Angular Velocity Calculator
A wheel at 1800 rpm spins at 188.5 rad/s, which is 30 turns per second and a period of 0.033 seconds per turn. Its rim moves at 188.5 inches per second for every inch of radius, so a 10 inch wheel turning 1800 rpm has an edge speed of 1,885 in/s.
Enter a spin rate and pick its unit, or a linear speed with a radius. The calculator converts between rpm, rad/s, turns per second and rim speed. Everything runs in your browser.
| Angular velocity | |
| In rpm | |
| In rad/s | |
| Turns per second | |
| Period, seconds per turn | |
| Rim speed at radius 1 |
All calculations run in your browser, and nothing you type is uploaded or stored.
Enter a spin rate in rpm or rad/s, or a linear speed with a radius, and the calculator converts between every common unit of rotation.
Key takeaways
- rad/s = rpm x 2 pi / 60; one rpm is 0.1047 rad/s
- Rim speed: v = r x omega, so each inch of radius adds 12.57 in/s at 188.5 rad/s
- Period and frequency flip: T = 1 / f, omega = 2 pi x f
- Earth spins at 7.292 x 10^-5 rad/s, giving 465.1 m/s at the equator
On this page
What angular velocity is
Angular velocity is how fast something spins, measured as an angle per unit of time. It is written omega and its SI unit is radians per second, per the NIST reference on SI units. One full turn is 2 pi radians, so a wheel making one turn per second spins at 6.283 rad/s.
The US habit is rpm, and the two units convert with one line: rad/s = rpm x 2 pi / 60, and rpm = rad/s x 60 / 2 pi. Each rpm is 0.1047 rad/s, which is why the second hand of a clock, at exactly 1 rpm, reads 0.1047 rad/s.
Two directions of the same formula matter: the spin rate itself, and the speed of a point on the rim, v = r x omega. This page handles both, plus the period (seconds per turn) and frequency (turns per second) that describe the same motion in different clothes.
One vocabulary note before the numbers. Angular velocity and angular speed mean the same thing in most shop and homework contexts. Physicists reserve velocity for the version that carries a turning direction, clockwise or counterclockwise, and use speed for the size alone. Every formula on this page works with the size alone, so enter your rpm and read the rad/s out.
Converting rpm to rad/s, with worked numbers
Take a bench grinder at 1800 rpm. Omega = 1800 x 6.283 / 60 = 188.5 rad/s. The same wheel turns 1800 / 60 = 30 revolutions each second, and each revolution takes 1 / 30 = 0.033 seconds.
A 12 inch vinyl record at 33-1/3 rpm spins at 33.333 x 6.283 / 60 = 3.49 rad/s. A point at the outer groove, 6 inches from the center, moves at 6 x 3.49 = 20.94 in/s, which is 53.2 cm/s. That falling groove speed across the record's radius is the reason a record's inner grooves sound different from its outer ones.
| Spin rate | rad/s | Revolutions per second | Seconds per turn |
|---|---|---|---|
| 1 rpm | 0.10 | 0.017 | 60.000 |
| 33.33 rpm | 3.49 | 0.555 | 1.800 |
| 45 rpm | 4.71 | 0.750 | 1.333 |
| 78 rpm | 8.17 | 1.300 | 0.769 |
| 300 rpm | 31.42 | 5.000 | 0.200 |
| 1800 rpm | 188.50 | 30.000 | 0.033 |
| 3600 rpm | 376.99 | 60.000 | 0.017 |
| 7200 rpm | 753.98 | 120.000 | 0.008 |
| 10000 rpm | 1047.20 | 166.667 | 0.006 |
Read the seconds-per-turn column before quoting any spin rate: a 7200 rpm platter finishes a turn in 0.00833 seconds, and anything vibrating faster than that will blur on camera without a fast shutter.
Linear speed at the rim
Multiply angular velocity by the radius and you get the speed of a point at that radius: v = r x omega. This is the conversion that turns a spin rate into a belt speed, a saw blade tip speed, or a road speed.
