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…
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.
View the full transcript and gain access to JoVE Core videos
Q1: What is angular displacement in circular motion?
Angular displacement is the change in angular position of a particle moving in a circular path, measured as Δθ in radians. One radian equals the angle where the arc length matches the radius, equivalent to 57.3 degrees. Angular displacement quantifies how far an object has rotated around a fixed center point.
Q2: How is angular velocity different from linear velocity?
Angular velocity measures the rate of change of angular displacement in radians per second, represented by omega. All points on a rotating object share the same angular velocity. However, linear velocity differs because points closer to the rotation center travel shorter distances than points farther away, resulting in different linear speeds despite identical angular velocities.
Q3: What are the units and measurement conventions for angular velocity?
Angular velocity is measured in radians per second (rad/s), also called the rotation rate. Counterclockwise rotations are defined as positive, while clockwise rotations are negative. When rotation rate is given in revolutions per second, multiply by 2π to convert to radians per second, since one complete revolution equals 2π radians.
Q4: Why do all points on a rotating string have the same angular velocity?
All points on a rotating rigid body sweep out the same angle in the same time interval, making their angular velocities identical. This is a fundamental property of rigid body rotation: the angular position changes uniformly across the entire object regardless of distance from the rotation axis.
Q5: How is angular position defined in uniform circular motion?
Angular position, denoted θ, is the angle swept by a position vector from the circle's origin to the particle, measured in radians. The angle increases counterclockwise as the particle moves along its circular path. Angular position serves as the reference frame for measuring angular displacement and angular velocity.
Q6: What is the relationship between arc length and angular displacement?
A radian is defined as the angle subtended when the arc length equals the radius. This dimensionless relationship means radian measure is a ratio of length measurements. Since 2π radians equal 360 degrees, the radian provides a natural unit connecting angular displacement to the circular path's geometry.
Q7: How does instantaneous angular velocity differ from average angular velocity?
Instantaneous angular velocity is the limit of average angular velocity as the time interval approaches zero (Δt → 0). This calculus-based definition provides the angular velocity at a specific moment, enabling analysis of rotation with changing angular velocity, which extends beyond uniform circular motion.