Rpm To Rads
A shaft turning at 1,000 RPM moves at about 104.7 radians per second.
Quick answer: Multiply the RPM by 0.10472, which is 2 pi divided by 60. For example, 1,000 RPM is 104.72 rad/s, 10,000 RPM is 1,047.2 rad/s and 100,000 RPM is 10,471.98 rad/s.
The RPM to rad/s formula
Angular speed in rad/s equals RPM times 2 pi divided by 60.
The factor 2 pi divided by 60 is about 0.10472.
Take 1,000 RPM. 1,000 × 0.1047198 gives 104.72 rad/s.
Take 10,000 RPM. 10,000 × 0.1047198 gives 1,047.2 rad/s.
Why 2 pi over 60
One turn is 2 pi radians and one minute is 60 seconds. So each RPM equals 2 pi divided by 60 radians per second (Lucidar: rpm to m/s converter).
That gives the factor of about 0.10472.
How to convert RPM to rad/s step by step
- Read the speed in RPM.
- Multiply it by 0.10472.
- Round to the digits you need.
- Write rad/s after the answer.
Example 1: 1,000 RPM
- Start with 1,000 RPM.
- 1,000 × 0.1047198 = 104.72.
- The angular speed is 104.72 rad/s.
1,000 RPM = 104.72 rad/s
Example 2: 10,000 RPM
- Start with 10,000 RPM.
- 10,000 × 0.1047198 = 1,047.2.
- The angular speed is 1,047.2 rad/s.
10,000 RPM = 1,047.2 rad/s
Example 3: 100,000 RPM
- Start with 100,000 RPM.
- 100,000 × 0.1047198 = 10,471.98.
- The angular speed is 10,471.98 rad/s.
100,000 RPM = 10,471.98 rad/s
Example 4: 5,000 RPM
- Start with 5,000 RPM.
- 5,000 × 0.1047198 = 523.6.
- The angular speed is 523.6 rad/s.
5,000 RPM = 523.6 rad/s
Example 5: 50,000 RPM
- Start with 50,000 RPM.
- 50,000 × 0.1047198 = 5,235.99.
- The angular speed is 5,235.99 rad/s.
50,000 RPM = 5,235.99 rad/s
Example 6: 2,000 RPM
- Start with 2,000 RPM.
- 2,000 × 0.1047198 = 209.44.
- The angular speed is 209.44 rad/s.
2,000 RPM = 209.44 rad/s
Round only at the end. Rounding in the middle of a sum adds small errors that grow.
RPM to rad/s conversion table
| Revolutions per minute (RPM) | Radians per second (rad/s) | Degrees per second (deg/s) |
|---|---|---|
| 100 RPM | 10.472 rad/s | 600 Degrees per second (deg/s) |
| 250 RPM | 26.18 rad/s | 1,500 Degrees per second (deg/s) |
| 500 RPM | 52.36 rad/s | 3,000 Degrees per second (deg/s) |
| 1,000 RPM | 104.72 rad/s | 6,000 Degrees per second (deg/s) |
| 2,000 RPM | 209.44 rad/s | 12,000 Degrees per second (deg/s) |
| 5,000 RPM | 523.6 rad/s | 30,000 Degrees per second (deg/s) |
| 10,000 RPM | 1,047.2 rad/s | 60,000 Degrees per second (deg/s) |
| 20,000 RPM | 2,094.4 rad/s | 120,000 Degrees per second (deg/s) |
| 50,000 RPM | 5,235.99 rad/s | 300,000 Degrees per second (deg/s) |
| 100,000 RPM | 10,471.98 rad/s | 600,000 Degrees per second (deg/s) |
| 250,000 RPM | 26,179.94 rad/s | 1,500,000 Degrees per second (deg/s) |
| 500,000 RPM | 52,359.88 rad/s | 3,000,000 Degrees per second (deg/s) |
The highlighted row is the default of the calculator. The last column gives Degrees per second (deg/s).
A quick mental shortcut for RPM to rad/s
Multiply by 0.1 for a rough answer, then add 5 percent. 600 RPM is about 60 plus 3, which is 63 rad/s.
The exact value is 62.83 rad/s.
Motors and fans
A fan at 1,500 RPM turns at about 157 rad/s. A hard drive at 7,200 RPM turns at about 754 rad/s.
Radians per second is the unit that physics formulas expect.
Formulas that need rad/s
Linear speed at a radius is angular speed times radius. Power is torque times angular speed.
Both formulas give wrong answers if you use RPM directly.
Degrees per second
One RPM is 6 degrees per second, because 360 degrees in 60 seconds is 6. The last column shows this.
Use it when you think in degrees instead of radians.
Quick conversion answers
| Conversion | Result |
|---|---|
| 1,000 RPM | 104.72 rad/s |
| 10,000 RPM | 1,047.2 rad/s |
| 100,000 RPM | 10,471.98 rad/s |
| 1,000,000 RPM | 104,719.76 rad/s |
Where this conversion is used
Engineers feed motor speeds into physics formulas.
Students link RPM and angular velocity.
Common mistakes to avoid
- Using RPM in v = omega r. Convert to rad/s first.
- Dividing instead of multiplying. rad/s is the smaller number.
- Using 2 pi alone. You also need to divide by 60.
- Mixing rad/s and Hz. One Hz is 2 pi rad/s.
Related converters
More values at a glance
20 RPM is 2.094 rad/s. 300 RPM is 31.42 rad/s. 750 RPM is 78.54 rad/s.
1,500 RPM is 157.1 rad/s. 3,000 RPM is 314.2 rad/s. 4,000 RPM is 418.9 rad/s.
15,000 RPM is 1,571 rad/s. 25,000 RPM is 2,618 rad/s. 200,000 RPM is 20,944 rad/s.
Common motor and drive speeds
60 RPM is 6.283 rad/s. 300 RPM is 31.42 rad/s. 900 RPM is 94.25 rad/s.
1,450 RPM is 151.8 rad/s. 1,800 RPM is 188.5 rad/s. 3,000 RPM is 314.2 rad/s.
3,600 RPM is 377 rad/s. 7,200 RPM is 754 rad/s. 15,000 RPM is 1,571 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 Radians per second is 1 RPM?
1 RPM is 0.1047198 rad/s.
How many Radians per second is 10,000 RPM?
10,000 RPM is 1,047.2 rad/s.
How many Revolutions per minute is 1 rad/s?
1 rad/s is 9.549297 RPM.
How many rad/s is 3,000 RPM?
3,000 RPM is about 314.16 rad/s.
How do I turn rad/s into RPM?
Multiply the rad/s by 9.5493.
Why does the factor include 60?
There are 60 seconds in a minute.
Sources
- Lucidar: rpm to m/s converter (lucidar.me)