A 10 inch table saw blade at 4000 rpm spins at 419 rad/s. Its rim, at r = 5 inches, moves at 5 x 419 = 2094 in/s. Divide by 12 for 174.5 feet per second, or about 119 mph at the teeth. That number is why blade guards and riving knives exist.
A car wheel rolling at 60 mph with a 25 inch diameter tire spins at omega = v / r. Speed first: 60 mph is 88 ft/s. Radius is 12.5 in = 1.042 ft, so omega = 88 / 1.042 = 84.5 rad/s, which is 807 rpm. The calculator above runs this direction too, from a known linear speed to the spin rate.
The Earth as a spinning object
The Earth is the largest spinning object anyone measures, and its angular velocity is a fixed, known number. The NASA planetary fact sheet lists Earth's sidereal rotation period as 23.9333 hours, which is 86,164 seconds. Omega = 2 pi / 86,164 = 7.292 x 10^-5 rad/s.
At the equator, radius 6,378 km, that spin carries you at 7.292 x 10^-5 x 6,378,137 = 465.1 m/s, about 1,040 mph. At 40 degrees latitude the radius of your circle shrinks by the cosine, cos(40) = 0.766, so the speed falls to 465.1 x 0.766 = 356 m/s, or 797 mph.
Two details keep this example honest. The rotation period relative to the stars is 23 hours 56 minutes, not the 24 hour solar day, and the speed is relative to Earth's center, not the ground you stand on. A GPS satellite accounts for both to keep its clock honest.
Period, frequency and turns
Angular velocity, period and frequency are three views of one number. Period T is seconds per turn, frequency f is turns per second, and omega = 2 pi x f = 2 pi / T.
A fan spinning at 300 rpm has f = 5 turns per second, T = 0.2 s per turn, and omega = 2 pi x 5 = 31.42 rad/s. A hard drive at 7200 rpm has f = 120 and T = 0.00833, and omega = 754 rad/s. Notice that omega is just 2 pi times the frequency; the radians only exist to make the rim speed formula come out in meters per second.
| Object | Angular velocity | Where the number shows up |
|---|---|---|
| Second hand of a clock | 0.1047 rad/s | One turn per minute, the definition of the rpm unit |
| Vinyl record at 33-1/3 | 3.49 rad/s | Album side length and groove speed calculations |
| Clothes washer spin cycle | 58 to 84 rad/s | Water extraction, typically 550 to 800 rpm |
| Car engine at 60 mph highway cruise | 209 rad/s | 2000 rpm in most passenger cars |
| Hard drive platter, 7200 rpm class | 754 rad/s | Latency and data transfer rate specs |
| Dental drill | up to 41,888 rad/s | 400,000 rpm low speed handpiece rating |
The table above is worth a scroll on its own: the same unit, rad/s, covers eight orders of magnitude, from a clock hand to a dental drill. Any calculator that claims to do angular velocity has to handle that range without losing the small decimals.
Mistakes that corrupt the conversion
| Mistake | Effect | Fix |
|---|---|---|
| Dividing by 2 pi instead of multiplying | 1800 rpm becomes 28.6 rad/s instead of 188.5, a factor of 6.6 error | Memorize the shape: rpm x 2 pi / 60, pi always on top |
| Using diameter for radius in v = r x omega | Doubles every rim speed, so a saw tip reads 238 mph | Halve the diameter first |
| Mixing revolutions with radians | Turning a 3.49 rad/s record into 3.49 turns per second, off by 6.3 times | A revolution is 2 pi radians, convert through rpm |
| Forgetting the cosine at latitude | Quotes equatorial speed for New York, a 23 percent overshoot | Multiply by cos(latitude) for the local circle radius |
| Minutes versus seconds | 300 rpm typed as 300 rad/s inflates the answer by 58 percent... corrected below | Convert the time unit once, at the top |
One correction worth spelling out: 300 rpm is 31.42 rad/s, so typing 300 rad/s instead inflates the answer by a factor of 300 / 31.42 = 9.55, not 58 percent. The table row above keeps its point, and the fix stays the same: convert the time unit once, before anything else.
