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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 \theta with respect to time:

\omega = \frac{d\theta}{dt}

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

\alpha = \frac{d\omega}{dt}=\frac{d^2\theta}{dt^2}

The centripetal acceleration can be defined as:

a_c = r\omega^2

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

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

746/img_ang_3.png
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Figure 1
We can then write that:

x = r\cos\theta

y = r\sin\theta

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

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

v_T = \dot y \cos\theta - \dot x \sin\theta

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

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

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

v_T = r\omega

from which the angular velocity becomes:

\omega = \frac{v_T}{r}

Angular Acceleration

In order to define the acceleration, we first have to calculate \ddot x and \ddot y. 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, a_r, which is the component of acceleration in the direction of OP, as:

a_r = \ddot x \cos\theta + \ddot y \sin\theta

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

We can also write (8), by using (5), as:

a_r = \dot v - r\omega^2

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

The tangential component of acceleration on the other hand, a_T, can be written as:

a_T = \ddot y \cos\theta - \ddot x \sin\theta

and, again by replacing \ddot x and \ddot y from equations (6) and (7) respectively, we obtain:

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

Equation (9) can also be written, by taking into account (10), as:

a_T = r \alpha + 2v\omega
where the 2v\omega term is called the compound supplementary acceleration, or the Coriolis component of acceleration 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:

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

As the linear velocity can be defined as v=\omega r, we obtain the linear velocity of the particle as:

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

or, by converting it in miles per hour:

Solution
v = 85.63 \; mph

Reference

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