Angular Acceleration Calculator (α from speed change and time, or torque and inertia)

Understanding the angular acceleration calculator — concept and principle

Angular acceleration tells you how quickly a rotating object speeds up or slows down. This calculator finds it from a change in angular speed over time, or from torque and moment of inertia, and also gives the tangential acceleration at a radius and the number of turns during the change.

Speeding up by 157.08 rad/s in 3 s is 52.36 rad/s²; a point 0.1 m from the axis accelerates at 5.236 m/s² along its path.

How it is calculated

From speeds: α = Δω ÷ Δt, in rad/s²; 1 rpm = 2π ÷ 60 rad/s.
From torque: α = τ ÷ I (Newton's second law for rotation).
Tangential acceleration at radius r: at = α × r.
Angle turned at constant α = (ω₀ + ω) ÷ 2 × t, shown in full turns.
A negative α means the rotation is slowing down.

The principle

Rotational motion mirrors straight-line motion: angle plays the role of distance, angular speed the role of speed, angular acceleration the role of acceleration, torque the role of force and moment of inertia the role of mass. So just as a = Δv ÷ Δt and F = ma, we have α = Δω ÷ Δt and τ = Iα. A point farther from the axis moves a longer arc for the same angle, so its tangential acceleration grows with radius.

Worked example

Default inputs: from rest to 1,500 rpm in 3 s, radius 0.1 m.

1,500 rpm = 1,500 × 2π ÷ 60 ≈ 157.08 rad/s.
α = 157.08 ÷ 3 ≈ 52.36 rad/s², which is 500 rpm per second or 3,000°/s².
Tangential acceleration at 0.1 m = 52.36 × 0.1 ≈ 5.236 m/s².
Average speed = 750 rpm, so turns during 3 s = 750 × 3 ÷ 60 = 37.5.

(The torque inputs, 20 N·m and 0.5 kg·m², are used only when you choose that method; they would give α = 40 rad/s².)

Each point is the calculator re-run with only “Time for the change (s)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Time for the change (s)	Angular acceleration α (rad/s²)
0.75	209.4
1.5	104.7
2.25	69.81
3	52.36
3.75	41.89
4.5	34.91
6	26.18

How to read the result

Compare the angular acceleration with what a motor and load can actually deliver: the torque needed is Iα plus any friction. High tangential accelerations at large radii can stress components. Check units carefully: rpm, rad/s and degrees/s are all common.

Common mistakes

Forgetting to convert rpm to rad/s.
Using the moment of inertia of the motor alone without the load.
Confusing angular acceleration with centripetal acceleration (ω²r).
Ignoring friction and load torque when computing α from τ.

Keep in mind: Results assume constant angular acceleration and rigid bodies. Real machine design needs relevant standards and a qualified engineer.

Frequently asked questions

How do you calculate angular acceleration?

Divide the change in angular velocity by the time, or divide net torque by the moment of inertia.

How do I convert rpm to rad/s?

Multiply by 2π ÷ 60, about 0.10472.

What is the difference between tangential and centripetal acceleration?

Tangential acceleration changes the speed along the circle (αr); centripetal acceleration changes direction (ω²r).

Sources: Halliday, Resnick & Walker, Fundamentals of Physics (rotation) · OpenStax, University Physics Volume 1, Chapter 10 Fixed-Axis Rotation

Checked 9 Nov 2026. OpenStax, University Physics Volume 1 (2016), §10.2 “Rotation with Constant Angular Acceleration” (α = dω/dt, Eqs. 10.9–10.11); Newton’s second law for rotation τ = Iα (standard formula).

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