10.2:
Angular Velocity and Acceleration
Consider a ball tied to a string rotating in a circular trajectory. The rate of change of angular displacement is called its angular velocity. Like linear velocity, angular velocity is also a vector quantity, and a rotation in a clockwise direction is considered as the negative direction. Consider a ball rotating at high speed, the rate of change of angular displacement will be high, and hence the angular velocity will be high.
The angular velocity value at any time during the motion is called its instantaneous angular velocity, and it is expressed as a derivative of θ with respect to time.
The rate at which the angular velocity of an object changes is called its angular acceleration, denoted by the letter α with units of radians per second square.
Angular acceleration is a vector quantity and is considered positive when the angular velocity increases and vice-versa. Since two points on the string have the same angular velocity, their angular acceleration is also the same.
10.2:
Angular Velocity and Acceleration
We previously discussed angular velocity for uniform circular motion, however not all motion is uniform. Envision an ice skater spinning with their arms outstretched; when they pull their arms inward, their angular velocity increases. Additionally, think about a computer's hard disk slowing to a halt as the angular velocity decreases. The faster the change in angular velocity, the greater the angular acceleration. The instantaneous angular acceleration is defined as the derivative of angular velocity with respect to time. The units of angular acceleration are (rad/s)/s, or radians per second squared.
We can relate the tangential acceleration of a point on a rotating body at a distance from the axis of rotation in the same way that we relate the tangential speed to the angular velocity. Thus, tangential acceleration is the radius times the angular acceleration.
The following points represent a problem-solving strategy that can be applied to rotational kinematics:
- Examine the situation to determine that rotational kinematics (rotational motion) is involved.
- Identify exactly what needs to be determined in the problem (identify the unknowns). A sketch of the situation is useful.
- Make a complete list of what is given or can be inferred from the problem as stated (identify the knowns).
- Solve the appropriate equation or equations for the quantity to be determined (the unknown). It can be useful to think in terms of a translational analog. Substitute the known values along with their units into the appropriate equation and obtain numerical solutions complete with units. Be sure to use units of radians for angles.
- Finally, check the answer to see if it is reasonable.
This text is adapted from Openstax, University Physics Volume 1, Section 10.1: Rotational Variables.