RPM, rad/s and deg/s: three ways to measure rotation
What revolutions per minute, radians per second and degrees per second each measure, how they convert, and when to use which.
Three units, one quantity
Angular velocity describes how fast something turns, and it can be written three ways. Revolutions per minute counts whole turns in a minute and is common for engines, fans and record players. Radians per second is the physicist's unit, because the equations of rotational motion, energy and centripetal force all expect radians. Degrees per second uses the familiar 360 degree circle and is handy for describing how quickly an angle sweeps. All three measure the same spinning, just against different yardsticks for the angle and the time.
How the conversions work
The bridge between the units is the definition of a radian. One full revolution equals 2 x pi radians and also equals 360 degrees. To go from RPM to radians per second, multiply by 2 x pi to convert turns to radians, then divide by 60 to turn minutes into seconds. To reach degrees per second from radians per second, multiply by 180 and divide by pi. Because pi is irrational, conversions that involve it never land on tidy decimals, which is why 60 RPM becomes 6.283185307 rad/s rather than a round number.
From spin to linear speed
Angular velocity on its own does not tell you how fast a point on the object is moving through space. That linear or tangential speed depends on the radius, the distance from the axis to the point. Multiply the angular velocity in radians per second by the radius in metres to get tangential speed in metres per second. The rim of a large wheel moves faster than a point near its hub even though both share the same RPM, which is why the outer edge of a grinding wheel or a merry-go-round feels so much quicker.
Choosing the right unit for the job
Use RPM when reading a spec sheet or a tachometer, since machinery is almost always rated that way. Switch to radians per second the moment you plug into a physics formula, because mixing degrees or revolutions into those equations gives wrong answers. Degrees per second is most natural for animation, gimbals and anything where you think in terms of a sweeping angle. Converting freely between the three, as this calculator does, lets you keep each part of a problem in its most convenient unit.