P01-200 – Angular Position & Displacement

Angular Position and Angular Displacement

Angular Position:

Consider a particle moving around a circular path of radius $R$, for example like a horse on the outer edge of a merry-go-round.

In this type of motion, the distance from the center of the circular path remains the same at all times and if we wish to describe the location of the particle, we then typically indicate its angular position $\theta $. This position is the angle, in radians, measured from some reference often chosen to the horizontal $x$-axis as shown below (as an homage to the unit circle). This reference, however, can be chosen anywhere and does not have to correspond to the $x$-axis.

Knowledge of what this angle is at any time then provides with a unique, unambiguous, position $\theta \left(t\right)$ that tells you exactly where the particle is located.

The angle $\theta $ measured with respect to a reference position gives the angular position of the particle and the distance $s\left(t\right)$ travelled by the particle from the reference position is equal to the arc length $s$ and is given by

\boxed{s\left(t\right)=R\cdot \theta \left(t\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \left(m\cdot rad\equiv m\right)}

where $\theta $ is measured in radians.

Note: the angle swept is independent of the particle’s distance from the center.

Angular Displacement:

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

The angular displacement of the particle is equal to the change in its angular position and we write

\boxed{\mathrm{\Delta }\theta ={\theta }_f\ -{\theta }_i}

Example: in the figure below, the angular position of the particle at time $t_1$ is ${\theta }_1=\pi /4\ \ rad$ while its angular position at time $t_2$ is ${\theta }_2=3\pi /4\ \ rad$.

Overall, from $t_1$ to $t_2$, the angular displacement of the particle is equal to

\mathrm{\Delta }\theta ={\theta }_2-{\theta }_1=\frac{3\pi }{4}-\frac{\pi }{4}=\frac{\pi }{2}\ \ rad

Note that as long as the particle remains on a circular path of radius $R$ – without moving closer or further from the center $O$ – its position is entirely defined by the angle $\theta $ measured with respect to the chosen reference position. That is, if you are given $\theta $ then you know exactly where the particle is.