Combining Rotational Kinematic Equations

The four rotational kinematic equations each connect four of five variables: $\Delta \theta$, $\omega_i$, $\omega_f$, $\alpha$, and $t$. Each equation omits exactly one variable.

$$\omega_f = \omega_i + \alpha t$$

This omits $\Delta \theta$.

$$\Delta \theta = \omega_0 t + \frac{1}{2}\alpha t^2$$

This omits $\omega_f$.

$$\omega_f^2 = \omega_i^2 + 2\alpha\Delta \theta$$

This omits $t$.

$$\Delta \theta = \frac{\omega_i + \omega_f}{2}t$$

This omits $\alpha$.

Selection Strategy: Find the variable that is neither given nor asked for. Choose the equation that omits it.

Suppose a problem gives $\omega_i$, $\alpha$, and $t$, and asks for $\omega_f$. The variable $\Delta\theta$ is not given and not needed. Select the first equation since it omits $\Delta \theta$.

If instead $\Delta \theta$ is given but $t$ is unknown, select the third equation. It omits $t$ and directly yields $\omega_f$.