Angular Frequency Calculator

Angular Frequency Calculator finds ω in rad/s from angle and time, frequency, period, rpm, speed and radius, a spring and mass, or a simple pendulum, then converts it to Hz or rpm.

What the Angular Frequency Calculator Finds

Angular frequency, written ω (lowercase omega), tells you how many radians of angle or phase pass each second. One full turn or one full cycle is 2π radians, so ω is the ordinary frequency multiplied by 2π.

The Angular Frequency Calculator starts from whatever you already know. A spinning wheel can go in as angle and time, as rpm, or as rim speed and radius.

An oscillation can go in as frequency or time period. A mass on a spring or a simple pendulum gives its natural angular frequency from its own properties.

The SI unit of angular frequency is the radian per second (rad/s). The NIST edition of the SI brochure keeps hertz for frequency and rad/s for angular frequency and angular velocity, because the two numbers differ by a factor of 2π. Reading one as the other puts every later answer off by that same factor.

Angular Frequency Calculator Formulas

Each choice under What to Calculate uses one formula. Every entry is converted to SI units first, and ω is then worked out in rad/s. The Step-by-Step Working panel prints the same substitution with your numbers, so it can be checked line by line against homework or lab notes.

Angle and Time

The Angular Frequency Calculator opens in this mode. For a rotating object, angular frequency is the angle turned divided by the time taken, which is also the formula for angular speed. The angle can be typed in radians, degrees or revolutions, and the time in anything from milliseconds to days.

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

A tire that turns 200 rad in 40 s has ω = 200 ÷ 40 = 5 rad/s. That is 0.79577 Hz, or 47.746 rpm. The result is an average over those 40 s, and it matches the value at every instant only when the spin is steady.

The same formula runs backward in two more modes. Angle from ω and Time multiplies the two, so 15 rad/s held for 40 s sweeps 600 rad, which is 95.493 full turns.

Time from ω and Angle divides instead, so 200 rad at 15 rad/s takes 13.333 s. Problems that start from arc length or include acceleration belong in the Angular Displacement Calculator.

Frequency, Period and RPM

An oscillation or a wave completes f cycles each second, and each cycle is 2π rad. The period T is the time for one cycle, so the period frequency formula is f = 1 ÷ T. Either one is enough to calculate angular frequency.

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

US mains power at 60 Hz gives ω = 2π × 60 = 376.99 rad/s. A signal with a period of 0.02 s gives ω = 2π ÷ 0.02 = 314.16 rad/s, which is 50 Hz mains. Frequency can be entered in Hz, kHz, MHz or GHz, and the period in units down to microseconds.

Shaft and motor speeds are usually quoted in revolutions per minute, written n here. One revolution is 2π rad and one minute is 60 s, so every rpm is worth 2π ÷ 60 ≈ 0.10472 rad/s of angular speed.

$$\omega = \frac{2\pi n}{60}$$

A motor at 1,800 rpm turns at 188.5 rad/s, the same as 30 revolutions per second. The rotation rate field also accepts rev/s and degrees per second, which suits turntable and gyroscope readings.

Speed and Radius

A point on a spinning body moves along a circle. Its tangential speed is the angular frequency times its distance from the axis, v = ωr. Used as an angular frequency calculator from velocity, the tool turns that relation around.

$$\omega = \frac{v_t}{r}$$

A rim point moving at 10 m/s on a 0.25 m radius gives ω = 40 rad/s, or 381.97 rpm. The speed has to be measured at the same radius you enter. In this mode the third result card shows the centripetal acceleration, 400 m/s² here, and the load of 40.789 g.

Natural Angular Frequency

Leave a spring or a pendulum alone after a push and it swings at its own rate. That rate is the natural angular frequency. It depends on the system itself, not on how hard it was pushed. The angular frequency calculator has two modes for it, Spring and Mass and Simple Pendulum.

$$\omega = \sqrt{\frac{k}{m}}$$

Here k is the spring constant and m is the mass on the spring. An 80 N/m spring carrying 0.2 kg gives ω = √400 = 20 rad/s and a period of 0.31416 s.

OpenStax University Physics derives this result and shows that a hanging spring oscillates at the same rate as a horizontal one. If you only know a force and the stretch it causes, get k first from the Hooke’s Law Calculator.

$$\omega = \sqrt{\frac{g}{L}}$$

For a simple pendulum, L is the length and g is gravity, which stays at the standard 9.80665 m/s² until you change it. A 1 m pendulum gives 3.1316 rad/s and a period of 2.0064 s. The small-angle derivation holds for swings under about 15°, and the mass of the bob does not appear in it at all.

Converting ω to Hz, RPM and Degrees

To convert angular frequency to Hz, divide by 2π. To go from Hz to angular frequency, multiply by 2π. With ω = 15 rad/s, the Frequency from ω mode returns f = 15 ÷ 6.28319 = 2.3873 Hz and a period of 0.41888 s.

The Angular Frequency Calculator runs these conversions on every result. The first card covers timing, with frequency, period and the time for half a cycle. The second covers rotation units such as rpm, degrees per second and turns per hour, which read better for slow motion like a clock hand.

