P01-210 – Average Angular Velocity

Average Angular Velocity

Consider a particle moving around a circular path of radius $R$ with an angular position ${\theta }_i$ at time $t_i$ and angular position ${\theta }_f$ at time $t_f>t_i$.

The average angular velocity ${\omega }_{avg}$ of the particle is equal to the ratio of its angular displacement $\mathrm{\Delta }\theta ={\theta }_f-{\theta }_i$ to the time $\mathrm{\Delta }t=t_f-t_i$ during which it covers that displacement.

\boxed{{\omega }_{avg}=\frac{\mathrm{\Delta }\theta }{\mathrm{\Delta }t}=\frac{{\theta }_f-{\theta }_i}{t_f-t_i}\ \ \ \ \ \ \ \ \left(rad/s\right)}

The average angular velocity therefore measures how fast the angular position changes over time on average. In other words, you can think of it as how fast, on average, the particle is moving from one angular position to another.

Example: if we consider the previous example again, we will find that in the time $\mathrm{\Delta }t=t_2-t_1$, the angular displacement of the particle is $\mathrm{\Delta }\theta $ as shown below.

Taking $t_1=1\ s$ and $t_2=3\ s$, we conclude that the average angular velocity of the particle is equal to

{\omega }_{avg}=\frac{\mathrm{\Delta }\theta }{\mathrm{\Delta }t}=\frac{\displaystyle{\frac{3\pi }{4}-\frac{\pi }{4}}}{3-1}=\frac{\pi }{4}\ \ rad/s

Note that this does not tell you if the angular velocity of the particle changes between $t_1$ and $t_2$ but simply tells you on average how fast it was rotating around the center. In order to take into account the fact that the angular velocity might change each instant of a given motion, we will have to later introduce the instantaneous angular velocity $\omega $.