Angular Velocity Calculator

The formula

Angular velocity (the Greek letter omega, ω) relates the linear speed of a point to the radius of the circle it travels on:

ω = v / r

From the angular velocity you can derive two everyday quantities, the rotational frequency f and the speed in revolutions per minute (RPM):

f = ω / (2π)

RPM = ω · 60 / (2π)

Where:

• ω = angular velocity, in radians per second (rad/s).
• v = linear (tangential) velocity, in metres per second (m/s).
• r = radius of the circular path, in metres (m).
• f = rotational frequency, in hertz (Hz), i.e. revolutions per second.
• 2π ≈ 6.2832 radians, the number of radians in one full turn.

One full revolution equals 2π radians, which is why frequency and RPM both divide ω by 2π.

Worked example

A wheel has a radius of 0.3 m and the rim moves at a linear speed of 6 m/s. The angular velocity is:

ω = v / r = 6 / 0.3 = 20 rad/s

Convert to frequency and RPM:

f = 20 / (2π) ≈ 3.18 Hz

RPM = 20 · 60 / (2π) ≈ 190.99 RPM

So the wheel turns almost 191 times every minute. If the radius shrank to 0.15 m at the same rim speed, ω would double to 40 rad/s, a smaller circle sweeps the same arc length in less angle, so it spins faster.

Quick reference

Common pitfalls

• Radius cannot be zero. ω = v / r is undefined when r = 0, so the calculator returns 0 until you enter a radius greater than zero.
• Radians, not degrees. Angular velocity here is in radians per second. To convert to degrees per second, multiply by 180/π ≈ 57.2958.
• Keep SI units. Velocity in m/s and radius in m give ω directly in rad/s. Convert km/h, RPM or centimetres first.
• ω vs. v. Linear velocity (v) is how fast the point moves along its arc; angular velocity (ω) is how fast the angle changes. They are linked by the radius: v = ω · r.