RPM To Angular Velocity
A wheel that turns at 60 RPM has an angular velocity of 6.283 radians per second, because it makes one turn each second.
Quick answer: Multiply the RPM by 2 pi and divide by 60. For example, 60 RPM is 6.283 rad/s, 600 RPM is 62.832 rad/s and 6,000 RPM is 628.32 rad/s.
The RPM to angular velocity formula
Angular velocity omega equals RPM times 2 pi divided by 60.
One turn each second is 60 RPM and 2 pi rad/s.
Take 60 RPM. 60 × 0.1047198 gives 6.283 rad/s.
Take 600 RPM. 600 × 0.1047198 gives 62.832 rad/s.
Angular velocity in plain words
Angular velocity is how fast an angle changes. One turn is 2 pi radians, so a speed in RPM becomes omega equals 2 pi times N divided by 60 (Lucidar: rpm to m/s converter).
Omega is the number that links turning speed to linear speed and to frequency.
How to convert RPM to angular velocity step by step
- Read the rotation rate in RPM.
- Multiply it by 2 pi, about 6.2832.
- Divide the result by 60.
- Write rad/s after the answer.
Example 1: 60 RPM
- Start with 60 RPM.
- 60 × 0.1047198 = 6.283.
- The angular velocity is 6.283 rad/s.
60 RPM = 6.283 rad/s
Example 2: 600 RPM
- Start with 600 RPM.
- 600 × 0.1047198 = 62.832.
- The angular velocity is 62.832 rad/s.
600 RPM = 62.832 rad/s
Example 3: 6,000 RPM
- Start with 6,000 RPM.
- 6,000 × 0.1047198 = 628.32.
- The angular velocity is 628.32 rad/s.
6,000 RPM = 628.32 rad/s
Example 4: 300 RPM
- Start with 300 RPM.
- 300 × 0.1047198 = 31.416.
- The angular velocity is 31.416 rad/s.
300 RPM = 31.416 rad/s
Example 5: 3,000 RPM
- Start with 3,000 RPM.
- 3,000 × 0.1047198 = 314.16.
- The angular velocity is 314.16 rad/s.
3,000 RPM = 314.16 rad/s
Example 6: 120 RPM
- Start with 120 RPM.
- 120 × 0.1047198 = 12.566.
- The angular velocity is 12.566 rad/s.
120 RPM = 12.566 rad/s
Round only at the end. Rounding in the middle of a sum adds small errors that grow.
RPM to angular velocity conversion table
| Revolutions per minute (RPM) | Angular velocity (rad/s) | Revolutions per second (rev/s) |
|---|---|---|
| 6 RPM | 0.6283 rad/s | 0.1 Revolutions per second (rev/s) |
| 15 RPM | 1.571 rad/s | 0.25 Revolutions per second (rev/s) |
| 30 RPM | 3.142 rad/s | 0.5 Revolutions per second (rev/s) |
| 60 RPM | 6.283 rad/s | 1 Revolutions per second (rev/s) |
| 120 RPM | 12.566 rad/s | 2 Revolutions per second (rev/s) |
| 300 RPM | 31.416 rad/s | 5 Revolutions per second (rev/s) |
| 600 RPM | 62.832 rad/s | 10 Revolutions per second (rev/s) |
| 1,200 RPM | 125.66 rad/s | 20 Revolutions per second (rev/s) |
| 3,000 RPM | 314.16 rad/s | 50 Revolutions per second (rev/s) |
| 6,000 RPM | 628.32 rad/s | 100 Revolutions per second (rev/s) |
| 15,000 RPM | 1,570.8 rad/s | 250 Revolutions per second (rev/s) |
| 30,000 RPM | 3,141.59 rad/s | 500 Revolutions per second (rev/s) |
The highlighted row is the default of the calculator. The last column gives Revolutions per second (rev/s).
A quick mental shortcut for RPM to angular velocity
One RPM is a little over a tenth of a rad/s. So 100 RPM is a little over 10 rad/s.
The exact value is 10.47 rad/s.
From omega to speed at a radius
Multiply omega by the radius in metres to get the linear speed at that radius. A point 0.5 m from the axis of a 60 RPM wheel moves at about 3.14 m/s.
This is the same as the rim speed.
Omega and frequency
Frequency in hertz is omega divided by 2 pi. So 60 RPM is 1 Hz.
The last column gives the speed in revolutions per second, which is the same number.
Sign and direction
Angular velocity can be positive or negative to show direction. This page gives the size of the speed only.
