Angular Displacement

Angular displacement of an object moving around a circular path is defined as the angle traced out by the radius vector at the axis of the circular path in a given time.

Angular Displacement Formula

An object traveling in a circular path of radius $r$ covers an arc of length $ \displaystyle l$ on its circumference, then the angular displacement of the object on that circular path is given by:

$ \displaystyle \theta =\frac{l}{r}$

Consider an object moving along a circular path of radius $r$, in an anticlockwise direction in the plane of paper, with the center at $O$. Let the axis of circular motion pass through $O$, perpendicular to the plane of paper. Let the position of the object change from $P$ to $Q$ in time $t$, where $\angle POQ = \theta$ as shown in the figure below. Therefore, during the time interval $t$, the radius vector traces out an angle $\theta$ at the axis of the circular path.

Here $\displaystyle \theta $ is known as the angular displacement of the object in time $\displaystyle t$.

\[
\text{Since, angle} = \frac{\text{arc}}{\text{radius}},
\]

\[
\therefore \theta = \frac{PQ}{r} \tag{1}
\]

In equation (1), the angle $\theta$ represents the magnitude of angular displacement and is expressed in radians (denoted by rad.).

Angular displacement is a vector quantity provided $\theta$ is small because the commutative law of vector addition for large angles is not valid, whereas for small angles, the law is valid.

The direction depends on the sense of rotation of the object and can be given by the Right-Hand Rule, which states that if the curvature of the fingers of the right hand represents the sense of rotation of the object, then the thumb, held perpendicular to the curvature of the fingers, represents the direction of the angular displacement vector.