Angular Velocity Converter

1. Pick the From unit your rate uses, for example rpm from a motor plate.
2. Pick the To unit you need. Choose rad/s for physics and engineering formulas.
3. Type the value. The answer, the exact factor, the reverse factor and the time for one turn update at once.

Angular velocity is the rate at which an object turns: the angle it sweeps per unit of time. It is written with the Greek letter omega. NIST SP 811 lists its SI unit as the radian per second, rad/s.

Any angular velocity unit is an angle unit divided by a time unit. The angle can be a radian, a degree or a full revolution. The time can be a second, minute, hour or day. That gives the 12 units in this tool, and every one of them is an exact multiple of rad/s.

Angular velocity is not the same as linear speed. Two points on a spinning disc share one omega, but the outer point moves faster, since v = omega x r. It is also not frequency. The SI brochure, NIST SP 330, keeps the hertz for frequency and the radian per second for angular velocity, and warns that mixing them causes an error of 2 pi.

The converter changes the angle part and the time part separately, using exact definitions, and passes through rad/s.

1. Express the From unit in rad/s. One rpm is 2 pi rad per 60 s, which is pi/30 rad/s.
2. Express the To unit in rad/s. One deg/s is pi/180 rad/s.
3. Divide the two and multiply by your value. rpm to deg/s is (pi/30) / (pi/180) = 6, so 1,500 rpm is 9,000 deg/s. Here pi cancels, and the factor is a whole number.

NIST SP 811 gives the decimal forms of two of these factors directly: 1 rev = 6.283185 rad and 1 rpm = 1.047198 x 10^-1 rad/s.

Motor to SI. A pump motor runs at 1,500 rpm. Multiply by pi/30: 1,500 x 0.1047198 = 157.08 rad/s. One turn takes 0.04 s.

Earth rate for navigation. The IERS gives Earth a mean angular velocity of 7.2921150 x 10^-5 rad/s. Multiply by 180/pi and by 3,600 to get 15.041 deg/h. That is a little more than 15 deg/h because Earth turns once in a sidereal day of about 86,164 s, not 86,400 s. For more on rates in degrees per hour, see the degree per hour converter.

Slow sky motion. The Sun moves about 0.9856 degrees per day along the ecliptic, since Earth orbits in 365.256 days. That is 0.9856 x pi/180 / 86,400 = 1.991 x 10^-7 rad/s. The degree per day converter covers this range in detail.

Turns per day. A body that makes 1 rev/d turns at 2 pi / 86,400 = 7.2722 x 10^-5 rad/s. For orbits counted in turns per day, the revolution per day converter is the single-pair tool.

The factors are fixed by definition, so differences come from the unit you pick and how the source defined it.

Which Day or Hour

This tool uses the 86,400 s day and the 3,600 s hour. NIST SP 811 also lists a sidereal day of about 86,164.09 s and a sidereal hour of about 3,590.170 s. A rate per sidereal day is about 0.27 percent faster in rad/s than the same number per ordinary day.

Radians vs Revolutions

One revolution is 2 pi rad, about 6.2832 rad. Choosing rad/min when the source means rpm makes the answer 6.28 times too small.

Average vs Instant Rate

Planet and satellite rates are usually mean values. Elliptical orbits speed up and slow down, and real motors vary with load.

Rounding Pi

Using 3.14 in place of pi is off by 0.05 percent. That is fine for a sketch, but it grows in chained calculations. The tool keeps full precision.

The three are linked but not interchangeable. Each answers a different question about a turning object.

NIST SP 330 recommends writing angular velocity in rad/s and frequency in Hz, never as a bare s^-1, so readers know which is meant. For cycles per second in hertz, use the frequency converter.

Machines and Drives

Motors, fans, pumps and engines are rated in rpm, but torque and power formulas need rad/s. Power in watts is torque in N m times omega in rad/s.

Sensors and Navigation

Gyroscopes and inertial units report deg/s or deg/h, while filters and control code often run in rad/s. The Earth rate, about 15.04 deg/h, is a common test value.

Astronomy and Orbits

Orbit tables give rates in degrees per day, radians per day or revolutions per day. Converting to rad/s puts them in the same system as orbital speed and gravity. Fast spinners, such as pulsars or lab rotors, are often quoted in rev/s; the revolution per second converter handles that pair.

Real rates from very slow to fast, computed from exact factors and the NASA and IERS figures cited here.

The range spans about 9 powers of ten, from Earth circling the Sun to a machine shaft. That spread is why so many units exist: each keeps its own range readable.

1. Treating Rpm as Rad/min

One rpm is 2 pi rad/min, not 1 rad/min. Leaving out 2 pi is the most frequent rotation error.

2. Writing Rad/s as Hz

A shaft at 10 rad/s turns at about 1.59 Hz, not 10 Hz. Hertz counts cycles; rad/s counts radians.

3. Using Degrees in a Radian Formula

v = omega x r and P = torque x omega give correct units only when omega is in rad/s.

4. Mixing Solar and Sidereal Time

Earth spins at 2 pi rad per 86,164 s, not per 86,400 s. The 2 pi rad/d figure is the rate of the Sun across the sky.

5. Rounding Tiny Rates to Zero

Planet rates in rad/s have five or more zeros after the decimal point. Use scientific notation to keep them.

6. Dropping the Sign

The sign carries the direction of rotation. Adding a clockwise and a counterclockwise rate without signs gives the wrong total.

Every factor is exact, so results are as accurate as your input and the figures you keep.

The radian per second, rad/s. NIST SP 811 lists it as the SI derived unit for angular velocity, which reduces to s^-1 in base units.

Multiply by pi/30, about 0.1047198. For example, 3,000 rpm is 314.16 rad/s.

Multiply by 30/pi, about 9.5493. A rate of 100 rad/s is 954.93 rpm.

One rad/s is 180/pi, about 57.2958 deg/s.

No. They share the base unit s^-1, but NIST SP 330 keeps them apart. Angular frequency in rad/s is 2 pi times the frequency in Hz.

Exactly 6 deg/s. One turn of 360 degrees in 60 seconds is 6 degrees each second.

Fifteen degrees per hour is one turn per 24 hour solar day. Earth turns once against the stars in about 23.93 hours, so its true rate is about 15.041 deg/h.

Pick the unit that keeps the numbers readable: deg/d or rad/d for planets, deg/h for Earth rate and gyro drift, and rpm for machines. Convert to rad/s only for formulas.

Yes. A negative value means rotation in the opposite sense. The sign carries through to every unit.

No. The conversion runs in your browser, and nothing is sent to our servers. Anything you choose to Save stays in this browser only.

• Exact conversion factors: revolution, rpm, degree, day and hour (NIST SP 811, Appendix B.8).
• SI derived units: radian and radian per second for angular velocity (NIST SP 811, Chapter 4).
• Hertz vs radian per second, and the factor of 2 pi (NIST SP 330, The International System of Units, Section 2).
• Mean angular velocity of the Earth and the sidereal day (IERS Earth Orientation Center, useful constants).
• Earth sidereal orbit period of 365.256 days (NASA NSSDCA Earth Fact Sheet).