; and - $f$ -
**Frequency**of oscillation.

The above relation establishes the connection between frequency, angular frequency, and period.

## How to find angular frequency

Let's work through some examples to understand better how to calculate angular frequency.

- If a wheel turns
**100 radians**in**15 seconds**, what is its angular frequency?

Given:

- Angular displacement $\Delta \theta = 100 \text{ rad}$.
- Time needed $\Delta t = 15 \text{ s}$.

To find:

- Angular frequency $\omega$ of the wheel's rotation.

We can use the angular frequency equation for rotating objects straight away:

- If a simple pendulum oscillates at a
**7 Hz**frequency, what is its angular frequency?

Given:

- Frequency $f = 7 \text{ Hz}$.

To find:

- Angular frequency $\omega$ of the oscillating pendulum.

Using the angular frequency formula for oscillating objects:

- Find the period of oscillation of an object if its angular frequency is
**15 rad/s**.

Given:

- Angular frequency $\omega = 15 \text{ rad/s}$.

To find:

- Period $T$ of the oscillation.

We can rewrite the angular frequency formula of an oscillating object to find the period from angular frequency:

Now you know how to find the angular frequency and how to convert the angular frequency to period. Test your understanding by now converting this angular frequency to frequency.

## How to use this angular frequency calculator

In line with our policy of simplicity, this angular frequency calculator is straightforward to use:

**For rotational motion**:

- Enter the
`angular displacement`

and`time`

in their respective fields - The calculator will immediately calculate the
`angular frequency`

.

**For oscillations:**

- Enter either the
`frequency`

or`period`

of oscillations, and watch as the calculator instantly gives you the`angular frequency`

.