Angular Acceleration Calculator
Calculate angular acceleration from torque and moment of inertia, or from change in angular velocity over time.
Shows resulting angle rotated.
Angular acceleration is the rate of change of angular velocity. Two ways to calculate it:
Method 1: From torque: α = τ / I
Method 2: From velocity change: α = Δω / Δt = (ω_final − ω_initial) / time
Where:
- α = Angular acceleration (rad/s²)
- τ = Torque applied (N·m)
- I = Moment of inertia (kg·m²)
- ω = Angular velocity (rad/s)
- t = Time (s)
Rotational kinematics equations (constant α, analogous to v = v₀ + at):
- ω_f = ω₀ + αt
- θ = ω₀t + ½αt²
- ω_f² = ω₀² + 2αθ
Analogy with linear motion:
| Linear | Rotational |
|---|---|
| F = ma | τ = Iα |
| a (m/s²) | α (rad/s²) |
| v (m/s) | ω (rad/s) |
| x (m) | θ (rad) |
| m (kg) | I (kg·m²) |
That table is worth more than it looks. Every kinematics result you already know carries straight across: swap the symbols and the algebra is identical. If you can do constant-acceleration problems in a straight line, you can do them in a circle.
The one place the analogy leaks is moment of inertia. Mass is a property of an object; moment of inertia is a property of an object and the axis you spin it about. The same flywheel has a different I about its centre than about its rim, and shifting mass outward raises I as the square of the radius. That is why a flywheel is heavy at the edge and thin in the middle, and why pulling your arms in on a spinning chair speeds you up: I drops, and with angular momentum conserved, ω has to rise to compensate.
Common moments of inertia
| Shape, about its centre | I |
|---|---|
| Solid disc or cylinder | ½ m r² |
| Thin ring or hoop | m r² |
| Solid sphere | ⅖ m r² |
| Rod about its middle | (1/12) m L² |
| Rod about one end | ⅓ m L² |
Note the last two: the same rod, four times harder to swing about its end than about its middle. Nothing about the rod changed except where you hold it.
Practical example: An electric motor applies 50 N·m of torque to a flywheel with I = 2.5 kg·m². The angular acceleration is: α = 50 / 2.5 = 20 rad/s²
Starting from rest, after 3 seconds: ω = 20 × 3 = 60 rad/s ≈ 573 RPM θ = ½ × 20 × 9 = 90 radians ≈ 14.3 full revolutions
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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