Angular Velocity Calculator gives ω in rad/s from an angle and time, a linear speed and radius, rpm, hertz or a period. It also covers the mean ω between two positions and ω₀ + αt.
Enter the angle turned and the time it took. Solve For also works backward.
Radius and Latitude
Radius feeds the linear speed card. Latitude is for a point on a spinning globe; leave it at 0 otherwise.
Step-by-Step Working
- Δα = 120°
- t = 2 s
- In SI units: Δα = 2.0944 rad, t = 2 s
- Formula: ω = Δα ÷ t
- ω = 2.0944 ÷ 2 = 1.0472 rad/s
Right-Hand Rule Diagram
Common ω Values
| Example | ω (rad/s) | rpm | Speed ratio |
|---|---|---|---|
| Earth around the Sun | 1.991 × 10⁻⁷ | 1.9013 × 10⁻⁶ | × 5,259,700 |
| Earth’s own rotation | 7.2921 × 10⁻⁵ | 0.00069634 | × 14,361 |
| Clock hour hand | 0.00014544 | 0.0013889 | × 7,200 |
| Clock minute hand | 0.0017453 | 0.016667 | × 600 |
| Clock second hand | 0.10472 | 1 | × 10 |
| Your value | 1.0472 | 10 | 1 |
| Vinyl LP, 33⅓ rpm | 3.4907 | 33.333 | × 0.3 |
| Vinyl single, 45 rpm | 4.7124 | 45 | × 0.22222 |
What the Angular Velocity Calculator Solves
The Angular Velocity Calculator finds ω, the rate at which something turns, in radians per second. Its Method dropdown holds eight ways in, grouped by the numbers you are likely to have on hand.
Angle Difference takes an angle change and a time, and Linear Velocity takes the speed of a point and its distance from the axis. Both have a Solve For menu, so they also return the angle, the time, the linear speed or the radius. Cross Product takes r and v as x, y, z components.
RPM or Frequency converts a rotation rate given in rpm, hertz or degrees per second, and Period uses the time for one turn. Two Positions gives the mean ω between two readings. Acceleration applies ω₀ + αt when the spin is speeding up or slowing down, and No Time finds the final ω from the angle turned.
A closed panel under the fields of the Angular Velocity Calculator holds an optional radius, latitude and elapsed time, and none of them changes ω. The radius and latitude fill the linear speed card, and the time fills the card for the angle turned.
Print / PDF saves the answer, the four cards and the working as one sheet, which covers searches for an angular velocity calculator pdf. Related tools for spin, torque and inertia are grouped under Rotational Motion Calculators.
What Is Angular Velocity (ω)?
Angular velocity is how fast an angle changes with time. It gives the rate of turning about an axis and, through its sign or direction, which way the turning goes. The short answer for an exam is the angle swept per unit time, measured in rad/s.
Is ω an angular velocity?
Yes, the lowercase Greek letter omega, ω, is the standard symbol for angular velocity in mechanics. ω is equal to the angle turned divided by the time taken. For steady rotation it also equals 2π divided by the period, 2π times the frequency, or v ÷ r.
The same letter stands for angular frequency in oscillations and waves, where nothing has to rotate. Both are measured in rad/s and both obey ω = 2πf, which is why the Angular Frequency Calculator shares some of these formulas and adds springs and pendulums.
What is angular speed?
Angular speed is the size of the angular velocity with the direction left out, so it is never negative. A wheel turning clockwise at 10 rad/s has an angular speed of 10 rad/s and an angular velocity of −10 rad/s, under the usual rule that counterclockwise is positive.
Another name for angular velocity is rotational velocity. When the figure is quoted in rpm, as it is for motors and engines, it is usually called rotational speed, since rpm carries no direction.
How is angular velocity measured?
On a machine it is measured with a tachometer or a shaft encoder, which count turns in a known time, or with a gyroscope. Analog Devices describes a MEMS gyroscope as a sensor of angular velocity that reads the Coriolis acceleration on a vibrating mass, with its sensitivity rated per degree per second.
Angular Velocity Formula
Every angular velocity formula comes from one definition, the change in angle divided by the change in time. OpenStax University Physics, section 10.1 gives the average form as Δθ ÷ Δt and the instantaneous form as the derivative dθ/dt.
$$\omega = \frac{\Delta \theta}{\Delta t}$$
How do I calculate angular velocity?
Convert the angle to radians, then divide by the time in seconds. The default entries are a 120° turn in 2 s. Since 120° is 2.0944 rad, ω = 2.0944 ÷ 2 = 1.0472 rad/s, which the first card also shows as 10 rpm and 60 °/s.
One full turn is 2π rad or 360°, so multiply degrees by π ÷ 180 and revolutions by 2π. The Angular Velocity Calculator accepts all three angle units and lists the converted values under Step-by-Step Working.
