Angular Acceleration Calculator
A wheel that reaches 1800 rpm from rest in 4 seconds has an angular acceleration of 47.12 rad/s^2, which is 450 rpm per second. During the spin-up it sweeps 377 radians, exactly 60 full turns, and its rim at a 4 inch radius accelerates at 188.5 in/s^2.
Enter two spin rates and the time between them, or an angle and a time from rest. The calculator returns angular acceleration in rad/s^2 with rpm per second beside it. Everything runs in your browser.
| Angular acceleration | |
| Same in rpm per second | |
| Angle swept | |
| Revolutions during the change | |
| Direction |
All calculations run in your browser, and nothing you type is uploaded or stored.
Enter spin rates in rpm or rad/s and a time, or an angle swept and a time, and the calculator returns angular acceleration in rad/s^2 with rpm per second beside it.
Key takeaways
- alpha = (omega1 - omega0) / t, with omega in rad/s
- Convert rpm once: rpm x 2 pi / 60 gives rad/s
- From rest with angle known: alpha = 2 x theta / t^2
- Negative alpha means the spin is slowing down
On this page
What angular acceleration is
Angular acceleration is how quickly a spin rate changes. When a fan speeds up, a saw blade coasts down, or a car wheel swings through a turn, the quantity that tracks the change is angular acceleration, written alpha and measured in radians per second squared (rad/s2).
The NIST reference on SI units lists the radian as the standard angle unit, which is why the formulas below use radians per second for spin rate. One full turn is 2 pi radians, about 6.283, and a radian is 57.3 degrees. US engineering practice still quotes spin rates in rpm, so this page shows both.
The base formula is one subtraction: alpha = (omega1 minus omega0) / t, where omega is angular velocity in rad/s and t is the elapsed time in seconds. The calculator above also handles the from-rest case with one angle, alpha = 2 x theta / t2, and converts rpm for you.
The formula, and rpm converted properly
Omega in radians per second is rpm times 2 pi divided by 60. A motor spinning at 1800 rpm runs at 1800 x 6.283 / 60 = 188.5 rad/s. Take the spin-up time in seconds, subtract the starting rate, and divide.
Worked example: a bench grinder reaches 1800 rpm from rest in 4 seconds. Omega goes from 0 to 188.5 rad/s, so alpha = (188.5 - 0) / 4 = 47.12 rad/s2. In rpm per second the same spin-up is 1800 / 4 = 450 rpm each second, which is the number most shop specs quote.
The angle turned during the spin-up comes from the average rate: theta = 0.5 x (0 + 188.5) x 4 = 377 rad, which is 377 / 6.283 = 60 revolutions. The grinder wheel turns a full 60 times while it gets up to speed, which surprises most people the first time they run the numbers.
| Quantity | Symbol | SI unit | Plain English |
|---|---|---|---|
| Angular displacement | theta | radian | How far it has spun, measured at the rim |
| Angular velocity | omega | rad/s | How fast it is spinning right now |
| Angular acceleration | alpha | rad/s2 | How quickly the spin rate is changing |
| Linear speed at rim | v | m/s | Speed of a point on the edge, v = r x omega |
Keep the three quantities straight and the rest of rotational motion follows. The table above is the whole vocabulary: displacement, rate, and the change in rate.
Three worked examples
A lab centrifuge spins up from 500 rpm to 3000 rpm in 5 seconds. That is 52.36 to 314.16 rad/s, so alpha = (314.16 - 52.36) / 5 = 52.36 rad/s2. The same answer in rpm per second is 2500 / 5 = 500.
A hard drive platter at 7200 rpm brakes to a full stop in 8 seconds after the power cut. Omega falls from 754.0 rad/s to 0, so alpha = -754.0 / 8 = -94.25 rad/s2, negative because the rate is dropping.
A turntable swings through a quarter turn, 90 degrees, in 1.2 seconds from rest. Theta is 1.571 rad, and from rest theta = 0.5 x alpha x t2, so alpha = 2 x 1.571 / (1.2 x 1.2) = 2.18 rad/s2. The three cases use the same formula with different knowns.
When a page or a spec sheet gives you one number only, the other two usually hide in plain sight. A 4 second spin-up to 1800 rpm is always 47.12 rad/s2, no matter what the brand name on the motor is.
Spin-up times and what they feel like
| Spin-up time to 600 rpm | Angular acceleration, rad/s2 | Revolutions during spin-up |
|---|---|---|
| 2 s | 31.42 | 10.0 |
| 4 s | 15.71 | 20.0 |
| 6 s | 10.47 | 30.0 |
| 10 s | 6.28 | 50.0 |
| 20 s | 3.14 | 100.0 |
Notice how the acceleration column falls while the time column rises, and how the revolutions column climbs as the square of the time. Doubling the spin-up time cuts alpha in half but doubles the revolutions turned during the spin, because the wheel is turning the whole time at an average rate.
Feel matters as much as the number. A ceiling fan that takes 20 seconds to reach speed has an alpha near 3 rad/s2, gentle enough that the blades seem to appear out of nowhere. A table saw that reaches 4000 rpm in under a second is above 400 rad/s2, and the wheel stores enough energy that the guard is not optional equipment.
From spin to edge speed
Angular acceleration converts to ordinary speed at the rim with one multiplication: linear acceleration a = r x alpha, where r is the radius in meters. A 0.1 m radius wheel accelerating at 47.12 rad/s2 speeds up its edge at 0.1 x 47.12 = 4.7 m/s2, which is about half the pull of gravity.
