Rotational Equations of Motion
With angular displacement, velocity and acceleration all defined, the final step is to assemble them into a working toolkit — four equations, each linking a different combination of these quantities under constant angular acceleration, ready to solve real spinning-machinery problems.
The four equations of uniform angular acceleration
When a rotating body has constant angular acceleration $\alpha$, its angular displacement $\theta$, initial and final angular velocities $\omega_1$ and $\omega_2$, and time $t$ are linked by four equations — direct rotational analogues of the four suvat equations of linear kinematics.
Each of these is obtained from its linear counterpart by relabelling every quantity with its rotational equivalent: $s \to \theta$, $u \to \omega_1$, $v \to \omega_2$, $a \to \alpha$, while $t$ stays as $t$ in both. Since the underlying algebra is identical, everything already known about choosing which suvat equation to use — picking the one that already contains the quantity you don't know, and the three you do — transfers across without modification.
| Linear (suvat) | Rotational analogue | Meaning |
|---|---|---|
| $v = u + at$ | $\omega_2 = \omega_1 + \alpha t$ | Final velocity from initial velocity, acceleration and time |
| $s = \dfrac{(u+v)}{2}t$ | $\theta = \dfrac{(\omega_1+\omega_2)}{2}t$ | Displacement from average of initial/final velocity and time |
| $s = ut + \tfrac{1}{2}at^2$ | $\theta = \omega_1 t + \tfrac{1}{2}\alpha t^2$ | Displacement from initial velocity, time and acceleration |
| $v^2 = u^2 + 2as$ | $\omega_2^2 = \omega_1^2 + 2\alpha\theta$ | Final velocity² without needing time |
As with the linear versions, all four equations only hold while $\alpha$ is genuinely constant — for a non-uniformly accelerating rotor, calculus (integrating $\alpha$ with respect to time) or graphical methods are needed instead, as covered in the previous lesson.
Exactly as with linear suvat problems, the fastest route through a rotational equations-of-motion question is to list what's known and unknown first, then pick whichever of the four equations already contains all the knowns and the one unknown you need — avoiding a two-step calculation. Always double-check that $\theta$ is being tracked in radians throughout, since these equations assume it.
Key takeaways
- The four equations of uniform angular acceleration are $\omega_2 = \omega_1 + \alpha t$, $\theta = \frac{(\omega_1+\omega_2)}{2}t$, $\theta = \omega_1 t + \frac{1}{2}\alpha t^2$, and $\omega_2^2 = \omega_1^2 + 2\alpha\theta$.
- Each is the direct rotational analogue of a linear suvat equation, obtained by the substitutions $s\to\theta$, $u\to\omega_1$, $v\to\omega_2$, $a\to\alpha$.
- They only apply while angular acceleration is constant (uniform); non-uniform cases need calculus or graphical methods instead.
- As with linear suvat, choose the equation that already contains the three known quantities and the one unknown, to avoid unnecessary extra steps.
- Angular displacement must be tracked in radians throughout for these equations to hold.
- These four equations form the working toolkit for solving realistic rotating-machinery problems, such as flywheels spinning up or rotors braking to a stop.