Free Angular Frequency Calculator

This free online angular frequency calculator works as both an angular velocity calculator and a rotational frequency calculator, enabling you to find omega (ω) for any rotating object or simple harmonic oscillator. Whether you need an oscillation frequency calculator for a pendulum or an RPM to rad/s calculator for a motor shaft, this tool covers both types of motion with ease. The following sections define angular frequency, present the essential formulas, and explain how to use the calculator effectively.

Angular Frequency — Definition and Units

Angular frequency (represented by the Greek letters Ω or ω) quantifies how swiftly an object rotates or oscillates in time. Its standard SI unit is the radian per second (rad/s), but it can also be expressed in revolutions per minute (rpm) or hertz (Hz) after a straightforward conversion. The angular frequency differs from ordinary frequency by the constant factor $2\pi$: $\omega = 2\pi f$, where $f$ is the frequency in hertz. Similarly, if the time period $T$ (the duration of one complete cycle) is known, the relationship is $\omega = \frac{2\pi}{T}$.

Formulas for Angular Frequency

The method used to calculate ω depends on whether the motion involves rotation around a fixed axis or oscillation about an equilibrium point.

Rotating Objects (Rotational Motion)

For a body spinning like a wheel or a planetary gear, the angular velocity is defined as the change in angular displacement per unit time:

$$\omega = \frac{\Delta \theta}{\Delta t}$$

where $\Delta \theta$ is the angle covered (in radians) and $\Delta t$ is the time elapsed (in seconds).

Example: If a car tire rotates through 200 radians in 40 seconds, its angular frequency is:

$$\omega = \frac{200\ \text{rad}}{40\ \text{s}} = 5\ \text{rad/s}$$

This result can also be expressed as approximately 47.75 rpm or 0.796 Hz by using the conversion factors later in this article.

Oscillating Objects (Simple Harmonic Motion)

For systems undergoing simple harmonic motion (SHM) — such as a mass‑spring arrangement, a pendulum, or an LC circuit — the angular frequency is derived from the ordinary frequency or the period:

$$\omega = 2\pi f = \frac{2\pi}{T}$$

Example: A pendulum oscillates with a frequency of 2 Hz. Its angular frequency is:

$$\omega = 2\pi \times 2\ \text{Hz} = 4\pi\ \text{rad/s} \approx 12.566\ \text{rad/s}$$

Conversely, if the period of the motion is 0.5 s, the same $\omega$ is obtained from $\omega = \frac{2\pi}{0.5\ \text{s}} = 4\pi\ \text{rad/s}$.

How to Use This Angular Frequency Calculator

1. Select the appropriate motion type — Rotating objects or Oscillating objects — using the switch or tabs provided.
2. Enter the known values. For rotation, input the angular displacement (in radians or degrees) and the time interval. For oscillation, input either the frequency (in Hz) or the time period (in seconds).
3. Choose your desired output unit for ω from the dropdown menu (rad/s, rpm, etc.). The calculator automatically performs the conversion.
4. Read the calculated angular frequency instantly. The tool also works in reverse: if you already have ω, you can enter it to solve for the missing parameter (displacement, time, frequency, or period).

Always verify that your input units are consistent. The dropdown helpers let you switch between radian and degree measures, as well as seconds, minutes, or hours for time.

Quick Unit Conversion Reference

The table below summarizes the most common conversions involving angular frequency:

Conversion	Multiplier
rad/s → rpm	$\displaystyle \frac{60}{2\pi} \approx 9.5493$
rpm → rad/s	$\displaystyle \frac{2\pi}{60} \approx 0.10472$
rad/s → Hz	$\displaystyle \frac{1}{2\pi} \approx 0.15915$
Hz → rad/s	$\displaystyle 2\pi \approx 6.2832$

For example, the 5 rad/s from the wheel example equals $5 \times 9.5493 \approx 47.75$ rpm, and the 4π rad/s from the pendulum corresponds to $f = \frac{\omega}{2\pi} = \frac{4\pi}{2\pi} = 2$ Hz — the original frequency, as expected.

Why Angular Frequency Matters

Angular frequency appears in many areas of physics and engineering: from analyzing rotating machinery to describing alternating currents, wave motion, and quantum mechanics. Having a dedicated tool that handles both rotational and oscillatory scenarios streamlines calculations and reduces errors.

Whether you are a student checking homework problems or a professional designing a mechanical system, this angular frequency calculator delivers quick, reliable results. Its ability to toggle between rad/s, rpm, and Hz makes it an indispensable resource for anyone dealing with periodic motion.