Quick answer
Angular acceleration α = (ω₂ − ω₁) ÷ t. Spinning up from rest to 31.4 rad/s in 5 seconds gives 6.28 rad/s², or 0.9995 rev/s².
Enter the initial and final angular velocity and the time. The calculator returns angular acceleration in rad/s² and rev/s².
Related tools: Angular Velocity Calculator: Angular velocity from rpm, frequency, or an angle and time. Acceleration Calculator: Average acceleration from change in velocity over time. Torque Calculator: Torque from force, lever arm and angle.
How to use the Angular Acceleration Calculator
- Enter initial angular velocity (rad/s).
- Enter final angular velocity (rad/s).
- Enter time (s) (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Initial angular velocity | A number, in rad/s | — |
| Final angular velocity | A number, in rad/s | — |
| Time | A number, in s | > 0 |
Formula and method
Formula
α = (ω₂ − ω₁) ÷ t
IEEE-754 double precision; shown to the decimals indicated.
Constant angular acceleration.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Initial angular velocity = 0; Final angular velocity = 31.4; Time = 5.
| Result | Value |
|---|---|
| Angular acceleration | 6.28 rad/s² |
| In rev/s² | 0.999493 |
Example 2
Inputs: Initial angular velocity = 100; Final angular velocity = 20; Time = 8.
| Result | Value |
|---|---|
| Angular acceleration | -10 rad/s² |
| In rev/s² | -1.591549 |
Example 3
Inputs: Initial angular velocity = 10; Final angular velocity = 10; Time = 2.
| Result | Value |
|---|---|
| Angular acceleration | 0 rad/s² |
| In rev/s² | 0 |
The first line is the headline result (angular acceleration). Below it the angular Acceleration Calculator lists in rev/s², which describe the same case from other angles. IEEE-754 double precision; shown to the decimals indicated.
Worked example
31.4 rad/s is about 5 revolutions per second (300 rpm). 31.4 ÷ 5 = 6.28 rad/s², one revolution per second per second.
Linking to torque
Torque = moment of inertia × α. Negative values mean slowing down. The tangential acceleration at radius r is α × r.
How the result changes with final angular velocity
Holding every other input at the first worked example, this table changes only final angular velocity. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Final angular velocity | Angular acceleration | In rev/s² |
|---|---|---|
| 15.7 | 3.14 rad/s² | 0.499747 |
| 31.4 | 6.28 rad/s² | 0.999493 |
| 62.8 | 12.56 rad/s² | 1.998986 |
| 125.6 | 25.12 rad/s² | 3.997972 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Time: entering "0" shows "Time must be greater than 0." beside that field and no result.
- Initial angular velocity: left empty shows "Enter initial angular velocity." beside that field and no result.
Choosing between the Angular Acceleration Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Angular Velocity Calculator](/science/angular-velocity-calculator/) | starting from | Angular velocity |
| [Acceleration Calculator](/science/acceleration-calculator/) | initial velocity, final velocity and time | Acceleration |
| [Torque Calculator](/science/torque-calculator/) | lever arm, force and angle between force and lever | Torque |
| [Magnitude of Acceleration Calculator](/science/magnitude-of-acceleration-calculator/) | x component and y component | Magnitude |
| [Engine Torque Calculator](/engineering/engine-torque-calculator/) | calculate and engine speed | Torque |
Background: gravity, rotation and optics
Gravity and rotation calculations use a few core relations: a circular orbit has speed v = √(GM ÷ r), torque equals moment of inertia times angular acceleration (τ = Iα), and a thin lens obeys 1/f = 1/dₒ + 1/dᵢ. The moment of inertia depends on both the shape and the axis of rotation.
Use SI units (metres, kilograms, seconds) and check the sign conventions of the formula before interpreting a negative result.
Assumptions and limitations
- Negative values mean slowing down
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for time and initial angular velocity to confirm this.
Frequently asked questions
What is the unit?
Radians per second squared.
How do I convert rpm to rad/s?
Multiply rpm by 2π/60: 300 rpm is 31.4 rad/s.
How does it relate to linear acceleration?
The tangential acceleration at radius r is a = αr, so a point 0.5 m out with α = 6.28 rad/s² accelerates at 3.14 m/s².
What does the Angular Acceleration Calculator calculate?
Angular acceleration from a change in angular velocity. It returns Angular acceleration and In rev/s².
What formula does the Angular Acceleration Calculator use?
α = (ω₂ − ω₁) ÷ t. Constant angular acceleration.
What do I need to enter in the angular Acceleration Calculator?
You enter Initial angular velocity (rad/s), Final angular velocity (rad/s) and Time (s).
How is the result rounded or limited?
IEEE-754 double precision; shown to the decimals indicated. Negative values mean slowing down.
What is an example of a angular Acceleration result?
With initial angular velocity 0, final angular velocity 31.4 and time 5, the angular acceleration is 6.28 rad/s².
Related calculators
- Angular Velocity Calculator — Angular velocity from rpm, frequency, or an angle and time.
- Acceleration Calculator — Average acceleration from change in velocity over time.
- Torque Calculator — Torque from force, lever arm and angle.
- Magnitude of Acceleration Calculator — Magnitude of an acceleration vector from its components.
- Dew Point Calculator — Dew point from temperature and relative humidity.
Sources
- OpenStax, University Physics Volume 1 (Rice University): mechanics
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.