Whatever you typed in is left off the cards, so no value is shown twice. In most modes that solve for ω, the answer itself can be switched from rad/s to rpm or °/s under Result Options.

The Common ω Values table then places your result among everyday rates, from a clock’s hour hand at 0.00014544 rad/s up to concert pitch A4 at 2,764.6 rad/s.

Angular Frequency vs Angular Velocity

Both are measured in rad/s, which is why the angular frequency calculator also works as an angular speed calculator for steady rotation. The difference is direction. Angular velocity is a vector that points along the rotation axis, and its sign tells clockwise from counterclockwise.

Angular frequency is a scalar, so it has a size and no direction. It equals the magnitude of the angular velocity, and that magnitude is what angular speed means. Every field in the calculator therefore takes a positive number, and direction is left for you to assign.

In an oscillation such as a mass on a spring nothing actually turns, and ω counts phase in radians per second instead. The Cycle Diagram shows the link between the two pictures. A radius sweeping the angle θ = ωt around a circle traces one full sine wave in one period T.

Angular speed is measured by timing rotation. A tachometer counts revolutions per minute, a gyroscope sensor reports degrees per second, and a stopwatch on a marked wheel gives angle over time. When the spin rate itself is changing, the Angular Acceleration Calculator finds how fast ω rises or falls.

Rim Speed, Rolling Speed and Wavelength

Angular frequency is not a speed in meters per second, but a radius turns it into one. Enter a radius under Result Options and the third card multiplies the two, v = ωr. At 5 rad/s and 0.5 m the rim moves at 2.5 m/s, with a centripetal acceleration of ω²r = 12.5 m/s².

In the rotating modes the fourth card is Rolling Speed, the same rim speed in everyday units. The tire example reads 9 km/h, with 5.5923 mph and 8.2021 ft/s beside it.

A wheel that rolls without slipping covers ground at exactly this speed, so the angular frequency calculator turns a rotation frequency into a road speed. The five rotating modes show this card in place of Wavelength (λ) and hide the Wave Speed (v) field.

For a spring or a pendulum the same field becomes the amplitude A. The card then gives the peak speed ωA as the mass passes the center, and the peak acceleration ω²A where it turns around at each end of the swing.

In the oscillating, spring and pendulum modes, the fourth card of the Angular Frequency Calculator treats your result as a wave. In physics, λ (lambda) is the wavelength, the distance from one crest to the next, and its SI unit is the meter.

Lambda is the Greek letter L, and its use for wavelength is a naming convention. To calculate lambda, divide the wave speed by the frequency.

$$\lambda = \frac{v}{f} = \frac{2\pi v}{\omega}$$

One lambda is also the distance the wave travels in one period. The Wave Speed (v) field under Result Options starts at 343 m/s, the speed of sound in air at 20 °C.

Concert pitch A4 at 440 Hz then has a wavelength of 0.77955 m and a wave number k = ω ÷ v = 8.0601 rad/m, so the card works as a frequency to distance calculator for sound. If the field is cleared, the card reads Enter a Wave Speed until a value goes back in.

For radio or light, change the wave speed to 299,792.458 km/s, and a 2.4 GHz Wi-Fi signal comes out at 12.491 cm. Radio waves, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays are the seven types of electromagnetic wavelength bands.

All seven bands follow the same rule, wave speed divided by frequency. For a wave, going from frequency to speed uses the reverse form, v = fλ, when the wavelength is known.

Angular Frequency and Hertz FAQ

What is the difference between angular frequency and normal frequency?

Normal frequency, also called linear frequency, counts whole cycles per second in hertz. Angular frequency counts radians per second. One cycle is 2π rad, so ω is always 6.2832 times the frequency. A 50 Hz supply and a 314.16 rad/s supply are the same supply, described in two units.

What is 1 Hz equal to?

One hertz is one complete cycle per second, so its period is exactly 1 s. In angular terms 1 Hz equals 2π ≈ 6.2832 rad/s. For a rotating shaft it equals one revolution per second, which is 60 rpm or 360 degrees per second.

How do you calculate frequency?

The basic formula for frequency is f = 1 ÷ T, where T is the time for one cycle. You can also divide rpm by 60, divide angular frequency by 2π, or divide wave speed by wavelength.

A fan at 300 rpm has f = 5 Hz, and a 343 m/s sound wave with λ = 0.5 m has f = 686 Hz. The Frequency from ω mode of the angular frequency calculator handles the 2π division.

Why is angular frequency important?

Sine and cosine take an angle, so equations of motion are written with ω, as in x = A cos(ωt + φ). Peak speed is then ωA and peak acceleration ω²A, with no stray 2π to carry. AC circuit work uses it the same way, since inductive reactance is ωL.

Rotation uses it too, because angular momentum is L = Iω. In physics problems the angular frequency calculator supplies the ω for that product, while the moment of inertia I has to come from the shape and mass of the object. The other rotational motion calculators cover torque, inertia and spin-up.