Angular Velocity and Acceleration
Angular velocity and acceleration, including centripetal and coriolis acceleration.
Key facts Angular velocity is the rate of change of the position-specific angle with respect to time:
Angular acceleration is the rate of change of angular velocity with respect to time:
The centripetal acceleration can be defined as:
The Coriolis component of acceleration, or compound supplementary acceleration can be defined as:
Angular velocity and acceleration are vector quantities that describe an object in circular motion. When an object, like a ball attached to a length of string, is rotated at a constant angular velocity, it is said to be in uniform circular motion. However, if the object is rotated at increasing or decreasing speeds, it can be said to be in a state of angular acceleration. The acceleration can have a centripetal component (acting inwards toward the axis of rotation) or a Coriolis component(acting perpendicular to the direction of velocity and the axis of rotation).
Angular VelocityIn order to define angular velocity, consider a particle moving in a reference plane, as diagramed in Figure 1. Let be the projection of on the axis, the projection of on the axis, the length of , and the angle. We can then write that:
By differentiating these equations with respect to time, we have:
and, by using the expressions of and from equations (2) and (3) respectively, we obtain: 5) in (4) we get that:
from which the angular velocity becomes:
Angular AccelerationIn order to define the acceleration, we first have to calculate and . To do this we differentiate equations (2) and (3) with respect to time, when we obtain that:
and, by using the expressions of and from equations (6) and (7) respectively, we get: 8), by using (5), as:
where is the rate of change of velocity, while the term is called the Centripetal Acceleration (). The tangential component of acceleration on the other hand, , can be written as:
and, again by replacing and from equations (6) and (7) respectively, we obtain: 9) can also be written, by taking into account (10), as:
where the term is called the compound supplementary acceleration, or the Coriolis component of acceleration .
Example - Linear velocity
Consider that the hard disk of a computer is circular and rotates with an angular velocity of 7200 revolutions per minute. Calculate the linear velocity (in miles per hour) of a particle which is found 2 inches away from the center of the hard disk.
The angular velocity of the hard disk expressed in SI units is:
As the linear velocity can be defined as , we obtain the linear velocity of the particle as:
or, by converting it in miles per hour: