P01-220 – Average Angular Acceleration

Average Angular Acceleration

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

The average angular acceleration ${\alpha }_{avg}$ of the particle is equal to the ratio of the change in its angular velocity $\mathrm{\Delta }\omega ={\omega }_f-{\omega }_i$ to the corresponding duration $\mathrm{\Delta }t=t_f-t_i$.

\boxed{{\alpha }_{avg}=\frac{\mathrm{\Delta }\omega }{\mathrm{\Delta }t}=\frac{{\omega }_f-{\omega }_i}{t_f-t_i}\ \ \ \ \ \ \ \ \left(rad/s^2\right)}

The average angular acceleration therefore measures how fast the angular velocity changes over time on average. In other words, you can think of it as how fast, on average, the rotational speed of the particle is increasing or decreasing every second.

Example: consider a particle with angular velocity ${\omega }_1=\pi \ rad/s$ at time $t_1=1\ s$ and angular velocity ${\omega }_2=5\pi \ \ rad/s$ at time $t_2=3\ s$.

The average angular acceleration of the particle is therefore equal to

{\alpha }_{avg}=\frac{\mathrm{\Delta }\omega }{\mathrm{\Delta }t}=\frac{5\pi -\pi }{3-1}=2\pi \ \ rad/s^2

This can be interpreted as the fact that, on average, the angular speed of the particle increased by $2\pi \ rad/s$ every second from $t_1$ to $t_2$.

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