Angular Kinematic Overview
Up until this unit we have only been studying translational motion. This is the motion of the center of mass of an object. This unit is all about rotational motion. All of the linear or translational variable we learned at the beginning of the year have rotational or angular counterparts. Angular displacement \(\Delta \theta\) is similar to a linear displacement \(\Delta x\) for example. One measures how far an object rotates and the other measures how far an object travels. All of the other variables relate in a similar way and are listed in the table below.
| Angular Quantities | Linear Quantities |
| \(\Delta \theta\) Angular Displacement | \(\Delta x\) Displacement |
| \(\omega\) Angular Velocity | \(v\) Velocity |
| \(\alpha\) Angular Acceleration | \(a\) Acceleration |
| \(t\), time | \(t\) time |
Because the quantities relate to each other in the same way, we can use the same equations with our new angular variable.
| Angular Equations | Linear Equations |
| \(\omega=\omega_0 + \alpha t\) | \(v=v_0 + a t\) |
| \( \Delta \theta =\omega_0 t +\frac{1}{2} \alpha t^2 \) | \( \Delta x =v_0 t +\frac{1}{2} a t^2 \) |
| \(\omega^2 = \omega_0^2 + 2\alpha(\Delta \theta)\) | \(v^2 = v_0^2 + 2a(\Delta x)\) |
| \(\Delta \theta = \frac{1}{2}(\omega_0+\omega)t\) | \(\Delta x = \frac{1}{2}(v_0+v)t\) |
Linear variables can be calculated by multiplying the angular variable by the radius. This only works as long as we measure the angle with radians.
| \( S = \Delta \theta r \) | Distance equals angular displacement times radius. |
| \(v=\omega r\) | Tangential velocity equals angular velocity times radius. |
| \(a = \alpha r\) | Tangential acceleration equals angular acceleration times radius. |