Angular Acceleration Equations

Equations for uniform angular acceleration

• The kinematic equations of motion for uniform linear acceleration can be rewritten for rotational motion. The key part is that every linear variable is simply swapped for its rotational equivalent.
• The four kinematic equations for uniform linear acceleration are: $$v = u + at$$ $$s = ut + \tfrac{1}{2}at^2$$ $$v^2 = u^2 + 2as$$ $$s = \frac{(u + v)t}{2}$$
• This leads to the four kinematic equations for uniform rotational acceleration: $$\omega_2 = \omega_1 + \alpha t$$ $$\Delta\theta = \omega_1 t + \tfrac{1}{2}\alpha t^2$$ $$\omega_2^{\,2} = \omega_1^{\,2} + 2\alpha\Delta\theta$$ $$\Delta\theta = \frac{(\omega_1 + \omega_2)t}{2}$$

Variable substitutions

• The five linear variables have been swapped for their rotational equivalents:
• Displacement: $s \to \theta$
• Initial velocity: $u \to \omega_1$
• Final velocity: $v \to \omega_2$
• Acceleration: $a \to \alpha$
• Time: $t \to t$ (time is the same in both cases)

Worked example: turntable

• A turntable spins at 45 RPM before being turned off. It decelerates at a constant rate of 0.8 rad s$^{-2}$. To find the angular displacement before it stops:
• Convert RPM to rad s$^{-1}$: $\omega_1 = \frac{2\pi \times 45}{60} = \frac{3\pi}{2}$ rad s$^{-1}$
• Use $\omega_2^{\,2} = \omega_1^{\,2} + 2\alpha\Delta\theta$ with $\omega_2 = 0$: $$\Delta\theta = \frac{\omega_1^{\,2}}{-2\alpha} = \frac{\left(\frac{3\pi}{2}\right)^2}{2 \times 0.8} = 13.88 \text{ rad}$$
• The number of rotations = $\frac{13.88}{2\pi} = 2.2$ rotations.