10.1:
Angular Velocity and Displacement
The angle made by a particle with respect to a straight line while moving in a circular path is defined as angular displacement. For instance, consider a ball tied to a string rotating in a circular trajectory. The change in angular position of the ball is measured as Δθ and has units of radians.
One radian is the angle at which the arc has the same length as the radius and is equal to 57.3°.
The rate of change of angular displacement of the ball is called its angular velocity. It is measured in radians per second and represented by the letter omega.
For a ball rotating in a circular path, notice that any two points on the string cover the same angular distance with time. Therefore, the angular velocity of the points on the string is the same. However, the linear velocity is different. Since the displacement covered for a point closer to the center of the string is smaller than for a point near the ball.
10.1:
Angular Velocity and Displacement
Uniform circular motion is motion in a circle at a constant speed. Although this is the simplest case of rotational motion, it is very useful for many situations and is used to introduce rotational variables. When a particle is moving in a circle, the coordinate system is fixed and serves as a frame of reference to define the particle’s position. Its position vector from the origin of the circle to the particle sweeps out the angle θ, which increases in the counterclockwise direction as the particle moves along its circular path. The angle θ is called the angular position of the particle. As the particle moves in its circular path, it also traces an arc length.
The angle θ, the angular position of a particle along its path, has units of radians (rad). There are 2π radians in 360°. Note that the radian measure is a ratio of length measurements and therefore is a dimensionless quantity. As a particle moves along its circular path, its angular position changes, and it undergoes angular displacements, Δθ.
The magnitude of the angular velocity, denoted by ω, is the rate of change of the angle θ as the particle moves in its circular path. The instantaneous angular velocity is defined as the limit in which Δt → 0 in the average angular velocity. The units of angular velocity are radians per second (rad/s). Angular velocity can also be referred to as the rotation rate in radians per second. In many situations, the rotation rate is defined in revolutions per second or cycles per second. To find the angular velocity, we must multiply revolutions/s by 2π, since there are 2π radians in one complete revolution. Since the direction of a positive angle in a circle is counterclockwise, we take counterclockwise rotations as being positive and clockwise rotations as negative.
This text is adapted from Openstax, University Physics Volume 1, Section 10.1: Rotational Variables.