The direction of that edge acceleration is tangent to the circle, not outward. The outward pull people feel on a merry-go-round is a reaction to the turn, not the acceleration itself, which is worth knowing before you quote a number at a playground argument.
Georgia State University's HyperPhysics walks through the same relation with graphs if you want the calculus behind it. The short version: alpha is the slope of the omega line, and omega is the slope of the theta line, one derivative apart at each step.
Where the mistakes come from
| Mistake | What it does | Fix |
|---|---|---|
| Using rpm directly | A 1800 rpm figure is 188.5 rad/s, not 1800, so every downstream number is off by a factor of 60 over 2 pi | Convert rpm to rad/s first: rpm x 2 x pi / 60 |
| Forgetting to square the time | Theta = alpha x t gives half the true angle, so a 90 degree turn comes out as 45 | Use theta = 0.5 x alpha x t2 for a spin from rest |
| Mixing rad and degrees inside one formula | 90 degrees entered as 90 radians makes the answer wrong by a factor of 57.3 | Convert degrees to radians once, at the top |
| Sign errors on spin-down | A braking wheel shows positive acceleration, so a 5 second stop reads as a speed gain | Make alpha negative when the spin rate falls |
| Averaging rpm the wrong way | Linear acceleration at the rim doubles alpha by mistake when users add r to it | Keep alpha in rad/s2 and multiply by r only for the edge speed |
Most errors trace back to one of two habits: leaving rpm unconverted, or losing track of the square on t. Both show up in the calculator the same way, as an answer that is 60 times or 2 times too big, so a quick sanity check against the worked examples above catches them.
Spin rates that show up in your week
A few stock numbers anchor the scale. A clothes dryer tumbles at about 50 rpm, a ceiling fan spins near 200, a car idles around 750, a highway cruise sits near 2,000, and a miter saw blade turns 4,000. Each converts with the same line, and each hides a rim speed worth knowing before you touch anything.
The dryer drum, roughly 27 inches across, spins its rim at 5.236 rad/s, and 5.236 x 13.5 = 70.7 in/s. The saw blade at 4,000 rpm and a 5 inch radius moves its teeth at 2,094 in/s, about 119 mph. Same formula, wildly different consequences.
Angular acceleration is the change between two of those spin rates, so the two pages share one vocabulary. When a saw brake stops a blade from 4,000 rpm in 2 seconds, the rates are 419 rad/s and 0, and alpha = (0 - 419) / 2 = -209 rad/s^2, the minus sign meaning the spin is ending.
Frequently asked questions
What is the formula for angular acceleration?
Alpha = (omega1 - omega0) / t. A wheel that goes from 0 to 188.5 rad/s in 4 s has alpha = 188.5 / 4 = 47.12 rad/s^2.
How do I convert rpm to rad/s?
Multiply rpm by 2 pi and divide by 60. 1800 rpm is 1800 x 6.283 / 60 = 188.5 rad/s. Going the other way, multiply rad/s by 60 / 6.283.
How do I find angular acceleration from rest with angle and time?
Alpha = 2 x theta / t^2. A quarter turn is 1.571 rad, so 1.571 rad in 1.2 s from rest gives alpha = 2 x 1.571 / 1.44 = 2.18 rad/s^2.
What is 1800 rpm in radians per second?
188.5 rad/s. The conversion is rpm x 2 pi / 60, and 1800 x 6.283 / 60 = 188.5.
What unit is angular acceleration measured in?
Radians per second squared, rad/s^2, per the SI unit system. Revolutions per minute per second, rpm/s, is the common shop alternative: 47.12 rad/s^2 equals 450 rpm/s.
How is angular acceleration different from linear acceleration?
Linear acceleration tracks a change in straight line speed, angular acceleration tracks a change in spin rate. They connect through the radius: a = r x alpha, so a 0.1 m wheel at 47.12 rad/s^2 accelerates its rim at 4.7 m/s^2.
Can angular acceleration be negative?
Yes, and it usually is during braking. A 7200 rpm platter stopping in 8 s has alpha = -94.25 rad/s^2, the minus sign meaning the spin rate is falling.
How many revolutions happen during a spin-up?
Theta = 0.5 x (omega0 + omega1) x t, then divide by 2 pi. From rest to 188.5 rad/s in 4 s, theta = 377 rad, which is 60 full revolutions.
What is the angular acceleration of a centrifuge going 500 to 3000 rpm in 5 s?
52.36 rad/s^2. The rates convert to 52.36 and 314.16 rad/s, so the change is 261.8 rad/s spread over 5 s.
How do I get angular acceleration in rpm per second?
Take the rpm change and divide by the seconds. 1800 rpm in 4 s is 1800 / 4 = 450 rpm/s, and the same spin is 47.12 rad/s^2.
Does radius change angular acceleration?
No. Alpha depends only on how the spin rate changes, which is the same for every point on a rigid wheel. Radius converts alpha into linear acceleration at the rim, a = r x alpha, but it does not change alpha itself.
How do I find time from angular acceleration?
t = (omega1 - omega0) / alpha. Going from rest to 94.25 rad/s at 47.12 rad/s^2 takes 94.25 / 47.12 = 2 s.
Is angular acceleration the same as centripetal acceleration?
No. Centripetal acceleration, v^2 / r or omega^2 x r, keeps a point moving in a circle at constant speed. Angular acceleration changes the speed of the spin itself, so a wheel at a steady 1800 rpm has zero angular acceleration but a large centripetal acceleration.
How many g is 47 rad/s^2 at the rim of a 0.1 m wheel?
4.7 m/s^2 divided by 9.81 is about 0.48 g, roughly half the pull of gravity, at the rim edge.
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