Where the number shows up in daily life
Angular velocity hides in more spec sheets than any other rotational number. Drill speeds are rpm, fan speeds are rpm, hard drives are rpm, washer spin cycles are rpm, record players are rpm, car tachometers read rpm. Every one of those conversations converts with the same line of arithmetic.
A cordless drill at 1500 rpm spins at 157 rad/s, and its half inch chuck moves a bit tip at 78.5 in/s. A ceiling fan at 200 rpm is 20.9 rad/s, and a 30 inch blade tip travels 524 in/s, which is 29.8 mph, fast enough to sting a knuckle. Run each number through the calculator once and the spec sheet stops being a foreign language.
Two cross checks keep any conversion honest. One rpm is always 0.1047 rad/s, so a 300 rpm answer of 31.42 rad/s can be verified by dividing 31.42 by 300 to get 0.1047. And rim speed always equals radius times omega, so a saw blade tip speed can be recomputed from its rpm and diameter in one step, no table needed.
Frequently asked questions
What is the formula for angular velocity?
omega = theta / t for steady spin, and v = r x omega links it to rim speed. A wheel turning 6.283 rad in 1 s has omega = 6.283 rad/s, one turn per second.
How do I convert rpm to rad/s?
Multiply by 2 pi and divide by 60. 1800 rpm x 6.283 / 60 = 188.5 rad/s. One rpm is 0.1047 rad/s.
How do I convert rad/s to rpm?
Multiply by 9.549, since 60 / 6.283 = 9.549. 188.5 x 9.549 = 1800 rpm.
What is 7200 rpm in rad/s?
754 rad/s. 7200 x 6.283 / 60 = 753.98, and the same spin is 120 turns per second with a period of 0.00833 s.
How fast is a point on the rim of a 10 inch blade at 4000 rpm?
2094 in/s at r = 5 in, which is 174.5 ft/s, about 119 mph. omega = 4000 x 6.283 / 60 = 419 rad/s and v = 5 x 419 = 2094.
What is the angular velocity of the Earth?
7.292 x 10^-5 rad/s, using the sidereal day of 86,164 s listed on NASA's planetary fact sheet. The solar day of 86,400 s gives 7.272 x 10^-5 rad/s, a 0.3 percent difference.
How fast does the Earth spin at 40 degrees latitude?
356 m/s, about 797 mph. Equatorial speed is 465.1 m/s and the local circle shrinks by cos(40) = 0.766, so 465.1 x 0.766 = 356.
What is one rpm in rad/s?
0.1047 rad/s. That is why the second hand of a clock, exactly 1 rpm, has omega = 0.1047 rad/s and a period of 60 s.
What is the difference between angular velocity and linear velocity?
Angular velocity is angle per time and is the same for every point on a rigid wheel. Linear velocity is distance per time and grows with radius, v = r x omega, so the rim moves fastest while the hub barely moves.
Is angular velocity the same as rpm?
They describe the same spin with different units. 1 rpm = 0.1047 rad/s and 1 rad/s = 9.549 rpm, so a 1800 rpm motor and a 188.5 rad/s motor are identical.
How do I find angular velocity from period?
omega = 2 pi / T. A ceiling fan with a period of 0.2 s per turn has omega = 6.283 / 0.2 = 31.42 rad/s, which is 300 rpm.
How many turns per second is 1800 rpm?
30. Divide rpm by 60 for turns per second, and each turn takes 1 / 30 = 0.033 s.
What is angular velocity of a 33 rpm record?
3.49 rad/s for the exact 33-1/3 speed. The outer groove at r = 6 in moves at 6 x 3.49 = 20.9 in/s, and the same record's inner groove at r = 2.5 in moves at 8.7 in/s.
Does angular velocity change with radius?
No. On a rigid wheel every point completes a turn in the same time, so omega is one number. Only the linear speed changes with radius, which is why the inner grooves of a record move slower than the outer ones.
What unit should angular velocity be reported in?
Radians per second in SI work, rpm in US shop talk, and degrees per second when a camera or game controller reports the rate. All three convert through one line of arithmetic.
Spotted a wrong number or a broken result?
More Math tools
Fee schedules, rates and rules change without notice. Always confirm important figures with the official source before you rely on them.