Angular Frequency Calculator – Calculate ω from Frequency, Period & RPM

  • Adam
  • November 27, 2025

Calculate angular frequency (ω) instantly from frequency, time period, or RPM. Free calculator with step-by-step solutions, formulas, and conversions for physics and engineering applications in rad/s.

Angular Frequency Calculator - Calculate Angular Frequency (ω)

An angular frequency calculator helps determine the angular frequency (ω) of oscillating or rotating systems, measured in radians per second. Angular frequency represents the rate of change of angular displacement over time and is fundamental to understanding periodic motion, simple harmonic motion, waves, and circular motion in physics and engineering applications.

Calculate Angular Frequency

Result:

What is Angular Frequency?

Angular frequency (denoted by the Greek letter ω, omega) is a scalar measure of rotation rate that represents how quickly an object rotates or oscillates in terms of angular displacement per unit time. It is measured in radians per second (rad/s) and describes the rate at which the phase of a sinusoidal waveform changes or the angular displacement changes in circular or oscillatory motion.

Key Characteristics of Angular Frequency:

  • Units: Radians per second (rad/s), the SI unit for angular frequency
  • Symbol: ω (omega), universally used in physics and engineering
  • Relationship to frequency: Angular frequency is 2π times the ordinary frequency (ω = 2πf)
  • Physical meaning: Represents the angle (in radians) swept per unit time in rotational motion or phase change per unit time in oscillations
  • Applications: Essential for analyzing simple harmonic motion, AC circuits, wave phenomena, and rotational dynamics

Angular Frequency Formula

Angular frequency can be calculated using several formulas depending on which parameters are known. The fundamental relationships connect angular frequency to ordinary frequency, time period, and rotational speed.

Angular Frequency from Frequency

The most common formula relates angular frequency to ordinary frequency (cycles per second):

ω = 2πf

Where:

  • ω = angular frequency (rad/s)
  • f = ordinary frequency (Hz or cycles per second)
  • 2π = conversion factor (approximately 6.2832 rad/cycle)

Angular Frequency from Time Period

Since frequency is the reciprocal of time period (f = 1/T), angular frequency can be expressed in terms of period:

ω = 2π / T

Where:

  • ω = angular frequency (rad/s)
  • T = time period (s), the time for one complete cycle
  • 2π = radians in one complete revolution

Angular Frequency from RPM

For rotating machinery and systems where speed is measured in revolutions per minute:

ω = 2πn / 60

or equivalently

ω = πn / 30

Where:

  • ω = angular frequency (rad/s)
  • n = rotational speed (RPM, revolutions per minute)
  • 60 = seconds per minute conversion factor

Related Formulas

Frequency from Angular Frequency:

f = ω / 2π

Period from Angular Frequency:

T = 2π / ω

Angular Displacement:

θ(t) = ωt

(for uniform angular motion)

Simple Harmonic Motion:

x(t) = A cos(ωt + φ)

(where A is amplitude and φ is phase)

How to Calculate Angular Frequency

Calculating angular frequency requires identifying which parameters are known and selecting the appropriate formula. Follow these systematic steps:

  1. Identify known parameters: Determine whether you have frequency (f), time period (T), rotational speed (RPM), or other oscillation characteristics.
  2. Select the appropriate formula: Use ω = 2πf for frequency, ω = 2π/T for period, or ω = 2πn/60 for RPM.
  3. Ensure consistent units: Convert all values to SI units (Hz for frequency, seconds for period, ensure RPM is properly specified).
  4. Substitute values: Plug the known values into the selected formula.
  5. Calculate the result: Perform the mathematical operations to determine angular frequency in rad/s.
  6. Verify reasonableness: Check that the result makes physical sense for the system being analyzed.

Calculation Example 1: From Frequency

Problem: An AC power supply operates at 60 Hz. Calculate the angular frequency.

Solution:

Given: f = 60 Hz

Using formula: ω = 2πf

ω = 2π × 60

ω = 120π rad/s

ω = 376.99 rad/s

The angular frequency of the 60 Hz AC supply is approximately 377 rad/s.

Calculation Example 2: From Time Period

Problem: A pendulum has a time period of 2 seconds. Find the angular frequency.

Solution:

Given: T = 2 s

Using formula: ω = 2π / T

ω = 2π / 2

ω = π rad/s

ω = 3.1416 rad/s

The pendulum oscillates with an angular frequency of π rad/s.

Calculation Example 3: From RPM

Problem: A motor shaft rotates at 1200 RPM. Calculate the angular frequency.

Solution:

Given: n = 1200 RPM

Using formula: ω = 2πn / 60

ω = 2π × 1200 / 60

ω = 2π × 20

ω = 40π rad/s

ω = 125.66 rad/s

The motor shaft has an angular frequency of approximately 125.66 rad/s.

