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 (rad/s)(\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 Δθ\Delta \theta is:

ω=ΔθΔt\omega = \frac{\Delta \theta}{\Delta t}

Where:

  • ω\omega - Angular frequency of rotation;
  • Δθ\Delta \theta - Angular displacement during rotation; and
  • Δt\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:

ω=2πT=2πf\omega = \frac{2 \pi}{T} = 2 \pi f

Where:

  • ω\omega - Angular frequency of oscillation;
  • TT - Period of oscillation, or time needed to complete one oscillation; and
  • ff - 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.

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

Given:

  • Angular displacement Δθ=100 rad\Delta \theta = 100 \text{ rad}.
  • Time needed Δt=15 s\Delta t = 15 \text{ s}.

To find:

  • Angular frequency ω\omega of the wheel's rotation.

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

ω=ΔθΔt=10015ω=6.667 rad/s\qquad \begin{align*} \omega &= \frac{\Delta \theta}{\Delta t} = \frac{100}{15}\\[1em] \omega &= 6.667 \text{ rad/s} \end{align*}
  1. If a simple pendulum oscillates at a 7 Hz frequency, what is its angular frequency?

Given:

  • Frequency f=7 Hzf = 7 \text{ Hz}.

To find:

  • Angular frequency ω\omega of the oscillating pendulum.

Using the angular frequency formula for oscillating objects:

ω=2πf=2π7ω=43.982 rad/s\qquad \begin{align*} \omega &= 2 \pi f = 2 \cdot \pi \cdot 7\\[1em] \omega &= 43.982 \text{ rad/s} \end{align*}
  1. Find the period of oscillation of an object if its angular frequency is 15 rad/s.

Given:

  • Angular frequency ω=15 rad/s\omega = 15 \text{ rad/s}.

To find:

  • Period TT of the oscillation.

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

ω=2πT    T=2πω=2π15T=0.418 s\qquad \begin{align*} \omega &= \frac{2 \pi}{T}\\[1em] \implies T &= \frac{2 \pi}{\omega} = \frac{2 \pi}{15}\\[1em] T &= 0.418 \text{ s} \end{align*}

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.
Krishna Nelaturu
For rotating objects
Angular displacement
rad
Time taken
sec
Angular frequency
rad/s
For oscillating objects
Time period
sec
Frequency
Hz
Angular frequency
rad/s
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