What this calculator does
Angular acceleration is how quickly a rotation rate changes, the rotational counterpart of ordinary acceleration. A wheel spinning up from rest, a drill reaching speed or a turbine slowing down all have one.
It connects directly to torque: angular acceleration equals torque divided by moment of inertia, which is the rotational version of F equals ma. That relationship is what determines how quickly any motor can spin up its load.
The formula
Subtract the initial angular velocity from the final one and divide by the time taken. A positive result means speeding up, a negative one means slowing down.
| Term | Meaning |
|---|---|
| Angular acceleration (α) | Rate of change of angular velocity, in radians per second squared. |
| Angular velocity (ω) | Rotation rate in radians per second. |
| Rotational analogue | Angular acceleration relates to torque and moment of inertia exactly as linear acceleration relates to force and mass. |
The inputs explained
| Field | What to enter |
|---|---|
| Initial angular velocity (ω₁) (rad/s) | The starting angular velocity in radians per second. Use zero when starting from rest. |
| Final angular velocity (ω₂) (rad/s) | The final angular velocity in radians per second. |
| Time interval (s) | The time taken for the change, in seconds. |
When to use it
Sizing a motor for spin-up time
A required angular acceleration combined with the load's moment of inertia gives the torque the motor must produce.
Analysing braking on a rotating machine
Deceleration is simply negative angular acceleration, and the same arithmetic applies.
Working a rotational dynamics problem
Angular acceleration is the bridge between torque and the resulting change in rotation.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does spin-up time affect the angular acceleration?
The same change in rotation rate achieved over different times.
| Time taken | Angular acceleration | Change in angular velocity |
|---|---|---|
| 1 s | 12.000 rad/s² | 12.000 rad/s |
| 2 s | 6.000 rad/s² | 12.000 rad/s |
| 4 s | 3.000 rad/s² | 12.000 rad/s |
| 8 s | 1.500 rad/s² | 12.000 rad/s |
Questions
How does this relate to torque?
Torque equals moment of inertia times angular acceleration, exactly parallel to force equals mass times acceleration. To spin something up faster you need either more torque or less rotational inertia.
What does a negative value mean?
That the rotation is slowing down. The sign simply indicates direction relative to the current rotation, not anything physically different.
Is angular acceleration the same everywhere on a rotating body?
Yes, for a rigid body every point shares the same angular acceleration. Linear acceleration differs from point to point, growing with distance from the axis.
Why radians rather than RPM?
Because radians make the relationship to torque and moment of inertia direct, with no conversion constants. RPM is convenient for describing machinery but awkward inside physics formulas.
For the rotation rate itself, see the angular velocity calculator. For the torque that produces the change, see the torque calculator.