# Angular Kinematics

- Angular displacement

$\Delta\theta \Longrightarrow \Delta x$ - Angular velocity

$\omega \Longrightarrow v$ - Angular acceleration

$\alpha \Longrightarrow a$

As we mentioned in Angular Displacement, every angular kinematic variable has a corresponding linear kinematic variable, and the angular kinematics equations are the same as the linear kinematics equations with the angular variables swapped in: thus $\omega={\Delta\theta\over \Delta t}$ instead of $v={\Delta x\over \Delta t}$ and so forth.

Thus, if the angular acceleration of a spinning object is constant, then we can use the same equations we saw in The Five Equations:

\omega_f&=\omega_i+\alpha\Delta t &\color{blue}{\hbox{no}\,\Delta \theta}\\

\Delta \theta&=\frac{1}{2}(\omega_i+\omega_f)\Delta t &\color{blue}{\hbox{no}\,\alpha}\\

\Delta \theta&=\omega_i\Delta t+\frac12\alpha\Delta t^2 &\color{blue}{\hbox{no}\,\omega_f}\\

\Delta \theta&=\omega_f\Delta t-\frac12\alpha\Delta t^2 &\color{blue}{\hbox{no}\,\omega_i}\\

\omega_f^2&=\omega_i^2+2\alpha\Delta \theta &\color{blue}{\hbox{no}\,\Delta t}\\

\end{align}$$

with the five angular kinematics variables

- the time Δt it takes to travel from
*i*to*f* - its initial angular velocity ω
_{i} - its final angular velocity ω
_{f} - its angular acceleration α
- the angular displacement Δθ through which the object has spun

As before, we must be given three of these five variables in order to solve for the rest.

If you've mastered linear kinematics problems, then angular kinematics problems should be comfortable. The trickiest new thing is to figure out what the angular displacements and angular velocities are— they can be described in many different ways as we have seen— and rewrite them in terms of the same units (degrees, radians, or revolutions, whichever is handiest).