Angular Frequency vs Frequency

Understanding the distinction between angular frequency and ordinary frequency is essential for correctly applying these concepts in physics and engineering:

Characteristic Angular Frequency (ω) Ordinary Frequency (f)
Definition Rate of change of angular displacement Number of complete cycles per unit time
Symbol ω (omega) f or ν (nu)
Units Radians per second (rad/s) Hertz (Hz) or cycles per second
Relationship ω = 2πf f = ω/2π
Physical Meaning Angle swept per unit time Complete oscillations per unit time
Application Preferred in mathematical physics equations Common in practical measurements

Applications of Angular Frequency

Angular frequency is a fundamental parameter in numerous fields of physics, engineering, and technology:

  • Simple Harmonic Motion: Describing oscillations of springs, pendulums, and vibrating systems using equations like x(t) = A cos(ωt + φ)
  • Wave Mechanics: Characterizing wave propagation with wave equations y(x,t) = A sin(kx - ωt) where ω determines temporal oscillation
  • AC Circuit Analysis: Analyzing alternating current circuits with impedance calculations using ω in expressions like Z = R + jωL
  • Rotational Dynamics: Describing rotating machinery, turbines, motors, and mechanical systems where ω represents rotation rate
  • Signal Processing: Analyzing periodic signals, Fourier transforms, and frequency domain representations in telecommunications
  • Quantum Mechanics: Representing energy states with E = ℏω where ℏ is reduced Planck's constant
  • Control Systems: Designing feedback control systems, analyzing stability, and determining system response characteristics
  • Electromagnetic Waves: Describing oscillating electric and magnetic fields in radio waves, microwaves, and optical systems

Angular Frequency in Simple Harmonic Motion

In simple harmonic motion (SHM), angular frequency plays a central role in determining all aspects of the oscillatory behavior:

Displacement Equation

x(t) = A cos(ωt + φ)

or equivalently

x(t) = A sin(ωt + φ)

Velocity and Acceleration in SHM

Velocity:

v(t) = -Aω sin(ωt + φ)

Maximum Velocity:

vmax = Aω

Acceleration:

a(t) = -Aω² cos(ωt + φ) = -ω²x

Maximum Acceleration:

amax = Aω²

Angular Frequency for Mass-Spring System

ω = √(k/m)

Where k is spring constant (N/m) and m is mass (kg)

Angular Frequency for Simple Pendulum

ω = √(g/L)

Where g is gravitational acceleration (9.8 m/s²) and L is pendulum length (m)

Frequently Asked Questions

What is the difference between angular frequency and regular frequency?
Angular frequency (ω) measures the rate of change of angular displacement in radians per second, while regular frequency (f) measures the number of complete cycles per second in Hertz. They are related by ω = 2πf. Angular frequency is 2π times larger than regular frequency because one complete cycle corresponds to 2π radians. For example, a 1 Hz signal has an angular frequency of 2π rad/s (approximately 6.28 rad/s).
How do you convert frequency to angular frequency?
To convert frequency to angular frequency, multiply the frequency by 2π: ω = 2πf. For example, a frequency of 50 Hz converts to ω = 2π × 50 = 314.16 rad/s. This conversion accounts for the fact that one complete cycle involves rotating through 2π radians (360 degrees). The factor 2π appears because angular frequency measures rotation in radians while ordinary frequency measures complete cycles.
What is the unit of angular frequency?
The SI unit of angular frequency is radians per second (rad/s). Since the radian is technically dimensionless (being a ratio of two lengths), angular frequency can also be expressed simply as s⁻¹ (inverse seconds or per second). However, rad/s is preferred to clearly distinguish angular frequency from ordinary frequency (Hz) and to emphasize that it measures angular displacement rate rather than cycle count.
How do you calculate angular frequency from RPM?
To calculate angular frequency from RPM (revolutions per minute), use the formula ω = 2πn/60 or ω = πn/30, where n is the RPM value. For example, 300 RPM converts to ω = 2π × 300/60 = 10π = 31.42 rad/s. This formula converts revolutions per minute to radians per second by accounting for 2π radians per revolution and 60 seconds per minute.
What is the angular frequency formula for simple harmonic motion?
For simple harmonic motion, angular frequency depends on the system. For a mass-spring system, ω = √(k/m) where k is spring constant and m is mass. For a simple pendulum, ω = √(g/L) where g is gravitational acceleration and L is length. For any SHM system, once you know the frequency f or period T, you can calculate ω = 2πf or ω = 2π/T.
Is angular frequency the same as angular velocity?
Angular frequency and angular velocity are numerically equal and both measured in rad/s, but they have slightly different conceptual meanings. Angular velocity typically refers to the rotation rate of a rigid body in space (with direction specified by the right-hand rule), while angular frequency refers to the temporal oscillation rate of a periodic phenomenon. In many physics contexts, especially for oscillations and waves, the terms are used interchangeably.
How do you find angular frequency from wavelength?
To find angular frequency from wavelength, you need to know the wave speed v. First calculate frequency using f = v/λ (where λ is wavelength), then convert to angular frequency using ω = 2πf = 2πv/λ. Alternatively, use the wave number k = 2π/λ and the dispersion relation ω = vk = 2πv/λ. For electromagnetic waves in vacuum, v = c (speed of light, 3×10⁸ m/s).
What is the relationship between angular frequency and period?
Angular frequency and period are inversely related through ω = 2π/T, where T is the period (time for one complete cycle). Rearranging gives T = 2π/ω. A higher angular frequency means a shorter period (faster oscillation), and a lower angular frequency means a longer period (slower oscillation). For example, if ω = π rad/s, then T = 2π/π = 2 seconds.
How does angular frequency relate to energy in oscillating systems?
In oscillating systems, total mechanical energy is proportional to the square of angular frequency: E = ½mA²ω² for a mass m oscillating with amplitude A. This shows that higher angular frequency results in greater energy for the same amplitude. In quantum mechanics, the energy of a photon relates to angular frequency by E = ℏω, where ℏ is the reduced Planck's constant (1.055×10⁻³⁴ J·s).
Can angular frequency be negative?
While angular frequency magnitude is always positive, in some mathematical contexts (particularly signal processing and complex analysis), negative angular frequencies are used to represent waves traveling in opposite directions or to distinguish between counter-rotating components. In physical systems, the actual oscillation rate is always positive, but the sign convention can indicate phase rotation direction or wave propagation direction in mathematical representations.

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