Set Solve For to Angle Change or Time to run the formula backward. The OpenStax flywheel turns at 45 rad/s, and over 30 s that gives Δα = 45 × 30 = 1,350 rad, about 215 revolutions. The Angular Displacement Calculator takes angle problems further.
What is the angular velocity class 11 formula?
Class 11 physics uses the same definition, ω = Δθ ÷ Δt, with ω = dθ/dt at an instant. Three results follow from it for uniform circular motion. One turn of 2π in a period T gives ω = 2π ÷ T, a frequency f gives ω = 2πf, and a point at radius r moves at v = rω. Each of the three has its own mode in the Angular Velocity Calculator.
How to calculate mean angular velocity?
Mean angular velocity is the net angle turned between two moments divided by the time between them. It ignores any speeding up or slowing down in between, so it matches the instantaneous value only when the spin is steady.
$$\omega = \frac{\theta_2 – \theta_1}{t_2 – t_1}$$
Two Positions mode in the Angular Velocity Calculator takes the start and end angles with their clock times. From 30° at 1 s to 150° at 4 s, the mean is 120° ÷ 3 s = 40 °/s, or 0.69813 rad/s. If the end angle is the smaller one, ω comes out negative and the result is labeled clockwise.
Enter the angles as running totals, not as readings that reset each lap. A wheel that passes 350° and reaches 10° on its next lap has turned to 370°, and typing 10° would be read as a swing of 340° backward.
How is angular velocity derived?
The link to linear speed is derived from arc length. A point at radius r that turns through θ radians moves a distance s = rθ along its circle. Dividing both sides by the time, with r fixed, gives v = rω, and so ω = v ÷ r.
RPM to Angular Velocity Formula
Motors, engines and drills are rated in revolutions per minute, while physics formulas need rad/s. One revolution is 2π rad and one minute is 60 s, which fixes the conversion factor at 2π ÷ 60.
$$\omega = \frac{2 \pi N}{60} = \frac{\pi N}{30}$$
How do I convert RPM to angular velocity?
To find angular velocity with revolutions per minute, multiply the rpm figure N by 0.10472. A shaft at 3,500 rpm turns at 3,500 × 0.10472 = 366.52 rad/s. RPM or Frequency mode loads with that example and adds 21,000 °/s, 58.333 rev/s and 17.143 ms per revolution.
Is rpm a measure of angular velocity? Yes, it measures the same thing in a different unit, a full turn per minute where rad/s uses a radian per second. It carries no direction, so strictly it states the angular speed.
How to convert angular velocity into RPM?
Multiply rad/s by 60 ÷ 2π, which is 9.5493. The fishing reel in OpenStax reaches 220 rad/s, and 220 × 9.5493 = 2,100.8 rpm. In the Angular Velocity Calculator, pick rad/s as the unit of Rotation Rate and the main answer switches to rpm.
$$N = \frac{60 \, \omega}{2 \pi}$$
Angular Velocity From Period or Frequency
A body that repeats its motion is often described by its period T, the time for one turn, or by its frequency f, the number of turns each second. Either one gives ω directly, because every turn is 2π radians.
$$\omega = \frac{2 \pi}{T} = 2 \pi f$$
Is angular velocity equal to 2π/t?
Only when t is the period, the time for exactly one revolution. The angle is then 2π, which is why angular velocity is 2π over T. For any other stretch of time, the formula is the angle actually turned divided by that time.
To calculate angular velocity from a period, enter it in the Period mode of the Angular Velocity Calculator. The default of 8 s gives ω = 6.2832 ÷ 8 = 0.7854 rad/s, or 7.5 rpm. The time units run from milliseconds to days, which suits both spindles and planets.
What is ω = 2πf?
It is the same rule written with frequency, since f = 1 ÷ T. A rotor at 50 Hz makes 50 turns each second and has ω = 2π × 50 = 314.16 rad/s, which is 3,000 rpm. Choose Hertz as the unit in RPM or Frequency mode to apply it.
How to find the angular velocity of a minute hand?
A minute hand takes 60 minutes, or 3,600 s, for one turn. Its angular velocity is 2π ÷ 3,600 = 0.0017453 rad/s, which is 0.1 °/s. The hour hand needs 12 hours and turns at 0.00014544 rad/s, while the second hand turns at 0.10472 rad/s, exactly 1 rpm.
Clock hands turn clockwise, so with counterclockwise taken as positive their angular velocity is negative. To check any of the three, type 1 h, 12 h or 60 s into Period mode and read the result.
Angular Velocity of the Earth
Earth turns once on its axis in 23.9345 hours, the sidereal rotation period on the NASA Earth Fact Sheet. The Angular Velocity Calculator, set to Period mode with that entry, gives ω = 7.2921 × 10⁻⁵ rad/s. The 24-hour day is slightly longer because Earth also moves along its orbit as it spins.
