Angular Velocity Converter (RPM to rad/s)
Convert angular velocity between RPM, rad/s, deg/s, rad/min and more. Enter a value in any unit and instantly get the equivalent in all other common angular speed units.
Convert Angular Velocity Units
Angular Velocity Formula
Where:
- \(\omega\) — angular velocity in radians per second (rad/s)
- \(N\) — rotational speed in revolutions per minute (RPM)
- \(2\pi\) — radians in one full revolution (a circle has \(360° = 2\pi \approx 6.2832\) radians)
- \(60\) — seconds in one minute
The formula converts RPM to rad/s by multiplying the number of revolutions per minute by the radians in each revolution and dividing by the seconds in a minute. Rad/s is the SI unit for angular velocity, which is why all other units in this converter are first changed into rad/s and then back out into the target unit.
The reverse conversion is just as simple:
\[ N_{\text{RPM}} = \omega_{\text{rad/s}} \times \frac{60}{2\pi} \]
Other useful relationships:
- \(1\ \text{RPM} = 0.104720\ \text{rad/s}\)
- \(1\ \text{rad/s} = 9.549297\ \text{RPM}\)
- \(1\ \text{deg/s} = 0.0174533\ \text{rad/s}\)
- \(1\ \text{RPM} = 6\ \text{deg/s}\)
Angular Velocity – Calculation Example
Example: an electric motor runs at 1750 RPM, a typical nameplate speed. What is its angular velocity in rad/s?
\[ \omega = 1750 \times \frac{2\pi}{60} = 1750 \times 0.10472 \approx 183.259571\ \text{rad/s} \]
Result: 1750 RPM ≈ 183.26 rad/s.
The same speed expressed in the other common units:
- 29.1667 rev/s (Hz)
- 10 995.57 rad/min
- 10 500 deg/s
- 630 000 deg/min
How to use the calculator
- Enter the angular speed \(\omega\) in the Angular speed (ω) field — for example the default value of 1750, a typical motor speed.
- Choose the unit of that value from the Unit dropdown: RPM (rev/min), rad/s, deg/s or rad/min.
- Read the result table Converted Angular Velocities, which shows the same angular velocity in all other common units, including rad/h, rev/s (Hz), deg/s, deg/min and deg/h.
What is angular velocity?
Angular velocity describes how fast an object rotates or revolves around an axis. While linear velocity is measured in distance per unit of time (m/s, mph), angular velocity is measured in angle per unit of time — most often radians per second (rad/s), the SI unit, or revolutions per minute (RPM), the unit commonly printed on motor nameplates, lathe spec sheets and fan curves.
One full revolution equals \(2\pi\) radians, and a minute contains 60 seconds. That is the entire idea behind the conversion: 1 RPM means the object sweeps \(2\pi\) radians every 60 seconds, or about 0.10472 rad/s. Going the other way, 1 rad/s corresponds to roughly 9.55 RPM — a number worth remembering when you move between engineering documents that mix the two conventions.
Angular velocity appears everywhere something spins: shafts, wheels, turbines, hard drives, centrifuges and orbiting satellites. It links directly to linear speed at the rim through \(v = \omega r\), where \(r\) is the radius, and to frequency through \(f = \omega / 2\pi\) (so rad/s divided by \(2\pi\) gives revolutions per second, i.e. hertz). If you also work with rotational power, combining angular velocity with torque via \(P = \tau\omega\) gives mechanical power in watts — see our Torque Converter for the other half of that calculation.
Interpreting the result
The calculator first converts your input into rad/s, then expresses it in every other unit so you can compare conventions at a glance. Note that rev/s (Hz) and RPM differ only by a factor of 60 — they measure the same thing on different time scales. Similarly, deg/s and rad/s differ by \(180/\pi \approx 57.2958\), because one radian is about 57.3 degrees.
