Angular Acceleration Calculator

angular acceleration = ( final angular velocity − initial angular velocity ) ÷ time rotation count = average angular velocity × time

ωi

Initial angular velocity, in rpm by default — the field also takes rad/s, °/s and rev/s. Leave it at 0 for anything starting from rest

ωf

Final angular velocity, in the same units. Smaller than the initial value is allowed and meaningful: the answer comes out negative, which is a deceleration

t

Time over which the change happens, in seconds — the field also takes minutes. It must be greater than zero, and the page assumes the change is spread evenly across it

α

Angular acceleration — in rad/s² as the primary result, and also in rpm/s and °/s². The sign follows the change in velocity: negative α means the rotation is slowing down

N

Turns made during the change, from the average angular velocity times the time — the number that says whether a machine has room to perform the ramp

Use this page when something spins and you need to know how hard it is being pushed — a motor sizing a load, a brake bringing a flywheel down, a centrifuge ramp, a hard drive spinning up, a wheel being accelerated. It is the rotational twin of the linear acceleration page, and the same intuition carries over: for a given change in speed, the shorter the time, the larger the acceleration and the larger the torque needed to produce it. Two habits make the answers more useful. First, read the rotation count as carefully as the acceleration, because it is the constraint that usually decides the design: a ramp that takes 125 turns needs a housing with 125 turns of clearance, and a brake that stops a shaft in a second has to absorb all the energy in that one second. Second, watch the sign, but do not read too much into it — a negative α says the rotation is slowing, not that it is going backwards, and the page cannot tell a controlled stop from a failure.

1. A washing machine drum, rest to 1,200 rpm in 8 seconds

   1. Change in angular velocity: 1,200 − 0 = 1,200 rpm
   2. In rpm per second: 1,200 ÷ 8 = 150 rpm/s
   3. In rad/s²: 1,200 rpm = 125.664 rad/s, and 125.664 ÷ 8 = 15.708 rad/s²
   4. In °/s²: 15.708 × 57.2958 = 900
   5. Average angular velocity: 600 rpm = 10 rev/s
   6. Turns during the ramp: 10 × 8 = 80

   This is the everyday case, and the useful number is the last one. Eighty turns in eight seconds is what the drum has to be able to do without the clothes tangling into a lump and stalling the motor, which is why a spin cycle ramps up rather than switching on at full speed. Note also how the three acceleration units divide the same physical fact: 15.708 rad/s² and 150 rpm/s are the same number, and the °/s² figure is only large because there are 360 degrees in a turn.

2. A hard disk platter, rest to 7,200 rpm in 10 seconds

   1. Change in angular velocity: 7,200 rpm
   2. In rpm per second: 7,200 ÷ 10 = 720 rpm/s
   3. In rad/s²: 7,200 rpm = 753.982 rad/s, and 753.982 ÷ 10 = 75.398 rad/s²
   4. In °/s²: 4,320
   5. Average angular velocity: 3,600 rpm = 60 rev/s
   6. Turns during the ramp: 60 × 10 = 600

   Same calculation, a machine that cares about the answer. 75.398 rad/s² is five times the washing machine's figure, and the platter passes through 600 turns getting there — which is why a drive that is spun down and up repeatedly wears its bearings, and why drives are left spinning rather than stopped between reads. The 10 second figure is deliberately slow for a modern drive; a laptop drive reaches its working speed in about three, and that is only possible because the platter is small and light enough that the torque is available.

3. A brake bringing a shaft from 1,500 rpm to rest in 4 seconds

   1. Change in angular velocity: 0 − 1,500 = −1,500 rpm
   2. In rpm per second: −1,500 ÷ 4 = −375 rpm/s
   3. In rad/s²: 1,500 rpm = 157.080 rad/s, so −157.080 ÷ 4 = −39.27 rad/s²
   4. In °/s²: −2,250
   5. Average angular velocity: 750 rpm = 12.5 rev/s
   6. Turns during the stop: 12.5 × 4 = 50

   The sign is the whole point of this example: the three acceleration outputs are all negative, and the rotation count is not, because the shaft is still turning forwards — it is turning more slowly. Fifty turns happen between the brake being applied and the shaft stopping, and all of the rotational energy in the shaft has to come out during those fifty turns. That is the difference between a brake and a clutch: a brake has to turn kinetic energy into heat, and the rate it does so is proportional to this acceleration, which is why the first second of an emergency stop is the one that cooks the disc.

