MotionLab

Angular Acceleration Calculator

How quickly something's spin rate is changing.

Initial angular velocity (rad/s)

Final angular velocity (rad/s)

Time (s)

Angular acceleration (rad/s²)

α = (ω₁ − ω₀) / t

What you entered

Initial angular velocity (rad/s): 0Final angular velocity (rad/s): 10Time (s): 5

1
α = (ω₁ − ω₀) ÷ t
(10 − 0) ÷ 5= 2 rad/s²

Result

Angular acceleration (rad/s²): 2

Angular velocity changing from 0 to 10 rad/s over 5s gives 2 rad/s².

Angular acceleration from a change in spin rate

Angular acceleration is the change in angular velocity divided by time, in radians per second squared.

Start (rad/s)	End (rad/s)	Time (s)	Angular acceleration (rad/s2)
0	10	5	2.0000
0	6.2832	2	3.1416
5	15	2	5.0000
0	100	10	10.0000
10	0	4	-2.5000

The last row is negative because the object is slowing, exactly as with linear deceleration - the sign carries the direction of the change. Angular acceleration relates to torque the way linear acceleration relates to force: torque equals moment of inertia times angular acceleration, which is the rotational form of Newton's second law. That is why a figure skater pulling their arms in speeds up without any torque at all - the moment of inertia falls instead.

The rotational parallel to linear acceleration

This is the exact rotational analog of a = Δv/t for linear motion — every linear motion formula has a rotational counterpart, just swapping distance for angle, velocity for angular velocity, and so on.

Connects to torque via moment of inertia

Just as force causes linear acceleration (F=ma), torque causes angular acceleration (τ=Iα), where I (moment of inertia) plays the role mass plays in linear motion — a spinning object's mass distribution, not just its total mass, determines how it responds to torque.

Frequently asked questions

A ceiling fan goes from rest to 120 RPM in 4 seconds. What's its angular acceleration?

ω = 120 × 2π/60 = 12.57 rad/s. α = Δω/Δt = 12.57/4 = 3.14 rad/s². It takes about 3 revolutions to reach full speed (θ = ½αt² = ½ × 3.14 × 16 = 25.1 rad ≈ 4 revolutions).

How is angular acceleration related to torque?

τ = Iα, just like F = ma. A flywheel with moment of inertia I = 2 kg·m² experiencing torque τ = 10 N·m has α = 10/2 = 5 rad/s². More mass distributed far from the axis means larger I and slower spin-up. See the torque calculator.

Can angular acceleration be negative?

Yes — negative angular acceleration means the object is slowing its rotation (angular deceleration). A spinning top winding down due to friction has negative α. The magnitude tells you how quickly it's decelerating.