Angular Velocity Calculator
Angular velocity measures how fast something rotates, expressed as the angle swept per unit of time in radians per second. It is the rotational equivalent of ordinary linear velocity and is central to any problem involving spinning objects: electric motors, wheels, fans, turbines, gears, and circular orbits. The standard formula is omega = 2πf, where f is frequency in hertz (revolutions per second), so one full revolution of 2π radians per second gives an angular velocity of 2π rad/s. Many practical devices are rated in revolutions per minute rather than hertz, so this calculator accepts both: enter frequency in Hz or RPM, and it converts automatically. The default example uses f = 50 Hz, the frequency of the AC mains supply in New Zealand, which gives omega = 2π times 50 = 314.16 rad/s. That matches a motor shaft speed of 3,000 RPM. You can also enter a radius to get the linear speed of a point at that distance from the axis, since v = omega times r. This is useful for working out the tip speed of a fan blade, the surface speed of a grinding wheel, or the speed at the rim of a flywheel. Results update as you type. All three outputs, rad/s, degrees per second, and RPM, are shown simultaneously, making it easy to cross-check between the different units that appear in textbooks and engineering specifications. Use this calculator for physics homework, electric motor analysis, or any rotating machinery problem.
omega = 2pi x f. From RPM: omega = RPM x 2pi / 60. Linear speed v = omega x r. Rounded for display.
How it works
From frequency f in Hz: ω = 2πf. From RPM: ω = RPM x 2π / 60. Converting to degrees per second: ω x 180 / π. Converting back to RPM: ω x 60 / (2π). Linear speed at radius r: v = ω r in metres per second.
Worked example
Mains frequency f = 50 Hz. Angular velocity = 2π times 50 = 314.16 rad/s. In RPM: 314.16 times 60 / (2π) = 3,000 RPM. In degrees per second: 314.16 times 180 / π = 18,000 deg/s. These match the default values pre-filled above. If a radius were entered (e.g. 0.1 m), linear speed would be 314.16 times 0.1 = 31.42 m/s.
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