The page assumes a constant angular acceleration — that the speed changes at a steady rate from start to finish. A real motor has a torque curve rather than a fixed acceleration, so the ramp is not a straight line on a speed chart, and the rotation count it gives is an average that will not match a machine with a soft start. Nothing here is about torque or moment of inertia: the angular acceleration is the kinematics, and converting it into the force a motor must supply needs the rotational version of Newton's second law, τ = Iα, plus the inertia of everything bolted to the shaft. There is no angular momentum and no energy: a brake that stops a shaft has to dissipate ½Iω² across the stop, and the page cannot tell you how much that is. It is also purely rotational — a wheel being accelerated while the vehicle it belongs to is also moving has a linear acceleration too, and the two are related by the radius, which this page does not ask for. It does not handle a change of direction: an angular velocity reversing sign passes through zero, and the constant-acceleration assumption is a poor fit for anything that does that. Finally, the page has no notion of where the acceleration comes from, so it will happily report an acceleration that no real motor or brake could deliver, and it says nothing about the vibration, the bearing loads or the heat that a real ramp at that rate would produce.

How do I calculate angular acceleration?

Subtract the starting angular velocity from the finishing one and divide by the time taken: α = (ωf − ωi) ÷ t. Going from rest to 3,000 rpm in 5 seconds is a change of 3,000 rpm over 5 s, which is 600 rpm/s, or 62.832 rad/s². If the answer comes out negative the rotation is slowing down rather than speeding up, and the page prints all three unit conventions at once so you can match whichever one your datasheet uses.

What is the formula in rad/s²?

The same one, with the speeds converted first: one revolution per minute is 2π ÷ 60 = 0.10472 rad/s, so 3,000 rpm is 314.159 rad/s and dividing by 5 s gives 62.832 rad/s². Going the other way, multiply rad/s² by 9.5493 to get rpm/s. The conversion never changes the physics — it only changes which of the three rows you read.

Why is the angular acceleration negative?

Because the final speed is lower than the initial one and the formula subtracts the two, so a deceleration comes out with a minus sign. A shaft going from 1,500 rpm to rest in 4 seconds has α = −39.27 rad/s². The sign says the rotation is slowing down, not that it is running backwards — the rotation count stays positive throughout, because the shaft is still turning forwards the whole time it is stopping.

What is the rotation count for?

It is the number of turns the object makes while its speed is changing, from the average angular velocity times the time. It is usually the constraint that decides whether a design works: a drum accelerating to 1,200 rpm over 8 seconds passes through 80 turns, and a machine with room for 50 turns simply cannot do that ramp. The same figure tells you how much braking surface a disc needs, since all of the rotational energy has to come out over those turns.

How does angular acceleration relate to torque?

Through the rotational form of Newton's second law, τ = Iα, where τ is the torque and I is the moment of inertia. This page gives you the kinematics — α and nothing else — and to find the torque a motor has to produce you need the inertia of everything on the shaft, which for a solid disc is ½mr² and for a ring is mr². In practice that is why the same acceleration takes a far bigger motor on a loaded centrifuge than on an empty one: the α is identical and the I is not.

Does it matter whether the speed change is even?

Yes, and this page assumes it is. A constant angular acceleration means a straight ramp from the starting speed to the finishing speed, and the rotation count is then simply the average of the two times the time. A real motor with a soft start changes speed unevenly, so the actual number of turns will differ — the page is the right tool for sizing and sanity checks, and the wrong one for predicting exactly where a real machine's shaft will be after a given ramp.

• Angular Acceleration (College Physics 2e, §10.1) — α = Δω ÷ Δt, the definition of angular acceleration, and the rotational kinematics equations built on it — OpenStax
• NIST Guide to the SI — the radian as the coherent unit of angle and the radian per second squared as the unit of angular acceleration, with the exact factors for degrees and revolutions — National Institute of Standards and Technology
• Angular acceleration — the relation to torque and moment of inertia that this page deliberately leaves out, and the sign convention for a decelerating rotation — Wikipedia