Angular Velocity Calculator
Angular velocity ω measures how fast something rotates, in radians per second. Enter a rotation rate (rpm or Hz), an angle swept over a time interval, or a linear speed at a known radius. The calculator returns ω in several units, the period, the linear rim speed v = ω·r, and, optionally, the rotational kinetic energy and angular momentum.
Formula
Worked example
A wheel spinning at 60 rpm has a frequency of 1 Hz, so ω = 2π·f = 2π × 1 ≈ 6.283 rad/s, with a period of 1 s per turn. A point on its rim 0.5 m from the axle moves at v = ω·r = 6.283 × 0.5 ≈ 3.142 m/s. Conversely, a 3.142 m/s rim speed at r = 0.5 m gives ω = v/r = 6.283 rad/s.
What angular velocity measures
Angular velocity, written with the Greek letter omega (ω), describes how fast an object rotates around an axis. It is defined as the angle swept per unit of time, ω = θ/t, where the angle θ is measured in radians and time t in seconds, giving units of radians per second. Because a radian is a pure ratio of arc length to radius, angular velocity is the same for every point on a rigid rotating body, the hub of a wheel and a point on its rim share one ω even though they trace very different distances. This makes ω the natural language for describing spin, from a record player and a ceiling fan to planets, gears, and turbines, independent of how big the object is.
Four ways in: rpm, hertz, angle over time, or linear speed
Rotation is often quoted in revolutions per minute (rpm) or in hertz (revolutions per second), and both convert cleanly to radians per second because one complete revolution is exactly 2π radians. From frequency the relationship is ω = 2π·f, and from rpm it is ω = 2π·rpm/60. If instead you know the angle swept and the time it took, use ω = θ/t after converting the angle to radians. The fourth route is a reverse solve: when you know the linear speed v of a point and its radius r from the axis, the angular velocity is ω = v/r. This calculator accepts any of these four inputs and reports ω in rad/s, rpm, hertz and degrees per second, along with the period T = 2π/ω, the time for one full turn.
Linear speed, kinetic energy and angular momentum
Angular velocity connects rotation to ordinary straight-line motion through v = ω·r: the linear speed of a point equals its angular velocity times its distance r from the axis. That is why the outer edge of a spinning disk moves faster than a point near the centre even though both share the same ω, and it is the key formula for sizing belts, gears, and the rim speed of grinding wheels or tyres. Turn on the advanced option to go further. Given the moment of inertia I (for a solid disc, I = ½·m·r²), the rotational kinetic energy is KE = ½·I·ω² in joules and the angular momentum is L = I·ω in kg·m²/s. These are the rotational analogues of ½·m·v² and m·v for straight-line motion, and they govern how much energy a flywheel stores and how a spinning object resists changes to its rate of spin.
Common rotation rates in rad/s
| Object | Rate (rpm) | Frequency (Hz) | Angular velocity (rad/s) |
|---|---|---|---|
| Vinyl LP record | 33.3 | 0.56 | 3.49 |
| Clock second hand | 1 | 0.017 | 0.105 |
| Ceiling fan (high) | 300 | 5 | 31.42 |
| Car engine at idle | 800 | 13.3 | 83.78 |
| Hard-drive platter (7200) | 7200 | 120 | 753.98 |
| Earth (one spin per day) | 0.000694 | 1.16e-5 | 7.29e-5 |
Converting everyday rotation rates to angular velocity using ω = 2π·rpm/60.
Frequently asked questions
What is the formula for angular velocity?
Angular velocity is ω = θ/t, the angle swept (in radians) divided by the time taken (in seconds), giving radians per second. For steady rotation it is equivalent to ω = 2π·f, where f is the frequency in hertz, since one full revolution equals 2π radians. If you know a point's linear speed v and its radius r, you can also use ω = v/r.
How do I convert rpm to rad/s?
Multiply the rpm value by 2π and divide by 60: ω = 2π·rpm/60. For example, 60 rpm equals 2π × 60 ÷ 60 = 2π ≈ 6.283 rad/s. To convert back, multiply rad/s by 60 and divide by 2π. This calculator also shows the equivalent frequency in hertz and degrees per second.
How do I find angular velocity from linear speed?
Divide the linear speed v of a point by its distance r from the axis: ω = v/r. For instance, a point moving at 3.142 m/s on a radius of 0.5 m has ω = 3.142 ÷ 0.5 = 6.283 rad/s. Select the "Linear speed and radius" method to solve it this way.
What is the difference between angular velocity and linear velocity?
Angular velocity (ω, in rad/s) is how fast an object rotates and is the same for every point on a rigid body. Linear velocity (v, in m/s) is how fast a particular point moves through space and depends on its distance r from the axis, related by v = ω·r. Points farther from the axis have a higher linear speed for the same ω.
How do I get rotational kinetic energy and angular momentum?
You need the moment of inertia I (in kg·m²). The rotational kinetic energy is KE = ½·I·ω² in joules, and the angular momentum is L = I·ω in kg·m²/s. Toggle the advanced option, enter I, and the calculator reports both alongside ω.