Large values in deg/min or deg/h can look surprising: the same 1750 RPM motor turns through 630 000 degrees per minute. That is not an error — degrees are simply a much smaller unit than revolutions. For most physics and engineering formulas, stay in rad/s; radians are the "natural" unit in which formulas like \(v = \omega r\) work without extra correction factors.
Tips, assumptions and limitations
- The calculator accepts only non-negative values. If you need to indicate rotation direction, keep the magnitude here and note the direction (clockwise or counterclockwise) separately; angular velocity as a vector carries a sign, but unit conversion only concerns the magnitude.
- Results are rounded to 6 decimal places. For very large inputs (tens of thousands of RPM or more), the trailing digits in deg/h may lose visible precision — keep more significant figures in your own work if that matters.
- The conversion assumes a constant angular speed. Real motors under varying load speed up and slow down; if you need average angular velocity, convert the average RPM, not instantaneous peaks.
- Watch out for the similar-looking unit rad/min: it is 60 times smaller than rad/s, and it is not the same as RPM (1 RPM ≈ 6.2832 rad/min).
- If your rotational data eventually turns into a surface or cutting speed, pair this tool with our Speed Converter for the linear side, or the Frequency Converter when working with cycles per second.
Frequently Asked Questions
How do I convert RPM to rad/s by hand without a calculator?
Multiply the RPM value by 2π and divide by 60. A quick mental shortcut: multiply RPM by 0.1047 (which is 2π/60 rounded). For example, 3000 RPM × 0.1047 ≈ 314.16 rad/s. To go the other way, multiply rad/s by 9.5493 to get RPM. Both shortcuts stay accurate to well within 0.01% for everyday engineering work.
Why do we divide by 60 when converting RPM to rad/s?
Because the units measure time differently. One revolution equals 2π radians, but RPM counts revolutions per minute while rad/s counts radians per second. The 60 converts minutes to seconds. A common mistake is multiplying by 2π alone and forgetting the 60 — that gives you rad/min, which is 60 times too large.
Is rad/s the same as hertz (Hz)?
No, they differ by a factor of 2π. Rad/s measures angular velocity, while hertz measures frequency (cycles per second). Convert with \( f = \omega / 2\pi \), so 1 rad/s ≈ 0.159 Hz. Only for very small angles would the two nearly coincide. RPM relates to both: \( 1\ \text{RPM} = 1/60\ \text{Hz} \approx 0.10472\ \text{rad/s} \).
What is a typical angular velocity for common machines?
A 1750 RPM induction motor (typical under load) is about 183 rad/s; a 3600 RPM motor is roughly 377 rad/s. A car engine at 2500 RPM runs near 262 rad/s. A hard drive at 7200 RPM spins at about 754 rad/s, while a ceiling fan at 200 RPM is only ~21 rad/s. Everyday machines span a huge range, so there is no single 'normal' value.
Can angular velocity be negative, and how do I handle rotation direction?
Yes — as a vector quantity, angular velocity carries a sign indicating direction (by convention, counterclockwise is positive). This converter works with magnitudes only, so enter the absolute value and track the direction separately. Direction matters in physics formulas involving torque or angular momentum, but it has no effect on unit conversion.
How do I convert RPM to degrees per second?
One revolution is 360° and one minute is 60 seconds, so multiply RPM by 6 to get deg/s — the 60 from the revolution and the 60 from the minute cancel neatly. For example, 1750 RPM = 10 500 deg/s. To convert deg/s to rad/s, multiply by \(\pi/180 \approx 0.01745\), since one degree is exactly that fraction of a radian.
I see 'rad/min' on a datasheet — is that just another way of writing RPM?
No, and mixing them up is a serious error. Rad/min is radians per minute: 1 rad/s = 60 rad/min, so 1 rad/min ≈ 0.01667 rad/s. Meanwhile 1 RPM ≈ 6.2832 rad/min because each revolution sweeps 2π radians. The two units differ by a factor of about 6.28, so always check the datasheet's notation carefully.