Angular Acceleration Calculator

What the angular acceleration calculator does

When a spinning thing speeds up or slows its rate of turn, that change is angular acceleration. This calculator finds it from a change in angular velocity over time, or from the tangential acceleration of a point and its distance from the centre. It can also solve for any of the other quantities in those relationships.

Below is what angular acceleration means, the equations behind it, how it connects to ordinary acceleration, and a worked example.

How to use it

1. Choose a method: from a change in angular velocity over time, or from tangential acceleration and radius.
2. Choose what to find within that method, then enter the known values.
3. Press Calculate for the result, or Reset to clear it.

What angular acceleration is

Angular acceleration is the rotational version of ordinary acceleration. Ordinary acceleration is how quickly velocity changes; angular acceleration is how quickly angular velocity changes, how fast the rate of spin itself is speeding up or slowing down. It is written with the Greek letter alpha, and its unit is radians per second squared.

You meet it whenever a rotation is not steady. A wheel spinning up from rest, a hard drive coming up to speed, a fan slowing to a stop after the power is cut, all have an angular acceleration while their turning rate changes. When the spin is steady, the angular acceleration is zero, just as a steady speed means no ordinary acceleration.

The equations it uses

The basic definition is the change in angular velocity divided by the time it took, with ω for angular velocity and t for time:

α = (ω_final − ω_initial) ÷ t

This is the rotational twin of the straight-line equation linking velocity, acceleration, and time, and the calculator rearranges it to find the initial or final angular velocity or the time. The other method connects angular acceleration to the tangential acceleration a_t of a point at radius r, the part of its acceleration along its circular path:

α = a_t ÷ r

which rearranges to a_t = αr and r = a_t ÷ α.

The link to tangential acceleration

The relationship a_t = αr mirrors the v = ωr link for speeds. As a wheel spins up, a point on it gains speed along its circular path, and that gain, the tangential acceleration, grows with how far out the point sits. A point on the rim picks up speed faster than one near the hub, even though both share the same angular acceleration.

It is worth noting that this tangential acceleration is only one part of what a point on a rotating object feels. There is also a centre-pulling acceleration that keeps the point on its curved path, present whenever it is moving at all. The tangential part, the one tied to angular acceleration, is specifically the change in how fast the point travels along its circle, separate from the turning that keeps it on the circle.

Units and precision

The calculator works in SI units: angular velocity in radians per second, angular acceleration in radians per second squared, time in seconds, tangential acceleration in metres per second squared, and radius in metres. Results are rounded to a few decimal places for display while the calculation runs at full precision.

A worked example: a spinning-up wheel

Suppose a wheel speeds up from rest to an angular velocity of 100 rad/s in 5 seconds.

The angular acceleration is α = (100 − 0) ÷ 5 = 20 rad/s². If a point sits 0.3 metres from the centre, its tangential acceleration, the rate it gains speed along its path, is a_t = αr = 20 × 0.3 = 6 m/s².

Questions people ask

What is angular acceleration?

It is the rate at which angular velocity changes, how fast a spin is speeding up or slowing down, measured in radians per second squared. A steady spin has zero angular acceleration.

How do you calculate angular acceleration?

Divide the change in angular velocity by the time, α = (ω_final − ω_initial)/t. From a point's tangential acceleration and radius, you can also use α = a_t/r.

What is tangential acceleration?

It is the part of a point's acceleration along its circular path, the change in how fast it travels. It equals the angular acceleration times the radius, a_t = αr.

How does it relate to ordinary acceleration?

It is the rotational counterpart. Ordinary acceleration changes a straight-line velocity; angular acceleration changes a rate of spin. The two are linked at any point by a_t = αr.

References

A quick note on where the physics comes from. Angular acceleration as the rate of change of angular velocity, and its link to tangential acceleration through a_t = αr, are standard rotational mechanics, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. The radian per second squared and the other SI units follow the US National Institute of Standards and Technology.

1. OpenStax, University Physics Volume 1, Section 10.2, Rotation with Constant Angular Acceleration. https://openstax.org/books/university-physics-volume-1/pages/10-2-rotation-with-constant-angular-acceleration
2. HyperPhysics, Georgia State University, Rotational Quantities. http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html
3. National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811