Consider a particle moving along a circular path when it moves from point A to point B as in figure below.
The reference point OA makes an angle with the line OB that represents the line joining the new position of the object and the center of the circle. The object has covered a distance S along the circle making an arc with θ.
Angular displacement is the angle swept by a line joining end of an object in a circular path with the center of the path when it moves from one point to another in a circular motion.
Angles in circular motion are usually expressed in radians θc.
That is:
it therefore follows that S = rꝊc
A radian is defined as an angle subtended at the center of a circle by an arc length equal to the radius of the circle.
Therefore angle θ subtended by the circumference at the center of a circle of radius r is therefore given by;
and we can write circumference in terms of π such that
We can relate the degrees from 2πc = 360o .
Problem
An object traces an arc of length 10.98 while attached to a cord of length 3.2 m that is fixed on a fixed surface on a flat smooth surface. Determine the angular diaplacement by the object.
Solution
We visualize the setup as in figure below:
from the equation S = rꝊc
Ꝋc = S/r = 10.98/3.2 = 3.43 radians
if we can express answers in degrees, then 3.43 radians = 196.52o .
Practice Question
An object moves a distance of 80.12π along a circular path of radius 3.8m. Determine it’s angular displacement.