One revolution is 2 pi radians, so revolutions per second convert to radians per second by multiplying by 6.283185. One rev/s is 6.2832 rad/s and also 60 RPM. The table below runs from 0.1 to 10 rev/s with all three columns, which covers the range where a rotating assembly is usually described in revolutions rather than in thousands of RPM.
Why the factor is 2 pi
A radian is the angle that subtends an arc equal to the circle’s own radius, and a full circle contains 2 pi of them. So a revolution is not a unit in its own right so much as a count of complete turns, and converting it to radians is just multiplying by how many radians a turn contains. There is no measurement error in the factor; the only inexactness is that pi has to be truncated somewhere.
Radians per second is what appears in any calculation involving torque and power, angular momentum or centripetal force, because those formulae are written in SI and SI counts angle in radians. Revolutions per second and RPM are what appear on a nameplate, because they are what a person can count. Converting between them is the first step in most rotating-machinery arithmetic and the easiest one to fumble, usually by dropping the 2.
Revolutions per second to radians per second
| rev/s | rad/s | RPM |
|---|---|---|
| 0.1 | 0.6283 | 6 |
| 0.2 | 1.2566 | 12 |
| 0.3 | 1.8850 | 18 |
| 0.4 | 2.5133 | 24 |
| 0.5 | 3.1416 | 30 |
| 0.6 | 3.7699 | 36 |
| 0.7 | 4.3982 | 42 |
| 0.8 | 5.0265 | 48 |
| 0.9 | 5.6549 | 54 |
| 1 | 6.2832 | 60 |
| 1.5 | 9.4248 | 90 |
| 2 | 12.5664 | 120 |
| 2.5 | 15.7080 | 150 |
| 3 | 18.8496 | 180 |
| 3.5 | 21.9911 | 210 |
| 4 | 25.1327 | 240 |
| 4.5 | 28.2743 | 270 |
| 5 | 31.4159 | 300 |
| 5.5 | 34.5575 | 330 |
| 6 | 37.6991 | 360 |
| 6.5 | 40.8407 | 390 |
| 7 | 43.9823 | 420 |
| 7.5 | 47.1239 | 450 |
| 8 | 50.2655 | 480 |
| 8.5 | 53.4071 | 510 |
| 9 | 56.5487 | 540 |
| 9.5 | 59.6903 | 570 |
| 10 | 62.8319 | 600 |
Method and precision
The factor is 2 pi, 6.283185307179586 to the precision carried here, and it is exact up to the truncation of pi itself. Radians per second are rounded to four decimal places and RPM to one, for display only; the converter above uses the untruncated constant. RPM is revolutions per second multiplied by exactly sixty.
Getting the factor of two right
The commonest error in this conversion is using pi instead of 2 pi, which halves the answer and is easy to miss because the result still looks like a plausible number. A quick sanity check: one revolution per second must come out at a little over six radians per second, because a circle is a little over six radii around. If your figure is a little over three, the 2 went missing.
The other trap is mixing the time base. Rotational speed on a nameplate is per minute; angular velocity in a formula is per second. Convert the units and the time base as separate steps rather than trying to do both in one multiplication, and check the result against the RPM column above.
The minute-based version of this conversion is in the RPM to hertz chart. Measuring equipment is in measuring equipment and rotating tools are in air tools.