What Are the Equations of Motion for Uniform Angular Acceleration?

For uniform angular acceleration, the four equations of motion relate angular displacement, initial and final angular velocity, angular acceleration, and time. They are the rotational equivalents of the linear equations for uniformly accelerated motion.

This is HL-only content from A.4 Rigid body mechanics.

Here, $\Delta\theta$ is angular displacement in radians, $\omega_i$ and $\omega_f$ are the initial and final angular velocities in $$, is in , and is time in seconds.

These equations apply only when $\alpha$ is constant. Their structure matches linear kinematics because angular acceleration is the rate of change of angular velocity:

$$\alpha=\frac{\Delta\omega}{\Delta t}.$$

The analogy is direct: replace linear displacement $s$, velocities (u,v), and acceleration $a$ with $\Delta\theta$, $\omega_i,\omega_f$, and . Since constant makes angular velocity change linearly with time, the average angular velocity is . Multiplying this average by time gives angular displacement, producing the first equation. Combining it with gives the other forms.

For example, a wheel has $\omega_i=4.0\text{ rad s}^{-1}$, accelerates uniformly at $3.0\text{ rad s}^{-2}$, and rotates for . Its final angular velocity is

$$\omega_f=4.0+(3.0)(5.0)=19\text{ rad s}^{-1}.$$

Its angular displacement is

$$\Delta\theta=(4.0)(5.0)+\frac{1}{2}(3.0)(5.0)^2=57.5\text{ rad}.$$

Choose a positive rotational direction before substituting. If a wheel rotating anticlockwise slows down, then with anticlockwise positive, $\omega$ is positive but $\alpha$ is negative. Angles must be in radians when linking rotational quantities to $s=r\theta$, $v=r\omega$, or $$. This prevents sign errors and incorrect use of degrees in rotational calculations.

A common misconception is that these equations apply whenever an object rotates. They apply only when angular acceleration is uniform; otherwise, graphical or calculus-based methods are required.

In an IB exam, identify the known quantities, select the equation that excludes the unneeded unknown, and maintain a consistent sign convention. For a Calculate question, show the equation, substitution, and final value.