Last updated: 2026-06-15

Rotational Kinematics

When something spins up or slows down at a steady rate — a hard drive reaching speed, a wheel braking, a turntable starting — its motion is governed by the rotational kinematics equations. They are the exact angular twins of the linear SUVAT equations, with angle in place of distance and spin rate in place of speed. This calculator solves any one of them for its unknown.

The five equations

With initial angular velocity $\omega_0$, final angular velocity $\omega$, constant angular acceleration $\alpha$, time $t$ and angular displacement $\theta$:

Each equation omits one variable, so you choose the one that contains what you want and the three things you already know.

Worked example

A wheel starts from rest ($\omega_0 = 0$) and accelerates at $\alpha = 2\ \text{rad/s}^2$ for $t = 5\ \text{s}$. Its final spin rate and the angle it turns through are

That $25\ \text{rad}$ is about four full turns, since one revolution is $2\pi \approx 6.28\ \text{rad}$.

The link to linear motion

Every angular quantity has a linear partner, joined by the radius $r$ of the path: arc length $s = r\theta$, tangential speed $v = r\omega$, and tangential acceleration $a = r\alpha$. So once you know the angular result, multiplying by the radius gives the straight-line motion of any point on the rotating body — the rim of a wheel, the tip of a blade, a seat on a fairground ride.

Why radians

These equations require angles in radians, because the link $s = r\theta$ only holds in radians. One revolution is $2\pi$ radians, or $360^\circ$. You may enter the angle in degrees or turns in this calculator and it converts to radians before solving, but the underlying relationships are radian-based.

When the equations do not apply

Like their linear counterparts, these formulas assume the angular acceleration is constant. If the torque varies during the motion — a fan spinning down against air resistance, for instance — the acceleration is not constant and these equations give only a rough average. That case needs calculus or a step-by-step numerical approach instead.