Angular Frequency Calculator
Find omega in rad/s from a frequency or period, or from the physics of a spring, a pendulum, an LC circuit or the mains supply.
〰️ What is Angular Frequency?
Angular frequency, written with the Greek letter omega, measures how fast the phase of an oscillation advances. It is expressed in radians per second. Because one complete cycle is 2 pi radians, omega is related to the ordinary frequency f in hertz and the period T in seconds by omega = 2 pi f = 2 pi / T. A 50 Hz signal therefore has an angular frequency of 314.16 rad/s, and a period of 20 ms. This angular frequency calculator converts between all of these quantities, including rotation rates in rpm, and then goes further by working out omega directly from the physics of the most common oscillating systems.
Physicists prefer omega because sine and cosine repeat every 2 pi radians, so a wave is neatly written as sin(omega t), and its derivatives carry no stray factors of 2 pi. Four systems are built into this page. A mass on a spring oscillates with omega = square root of (k / m), where stiffer springs and lighter masses give faster oscillation. A simple pendulum swinging through small angles has omega = square root of (g / L), which depends on the length and the local gravity but not on the mass. An LC circuit rings at omega = 1 / square root of (L C), the principle behind radio tuning. And mains electricity oscillates at 50 or 60 Hz, which sets the angular frequency of the voltage waveform. Each mode also draws a sinusoid chart so that you can see three cycles and check the period against the horizontal axis.
It is easy to mix up three similar ideas. Frequency f counts cycles per second in hertz. Angular frequency counts radians of phase per second, so it is always 2 pi times larger numerically. Angular velocity, as in the angular velocity calculator, describes how fast something rotates and has the same magnitude as omega for circular motion, but it is a vector with a direction. In oscillations nothing need rotate: omega is the rate at which the phase advances. Use the angular displacement calculator to turn omega and time into an angle turned.
Typical uses include checking the natural frequency of a suspension or a spring-loaded mechanism, timing a pendulum clock on different planets, tuning a resonant circuit, and understanding why the household supply voltage is described by 230 V RMS but peaks at 325 V.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — 50 Hz Mains
Find the angular frequency of a 50 Hz supply
Example 2 — An Engine at 1200 rpm
An engine idles at 1200 rotations per minute
Example 3 — A Mass on a Spring
A 0.5 kg mass on a 200 N/m spring oscillates with amplitude 0.05 m
Example 4 — A Pendulum on Mars
A pendulum 0.5 m long swings on Mars, where g = 3.721 m/s²
Example 5 — An LC Tank Circuit
A 10 mH inductor and a 100 µF capacitor are connected in a loop
Example 6 — 60 Hz, 120 V Mains
What is the voltage 2 ms into a cycle of 120 V RMS, 60 Hz supply?
❓ Frequently Asked Questions
🔗 Related Calculators
What is angular frequency?
Angular frequency, written omega, is how fast the phase of an oscillation advances, in radians per second. It equals 2 pi times the ordinary frequency f, so a 50 Hz signal has omega = 314.16 rad/s.
What is the formula for angular frequency?
omega = 2 pi f = 2 pi / T, where f is the frequency in hertz and T is the period in seconds. For a spring, omega = sqrt(k/m); for a pendulum, omega = sqrt(g/L); for an LC circuit, omega = 1/sqrt(LC).
How do you convert Hz to rad/s?
Multiply the frequency in hertz by 2 pi (6.28319). 50 Hz is 314.159 rad/s, 60 Hz is 376.991 rad/s and 1 Hz is 6.28319 rad/s.
How do you convert rpm to angular frequency?
Multiply rpm by 2 pi / 60, which is about 0.10472. 1200 rpm is 20 revolutions per second and 125.664 rad/s.
What is the angular frequency of a mass on a spring?
omega = sqrt(k/m). A 0.5 kg mass on a 200 N/m spring has omega = sqrt(400) = 20 rad/s, a frequency of 3.183 Hz and a period of 0.314 s.
What is the angular frequency of a pendulum?
For small swings omega = sqrt(g/L). A 1 m pendulum on Earth has omega = 3.132 rad/s and a period of 2.006 s. It does not depend on the mass or, for small swings, the amplitude.
What is the resonant angular frequency of an LC circuit?
omega = 1/sqrt(LC), with L in henries and C in farads. A 10 mH inductor with a 100 microfarad capacitor resonates at 1000 rad/s, which is 159.15 Hz.
What is the angular frequency of mains electricity?
It is 2 pi times 50 Hz = 314.16 rad/s in most of the world and 2 pi times 60 Hz = 376.99 rad/s in North America. Some aircraft and ships use 400 Hz, which is 2513.27 rad/s.
What is the difference between angular frequency and angular velocity?
They have the same units and the same size for circular motion. Angular velocity is a vector for rotation, and angular frequency is the scalar rate at which the phase of a periodic signal advances. A turntable at 33 rpm has both 3.46 rad/s.
Why do physicists use omega instead of f?
Sine and cosine repeat every 2 pi radians, so writing sin(omega t) avoids carrying a factor of 2 pi through derivatives. The derivative of sin(omega t) is omega cos(omega t) with no extra constants.