Angular Velocity Calculator
Angular velocity in rad/s, rpm and degrees per second, from angle and time, linear speed and radius, or period and frequency.
🔄 What is Angular Velocity?
Angular velocity is the rate at which an object turns, measured as the angle it sweeps per unit time. Its symbol is ω (omega) and its SI unit is the radian per second. A radian is the angle at which the arc length equals the radius, so one full turn is 2π radians, or about 6.283 rad. Angular velocity is the rotational counterpart of ordinary velocity: velocity tells you how quickly a position changes, angular velocity tells you how quickly an orientation changes. For rotation about a fixed axis it can be treated as a signed number, positive for one sense of rotation and negative for the other.
People quote spin rates in many units, which is why a converter is useful. Engines, drills, fans and hard drives are rated in revolutions per minute. Physics problems use radians per second. Pointing mechanisms and robots often use degrees per second, and signal and electrical work uses frequency in hertz. They all describe the same thing: 1 revolution per second is 1 Hz, 60 rpm, 360 degrees per second and 2π rad/s. This calculator converts between them in one step and also gives the period of one turn, T = 2π / ω.
There are four common ways to arrive at ω, and the calculator offers each. From the angle turned and the time taken, ω = Δθ / Δt. From the linear speed of a point at a known distance, ω = v / r. From a measured period or frequency, ω = 2π / T = 2π f. Or by converting a number you already have. Typical applications include sizing a motor, relating a wheel's rotation to a vehicle speed, checking a rotor or turntable speed, and describing circular motion in a lab. The related angular acceleration calculator covers how quickly ω changes, and the angular momentum calculator uses ω with the moment of inertia.
Two points are easy to mix up. Angular velocity is the same for every point on a rigid rotating body, but the linear speed v = ω r increases with distance from the axis. And the centripetal acceleration v2 / r, shown in the speed and radius mode, points toward the axis and is not the same as angular acceleration. For straight-line speeds, see the velocity calculator.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — 3000 rpm to rad/s
An engine turns at 3000 rpm
Example 2 — 90 Degrees per Second
A radar-style antenna turns at 90 degrees per second
Example 3 — From Angle and Time
A turntable turns through 720 degrees in 4 seconds
Example 4 — From Speed and Radius
A wheel of radius 0.3 m rolls with its rim moving at 25 m/s
Example 5 — From the Period
A fan blade completes one turn every 0.5 seconds, and its tip is 0.2 m from the axis
Example 6 — From the Frequency
A motor shaft turns at 60 Hz, and a point on it is 0.05 m from the axis
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for angular velocity?
Average angular velocity is the angle turned divided by the time, omega = delta theta / delta t, in rad/s. A turntable that turns 720 degrees in 4 s has omega = 12.566 rad / 4 s = 3.142 rad/s.
How do you convert rpm to rad/s?
Multiply by 2 pi / 60, which is about 0.10472. So 3000 rpm is 3000 x 0.10472 = 314.16 rad/s. To go back, multiply rad/s by 60 / (2 pi), about 9.5493.
How do you find angular velocity from linear speed?
Divide the linear speed by the radius, omega = v / r. A point moving at 25 m/s on a circle of radius 0.3 m has omega = 83.33 rad/s, which is 795.77 rpm.
How do you find angular velocity from frequency or period?
Use omega = 2 pi f = 2 pi / T. A 0.5 s period gives omega = 12.566 rad/s, and a 60 Hz frequency gives omega = 376.99 rad/s.
What is the relation between frequency and angular velocity?
Angular velocity in rad/s is 2 pi times the frequency in hertz. One revolution per second is 1 Hz, 6.283 rad/s, 360 degrees per second and 60 rpm.
What is the unit of angular velocity?
The SI unit is radian per second (rad/s). Common alternatives are revolutions per minute (rpm), degrees per second and revolutions per second. The radian is dimensionless, so rad/s is dimensionally the same as 1/s.
Is angular velocity a vector?
Yes. Its direction is along the rotation axis by the right-hand rule. For rotation about a fixed axis it is enough to give a signed number, positive for one sense and negative for the other.
What is the difference between angular velocity and angular speed?
Angular speed is the magnitude of angular velocity. Angular velocity also carries the direction of the axis and the sense of rotation, so it can be negative in a one-axis problem.
How is angular velocity different from linear velocity?
Angular velocity describes how fast the angle changes, and is the same for every point of a rigid body. Linear velocity v = omega r depends on the distance from the axis, so points farther out move faster.
How do I find the period from rpm?
Period is the time for one turn, T = 60 / rpm seconds. At 3000 rpm, T = 60 / 3000 = 0.02 s, and at 90 degrees per second (15 rpm) it is 4 s.
What is the centripetal acceleration for a given angular velocity?
It is omega squared times r, equal to v squared over r. A point at 0.3 m moving at 25 m/s has a centripetal acceleration of 625 / 0.3 = 2083.3 m/s squared, directed toward the axis.