Angular Velocity Calculator
A wheel spinning at 60 rpm has an angular velocity of about 6.28 rad/s. This angular velocity calculator finds omega, the rate at which an object rotates or sweeps out an angle, using whichever information you already have. Pick the angle and time mode when you know how far something turned in radians and how long it took, the linear speed mode when you know the tangential speed at a known radius, or the rpm mode when a motor or wheel spec is given in revolutions per minute. Whichever path you choose, the tool reports omega in radians per second as the primary answer and then converts that same value into revolutions per minute and degrees per second so you can read the result in the unit your problem needs. In the linear speed mode it also returns the rotation period, the number of seconds for one full turn. Every displayed figure is rounded to four decimal places, which keeps physics homework, engineering estimates, and quick sanity checks readable without hiding meaningful precision.
Quick answer
Angular velocity, written as the Greek letter omega, measures how fast an angle changes over time and is expressed in radians per second.
What this tells you
- •Angular velocity, written as the Greek letter omega, measures how fast an angle changes over time and is expressed in radians per second.
- •From an angle and time, omega equals the swept angle in radians divided by the elapsed time in seconds.
- •From linear motion, omega equals the tangential speed divided by the radius, since a larger radius covers the same speed with a smaller turn rate.
- •From a rotation spec, omega equals rpm multiplied by 2 times pi and divided by 60, because one revolution is 2 pi radians and one minute is 60 seconds.
- •One full revolution equals 2 pi radians, which is roughly 6.2832 radians or 360 degrees.
- •The rotation period T is the reciprocal relationship T equals 2 pi divided by omega, giving the seconds needed for one complete turn.
How to Use
- 11. Choose an input mode. Use Angle and time when you know the radians turned and the seconds taken, Linear speed and radius when you know the tangential speed and the circle radius, or RPM when you have a revolutions-per-minute value.
- 22. Enter the two required numbers for angle and time or linear modes, or the single rpm value for rpm mode. Time and radius must be greater than zero.
- 33. Make sure your angle is in radians for the angle mode. If you have degrees, convert first by multiplying degrees by pi and dividing by 180.
- 44. Press Calculate to read omega in radians per second as the main result.
- 55. Check the secondary results for the same rotation expressed in revolutions per minute and degrees per second, plus the rotation period when you use the linear speed mode.
How It Works
Formula
omega = theta / t | omega = v / r | omega = rpm * 2*pi / 60Angular velocity omega has three equivalent starting points that all resolve to radians per second. When you know an angle theta measured in radians and the time t in seconds it took to sweep that angle, omega equals theta divided by t. When you know the tangential speed v in metres per second at a radius r in metres, omega equals v divided by r, because the linear speed at the rim is the product of omega and the radius. When you have a rotation rate in revolutions per minute, omega equals rpm times 2 pi divided by 60, since each revolution is 2 pi radians and each minute holds 60 seconds. The calculator then converts omega back into rpm using rpm equals omega times 60 divided by 2 pi, into degrees per second using omega times 180 divided by pi, and, in the linear mode, into a period T equals 2 pi divided by omega. Time and radius must both be positive, and any non-finite input returns no result.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Angle and time
An object sweeps pi radians (about 3.1416) in 2 seconds. Dividing 3.1416 by 2 gives 1.5708 rad/s, which is a quarter turn every second, or 90 degrees per second, or 15 rpm.
Linear speed on a wheel
A point moving at 10 m/s around a circle of radius 2 m turns at 10 divided by 2, which is 5 rad/s. The period is 2 pi divided by 5, about 1.2566 seconds for each full revolution, which works out to roughly 47.75 rpm.
Motor rated in rpm
A motor spinning at 60 rpm turns once per second. Multiplying 60 by 2 pi and dividing by 60 gives 6.2832 rad/s, which equals 360 degrees per second, the signature of one full turn each second.
High-speed shaft in rpm
A shaft at 3000 rpm is a common industrial speed. Multiplying 3000 by 2 pi and dividing by 60 gives 314.1593 rad/s, the same as 50 revolutions every second and 18000 degrees per second.
RPM to Angular Velocity Quick Reference
Common revolution rates and their equivalent angular velocity in radians per second and degrees per second.
| RPM | Omega (rad/s) | Degrees per second | Period (s) |
|---|---|---|---|
| 1 | 0.1047 | 6.0000 | 60.0000 |
| 10 | 1.0472 | 60.0000 | 6.0000 |
| 60 | 6.2832 | 360.0000 | 1.0000 |
| 100 | 10.4720 | 600.0000 | 0.6000 |
| 1000 | 104.7198 | 6000.0000 | 0.0600 |
| 3000 | 314.1593 | 18000.0000 | 0.0200 |
Values are rounded to four decimal places. The period column is the seconds for one full revolution at that speed.
Angular velocity versus linear velocity
Angular velocity describes how fast something rotates, while linear velocity describes how fast a point on that rotating object travels through space. The two are tied together by the radius. At a fixed angular velocity, a point far from the center moves faster in a straight-line sense than a point near the center, because it covers a larger circle in the same amount of time. That is why the tip of a fan blade whistles through the air while the hub barely seems to move even though both share the exact same omega.
The bridge between them is the equation v equals omega times r. If you know two of the three quantities, you can solve for the third. This calculator uses that relationship directly in the linear speed mode, dividing the tangential speed by the radius to recover omega. It is the same physics that lets a small drive gear spin a large wheel more slowly, or that makes a record player groove near the edge pass under the needle faster than a groove near the label.
Radians sit at the heart of these formulas for a reason. A radian is defined so that an arc length equals the radius times the angle, which makes v equals omega times r exact rather than approximate. If you work in degrees or revolutions instead, you have to sprinkle conversion factors like pi over 180 or 2 pi throughout your math. Keeping omega in radians per second as the base unit, and converting only at the end for display, keeps the underlying calculation clean and less prone to unit mistakes.
Common mistakes
- Entering an angle in degrees while using the angle and time mode, which expects radians. Convert degrees to radians first by multiplying by pi and dividing by 180.
- Confusing rpm with rad/s. They measure the same rotation but differ by a factor of 2 pi over 60, so a raw rpm number is far larger than the matching rad/s value.
- Using a diameter where the formula wants a radius in the linear speed mode. The radius is half the diameter, and using the full diameter halves your omega.
- Forgetting that time and radius must be greater than zero. A zero or negative value returns no result because division by zero is undefined.
- Mixing unit systems, such as a speed in kilometres per hour with a radius in metres. Convert everything to consistent SI units before entering values.
Embed this calculator on your site
Drop this single line where you want the calculator to appear. It is responsive, mobile-friendly, resizes automatically, and is free to use with attribution.
<script src="https://calctide.com/embed.js" data-tool="angular-velocity-calculator" async></script>Preview the embed at /embed/angular-velocity-calculator/.