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# Angular Velocity and Acceleration

Angular velocity and acceleration, including centripetal and coriolis acceleration.
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## Overview

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 Velocity

In 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:

We can write the tangential component of velocity, , which is the component of velocity perpendicular to , as:

and, by using the expressions of and from equations (2) and (3) respectively, we obtain:

We define the angular velocity as the rate of change of with respect to time:

Using (5) in (4) we get that:

from which the angular velocity becomes:

## Angular Acceleration

In 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:

We can write the radial component of acceleration, , which is the component of acceleration in the direction of , as:

and, by using the expressions of and from equations (6) and (7) respectively, we get:

We can also write (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:

We define the angular acceleration as the rate of change of the angular velocity with respect to time:

Equation (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:

[imperial]
##### Example - Linear velocity
Problem
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.
Workings
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:

Solution

### Reference

For an application of angular velocity to mechanics, also see the reference page on Velocity and Acceleration of a Piston .