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# Angular Velocity and Acceleration

Angular velocity and acceleration, including centripetal and coriolis acceleration.

## Overview

Key facts

Angular velocity is the rate of change of the position-specific angle $\inline&space;\theta$ with respect to time:

$\omega&space;=&space;\frac{d\theta}{dt}$

Angular acceleration is the rate of change of angular velocity with respect to time:

$\alpha&space;=&space;\frac{d\omega}{dt}=\frac{d^2\theta}{dt^2}$

The centripetal acceleration can be defined as:

$a_c&space;=&space;r\omega^2$

The Coriolis component of acceleration, or compound supplementary acceleration can be defined as:

$a_C&space;=&space;2v\omega$

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 $\inline&space;P$ moving in a $\inline&space;XOY$ reference plane, as diagramed in Figure 1. Let $\inline&space;x$ be the projection of $\inline&space;P$ on the $\inline&space;OX$ axis, $\inline&space;y$ the projection of $\inline&space;P$ on the $\inline&space;OY$ axis, $\inline&space;r$ the length of $\inline&space;OP$, and $\inline&space;\theta$ the $\inline&space;\angle{POX}$ angle.

We can then write that:

$x&space;=&space;r\cos\theta$

$y&space;=&space;r\sin\theta$

By differentiating these equations with respect to time, we have:

$\dot&space;x&space;=&space;\frac{dx}{dt}&space;=&space;\dot&space;r&space;\cos\theta&space;-&space;r\dot&space;\theta&space;\sin\theta$

$\dot&space;y&space;=&space;\frac{dy}{dt}&space;=&space;\dot&space;r&space;\sin\theta&space;+&space;r\dot&space;\theta&space;\cos\theta$

We can write the tangential component of velocity, $\inline&space;v_T$, which is the component of velocity perpendicular to $\inline&space;OP$, as:

$v_T&space;=&space;\dot&space;y&space;\cos\theta&space;-&space;\dot&space;x&space;\sin\theta$

and, by using the expressions of $\inline&space;\dot&space;x$ and $\inline&space;\dot&space;y$ from equations (1) and (2) respectively, we obtain:

$v_T&space;=&space;r&space;\dot&space;\theta$

We define the angular velocity $\inline&space;\omega$ as the rate of change of $\inline&space;\theta$ with respect to time:

$\omega&space;=&space;\frac{d\theta}{dt}$

Using (4) in (3) we get that:

$v_T&space;=&space;r\omega$

from which the angular velocity becomes:

$\omega&space;=&space;\frac{v_T}{r}$

## Angular Acceleration

In order to define the acceleration, we first have to calculate $\inline&space;\ddot&space;x$ and $\inline&space;\ddot&space;y$. To do this we differentiate equations (1) and (2) with respect to time, when we obtain that:

$\ddot&space;x&space;=&space;\ddot&space;r&space;\cos\theta&space;-&space;\dot&space;r&space;\sin\theta&space;-&space;r\ddot&space;\theta^2&space;\cos\theta&space;-&space;r&space;\ddot&space;\theta&space;\sin\theta$

$\ddot&space;y&space;=&space;\ddot&space;r&space;\sin\theta&space;+&space;\dot&space;r&space;\dot&space;\theta&space;\cos\theta&space;+&space;r&space;\dot&space;\theta^2&space;\sin\theta&space;+r&space;\ddot&space;\theta&space;\cos\theta$

We can write the radial component of acceleration, $\inline&space;a_r$, which is the component of acceleration in the direction of $\inline&space;OP$, as:

$a_r&space;=&space;\ddot&space;x&space;\cos\theta&space;+&space;\ddot&space;y&space;\sin\theta$

and, by using the expressions of $\inline&space;\ddot&space;x$ and $\inline&space;\ddot&space;y$ from equations (5) and (6) respectively, we get:

$a_r&space;=&space;\ddot&space;r&space;-&space;r&space;\dot&space;\theta^2$

We can also write (7), by using (4), as:

$a_r&space;=&space;\dot&space;v&space;-&space;r\omega^2$

where $\inline&space;\dot&space;v$ is the rate of change of velocity, while the $\inline&space;r\omega^2$ term is called the Centripetal Acceleration ($\inline&space;a_c$).

The tangential component of acceleration on the other hand, $\inline&space;a_T$, can be written as:

$a_T&space;=&space;\ddot&space;y&space;\cos\theta&space;-&space;\ddot&space;x&space;\sin\theta$

and, again by replacing $\inline&space;\ddot&space;x$ and $\inline&space;\ddot&space;y$ from equations (5) and (6) respectively, we obtain:

$a_T&space;=&space;r&space;\ddot&space;\theta&space;+&space;2&space;\dot&space;r&space;\dot&space;\theta$

We define the angular acceleration $\inline&space;\alpha$ as the rate of change of the angular velocity $\inline&space;\omega$ with respect to time:

$\alpha&space;=&space;\frac{d\omega}{dt}&space;=&space;\ddot&space;\theta$

Equation (8) can also be written, by taking into account (9), as:

$a_T&space;=&space;r&space;\alpha&space;+&space;2v\omega$
where the $\inline&space;2v\omega$ term is called the compound supplementary acceleration, or the Coriolis component of acceleration $\inline&space;a_C$.

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:

$\omega&space;=&space;7200&space;\cdot&space;\frac{2\pi}{60}&space;=&space;240\pi&space;\;&space;\frac{rad}{s}$

As the linear velocity can be defined as $\inline&space;v=\omega&space;r$, we obtain the linear velocity of the particle as:

$v&space;=&space;240\pi\cdot&space;2\&space;=&space;1507.2&space;\;\frac{inch}{s}$

or, by converting it in miles per hour:

Solution
$v&space;=&space;85.63&space;\;&space;mph$

### Reference

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