Angular Velocity Calculator

What the angular velocity calculator does

Angular velocity is how fast something spins, measured as the angle it sweeps through each second. This calculator finds it two ways: from the angle turned in a given time, or from the linear speed of a point and how far that point sits from the centre. It also works the other way, finding the linear speed or the radius when you know the angular velocity.

Below is what angular velocity means, the equations behind it, how it connects to ordinary speed, and a worked example.

How to use it

1. Choose what to calculate: angular velocity from an angle and time, angular velocity from a linear speed and radius, or the linear speed or radius from the angular velocity.
2. Enter the values the method asks for, each with its own unit.
3. Press Calculate for the result, or Reset to clear it.

What angular velocity is

Angular velocity is the rotational cousin of ordinary velocity. Where ordinary velocity measures how fast you move along a line, angular velocity measures how fast you turn through an angle. It is written with the Greek letter omega, and its natural unit is radians per second, a radian being the angle that wraps one radius-length around the circle.

It is the right way to describe anything that spins as a whole: a wheel, a turntable, a planet on its axis. Every point on a spinning object shares the same angular velocity, since the whole thing turns through the same angle in the same time, even though points near the rim travel much farther than points near the centre. That shared turning rate is what angular velocity captures.

The equations it uses

The most direct definition is the angle θ turned divided by the time t it took:

ω = θ ÷ t

The other route uses the linear speed v of a point and its distance r from the centre, since a point farther out must move faster to keep up with the same rotation:

ω = v ÷ r

That second relationship rearranges to give the linear speed from the angular velocity, v = ωr, and the radius from both, r = v ÷ ω, which are the calculator's other two modes.

The link to linear speed

The equation v = ωr is the bridge between spinning and moving, and it carries a fact you can feel. Because the linear speed is the angular velocity times the radius, a point twice as far from the centre moves twice as fast, even though both share the same rate of turn. On a spinning playground roundabout, the rider at the edge is flung along far quicker than one near the middle.

This is why the outer edge of a wheel meets the road at the wheel's full travelling speed while the axle barely moves sideways, and why the tip of a propeller can approach the speed of sound while its hub turns gently. The same turning rate produces wildly different linear speeds depending only on how far out you stand.

Units and precision

The natural SI unit of angular velocity is the radian per second, and the calculator reports it there, alongside degrees per second for convenience. You can enter the angle in radians or degrees, time in seconds, minutes, or hours, and the linear speed and radius in their own units, with the calculator converting internally. Results are carried to several decimal places, finer than most measurements of rotation require.

A worked example: a rolling wheel

Take a car wheel of radius 0.3 metres, on a car travelling at 30 metres per second.

The rim of the wheel meets the road at the car's speed, so the angular velocity is ω = v ÷ r = 30 ÷ 0.3 = 100 rad/s. That works out to about 955 revolutions per minute. Turn it around with v = ωr and you recover the rim speed: 100 × 0.3 = 30 m/s, the speed the car is moving.

Questions people ask

What is angular velocity?

It is how fast something rotates, the angle it turns through per unit time, usually in radians per second. Every point on a rigid spinning object has the same angular velocity.

How do you calculate angular velocity?

Divide the angle turned by the time, ω = θ/t. If you know a point's linear speed and its distance from the centre, you can also use ω = v/r.

What is the difference between angular and linear velocity?

Angular velocity is the rate of turning, shared by the whole object. Linear velocity is how fast a particular point moves along its path, and it grows with distance from the centre, linked by v = ωr.

How does this relate to RPM?

RPM is revolutions per minute, another measure of rotation. One revolution is 2π radians, so radians per second can be converted to RPM by multiplying by 60 and dividing by 2π.

References

A quick note on where the physics comes from. Angular velocity as the rate of rotation, and its link to linear speed through v = ωr, are standard rotational mechanics, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. The radian per second and the other SI units follow the US National Institute of Standards and Technology.

1. OpenStax, University Physics Volume 1, Section 10.1, Rotational Variables. https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables
2. HyperPhysics, Georgia State University, Rotational Quantities. http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html
3. National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811