At the equator, 6,378.137 km from the axis, that small ω still means a linear speed of 465.1 m/s. Set the radius in the optional panel to see it. The centripetal acceleration there is only 0.033916 m/s², about 0.35% of standard gravity.
Away from the equator a point circles the axis at a smaller radius, r cos φ. Enter the latitude beside the radius and the card switches to ωr cos φ. With Earth’s mean radius of 6,371 km, a point at 60° moves at 232.29 m/s, half the 464.58 m/s of the equator.
Around the Sun, one orbit takes 365.256 days and ω is 1.991 × 10⁻⁷ rad/s. Multiplied by the orbit radius of 149.598 million km, that comes to about 29.8 km/s, in line with NASA’s mean orbital velocity of 29.78 km/s.
What is an example of angular velocity closer to home? The Common ω Values table in the Angular Velocity Calculator places your result among the everyday cases below and shows how many times faster or slower yours is.
| Example | ω (rad/s) | rpm |
|---|---|---|
| Earth around the Sun | 1.991 × 10⁻⁷ | 1.9013 × 10⁻⁶ |
| Earth’s own rotation | 7.2921 × 10⁻⁵ | 0.00069634 |
| Clock hour hand | 0.00014544 | 0.0013889 |
| Clock minute hand | 0.0017453 | 0.016667 |
| Clock second hand | 0.10472 | 1 |
| Vinyl LP | 3.4907 | 33.333 |
| Vinyl single | 4.7124 | 45 |
Angular Velocity to Linear Velocity
Every point on a rigid body shares one angular velocity, but points farther from the axis travel faster. The link between the two is the radius, and this is the angular velocity formula with radius that most problems call for.
$$v = \omega r$$
How to change angular velocity to velocity?
Multiply ω in rad/s by the radius in meters. In the Linear Velocity mode of the Angular Velocity Calculator, set Solve For to Velocity and enter both. The OpenStax flywheel at 45 rad/s moves a point 10 cm from its axis at 45 × 0.10 = 4.5 m/s.
Going the other way, ω = v ÷ r, and the default 12 m/s at 0.4 m gives 30 rad/s, or 286.48 rpm. Use the tangential speed, the part of the motion at right angles to the radius. Motion straight toward or away from the axis adds nothing to ω.
This mode leads its third card with centripetal acceleration, ω²r, which is 360 m/s² for the default, or 36.71 g. Path per turn, 2πr, is 2.5133 m. In the other modes the same card starts with the linear speed at the optional radius.
What is the difference between velocity and angular velocity?
Velocity is the rate of change of position. The formula for velocity is displacement divided by time, v = Δx ÷ Δt, in meters per second. Angular velocity is the rate of change of angle, in radians per second, and it is the same at every point of a rigid body.
| Quantity | Linear motion | Rotation |
|---|---|---|
| What changes | Position x, in m | Angle θ, in rad |
| Velocity | v = Δx ÷ Δt, in m/s | ω = Δθ ÷ Δt, in rad/s |
| Acceleration | a, in m/s² | α, in rad/s² |
| Link at radius r | v = ωr | ω = v ÷ r |
How is angular velocity related to velocity? Through v = ωr alone, so doubling the radius at the same ω doubles the speed. For straight-line problems built on distance and time, the Velocity Calculator is the better fit.
Angular Velocity Unit
The SI unit of angular velocity is the radian per second, rad/s, as listed in the NIST Guide to the SI. A radian is a ratio of two lengths, so rad/s reduces to 1/s, which is why ω × r comes out in plain m/s.
The SI unit of velocity, by comparison, is the meter per second. Angular velocity is also quoted in rpm, in revolutions per second, which equal hertz for a turning shaft, and in degrees per second for gyroscopes and slow mechanisms.
| Unit | In rad/s | In rpm |
|---|---|---|
| 1 rad/s | 1 | 9.5493 |
| 1 rpm | 0.10472 | 1 |
| 1 rev/s (1 Hz) | 6.2832 | 60 |
| 1 °/s | 0.017453 | 0.16667 |
The first result card of the Angular Velocity Calculator shows the answer in whichever of these units you did not type. The second card turns it into time, with seconds per revolution, the time for a quarter turn and the time per radian, and Period mode adds turns per day.
When the Spin Changes
A constant ω is the exception in real machines. When a motor starts or a brake grips, the angular velocity changes at a rate called the angular acceleration, α, which is measured in rad/s².
What is the difference between angular velocity and angular acceleration?