Choose a sign rule, such as counter clockwise being positive.
Quick conversion answers
| Conversion | Result |
|---|---|
| 60 RPM | 6.283 rad/s |
| 600 RPM | 62.832 rad/s |
| 6,000 RPM | 628.32 rad/s |
| 60,000 RPM | 6,283.19 rad/s |
Where this conversion is used
Physics students convert wheel speeds.
Engineers prepare motor speeds for equations.
Common mistakes to avoid
- Forgetting to divide by 60. The result will be 60 times too big.
- Using degrees. Omega in rad/s needs radians.
- Mixing omega and RPM in a formula. Use one unit.
- Ignoring direction. The sign shows the turning sense.
Related converters
More values at a glance
1.2 RPM is 0.1257 rad/s. 18 RPM is 1.885 rad/s. 45 RPM is 4.712 rad/s.
90 RPM is 9.425 rad/s. 180 RPM is 18.85 rad/s. 240 RPM is 25.13 rad/s.
900 RPM is 94.25 rad/s. 1,500 RPM is 157.1 rad/s. 12,000 RPM is 1,257 rad/s.
Common rotation rates
1 RPM is 0.1047 rad/s. 10 RPM is 1.047 rad/s. 30 RPM is 3.142 rad/s.
45 RPM is 4.712 rad/s. 78 RPM is 8.168 rad/s. 120 RPM is 12.57 rad/s.
500 RPM is 52.36 rad/s. 2,000 RPM is 209.4 rad/s. 5,000 RPM is 523.6 rad/s.
More worked values
1 RPM is 0.10472 rad/s. 2 RPM is 0.20944 rad/s. 4 RPM is 0.418879 rad/s.
8 RPM is 0.837758 rad/s. 15 RPM is 1.570796 rad/s. 25 RPM is 2.617994 rad/s.
40 RPM is 4.18879 rad/s. 75 RPM is 7.853982 rad/s. 100 RPM is 10.471976 rad/s.
Going the other way
To turn rad/s back into RPM, do the opposite step.
1 rad/s is 9.549297 RPM. 5 rad/s is 47.746483 RPM. 10 rad/s is 95.492966 RPM.
20 rad/s is 190.985932 RPM. 50 rad/s is 477.464829 RPM. 100 rad/s is 954.929659 RPM.
250 rad/s is 2,387.324146 RPM. 500 rad/s is 4,774.648293 RPM.
A full worked example
Take 7.5 RPM. Multiply by 0.10472 to get 0.785398 rad/s.
Now check it. Turn 0.785398 rad/s back into RPM and you should land on 7.5 again.
If you do not, look for a slipped decimal point or a swapped unit.
Rounding the answer
Keep about as many digits as your input has. For example, 12.3456 RPM is 1.292828 rad/s, which is 1.29 rad/s to 2 decimals.
Round only at the end. Rounding early can add a small error to the final number.
Small and large values
0.1 RPM is 0.010472 rad/s. 0.25 RPM is 0.02618 rad/s. 0.5 RPM is 0.05236 rad/s.
3 RPM is 0.314159 rad/s. 6 RPM is 0.628319 rad/s. 12 RPM is 1.256637 rad/s.
30 RPM is 3.141593 rad/s. 60 RPM is 6.283185 rad/s. 150 RPM is 15.707963 rad/s.
200 RPM is 20.943951 rad/s. 300 RPM is 31.415927 rad/s. 400 RPM is 41.887902 rad/s.
More values in the reverse direction
2 rad/s is 19.098593 RPM. 3 rad/s is 28.64789 RPM. 25 rad/s is 238.732415 RPM.
30 rad/s is 286.478898 RPM. 40 rad/s is 381.971863 RPM. 60 rad/s is 572.957795 RPM.
75 rad/s is 716.197244 RPM. 200 rad/s is 1,909.859317 RPM. 400 rad/s is 3,819.718634 RPM.
1,000 rad/s is 9,549.296586 RPM.
Frequently asked questions
How many Angular velocity is 1 RPM?
1 RPM is 0.1047198 rad/s.
How many Angular velocity is 600 RPM?
600 RPM is 62.8319 rad/s.
How many Revolutions per minute is 1 rad/s?
1 rad/s is 9.549297 RPM.
What is the angular velocity of 120 RPM?
About 12.566 rad/s.
How do I go from omega to RPM?
Multiply omega by 60 and divide by 2 pi.
What is 60 RPM in Hz?
1 Hz, because it is one turn each second.
Sources
- Lucidar: rpm to m/s converter (lucidar.me)