Angular velocity says how fast the angle changes, and angular acceleration says how fast the angular velocity changes. The first is in rad/s and the second in rad/s². With a steady α, the angular velocity after a time t follows the equation below.
$$\omega = \omega_0 + \alpha t$$
Acceleration mode in the Angular Velocity Calculator loads with a disc at 27.5 rad/s slowing at 10 rad/s². After 2 s it turns at 27.5 − 10 × 2 = 7.5 rad/s, having swept 35 rad. For the fishing reel in OpenStax section 10.2, a start from rest with α = 110 rad/s² for 2 s gives 220 rad/s.
Run the disc for 4 s and ω reaches −12.5 rad/s. The tool reports that it stopped at 2.75 s and then reversed, and the angle card changes to a net turn of 30 rad. To find α itself from two speeds or from a torque, go to the Angular Acceleration Calculator.
At the exact moment of stopping the answer is ω = 0. The OpenStax propeller slows from 30 rad/s at 6 rad/s² and is at rest after 5 s. The tool shows ω = 0 rad/s with 75 rad turned, and the period card reads No Period While at Rest.
The angular velocity squared formula
When the angle turned is known and the time is not, the equation with angular velocity squared applies. OpenStax lists it with the other equations for constant angular acceleration, where Δθ is the angle turned.
$$\omega^2 = \omega_0^2 + 2 \alpha \Delta \theta$$
No Time mode solves it. The disc at 27.5 rad/s slowing at 10 rad/s² turns at 7.5 rad/s after 35 rad, the same answer as before, and it would stop at 37.813 rad. For the reel, which turns through 220 rad, ω² = 0 + 2 × 110 × 220 = 48,400, so ω is 220 rad/s again.
Angular velocity squared also sets the centripetal acceleration, ω²r, and the rotational kinetic energy, ½Iω², so doubling the spin quadruples both. Angular momentum rises only in step with ω, since L = Iω. A flywheel at twice the speed therefore carries twice the angular momentum and four times the energy, a comparison the Angular Momentum Calculator works through.
Right-Hand Rule and Cross Product
Angular velocity is a vector that points along the axis of rotation, not around it. That direction is a convention set by the right-hand rule, and it lets the link to linear velocity be written as a cross product.
What is the right hand rule for angular velocity?
Curl the fingers of your right hand in the direction the body turns, and your thumb points along ω. For counterclockwise rotation seen from above, ω points up, out of the page, and counts as positive. Clockwise rotation points it into the page and makes it negative.
The Angular Velocity Calculator applies this sign rule. A negative angle change in Angle Difference mode, or a falling angle in Two Positions mode, returns a negative ω, and the Right-Hand Rule Diagram switches its caption to clockwise, ω into page.
The angular velocity cross product
The velocity of any point on a rotating body is the cross product of the angular velocity and the position vector of that point. Its direction is tangent to the circle the point moves on, at right angles to both ω and r.
$$\vec{v} = \vec{\omega} \times \vec{r}$$
The size of this product is ωr sin θ, which becomes ωr when r is measured straight out from the axis. Turned around for circular motion, ω = (r × v) ÷ r², and its magnitude is the v ÷ r that Linear Velocity mode computes.
Cross Product mode in the Angular Velocity Calculator does this with components. It loads with r = (3, 4, 0) m and v = (−12, 9, 0) m/s. These give r × v = (0, 0, 75) m²/s and r² = 25 m², so ω = (0, 0, 3) rad/s, pointing along +z.
Change v to (−9, 13, 0) m/s and ω stays at 3 rad/s. The speed is now 15.811 m/s, but 5 m/s of it lies along r, and that radial velocity adds nothing. Measure r straight out from the axis, since a component of r along the axis would lower the result.
What is the 3 finger right-hand rule?
It is a second way to read any cross product a × b = c. Point your right index finger along the first vector and your middle finger along the second, and your thumb gives the result. For v = ω × r, the index finger goes along ω and the middle finger along r, which leaves the thumb along v.
ω in Waves: v, λ and f
Questions about ω often lead into waves, because a wave has an angular frequency ω = 2πf even though nothing rotates. The wave quantities sit together in one equation from OpenStax section 16.1.
$$v = f \lambda$$
What do v, λ, and f stand for?
v is the wave speed, λ is the wavelength and f is the frequency in hertz. Lambda is just the wavelength, the length of one full cycle of the wave. Frequency is not 1 ÷ λ, since it equals v ÷ λ, and the quantity that equals 1 ÷ f is the period T.
When frequency increases, wavelength decreases, as long as the wave speed stays the same. To convert Hz to nm for light, divide the speed of light by the frequency. A 5 × 10¹⁴ Hz wave has λ = 299,792,458 ÷ (5 × 10¹⁴) = 5.9958 × 10⁻⁷ m, which is 599.58 nm.
The seven types of electromagnetic waves, in order from longest wavelength to shortest, are radio waves, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays, as NASA lists them. Frequency and ω rise in that same order.