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Understanding Angular Velocity
To understand angular velocity, it is helpful to first introduce two concepts:
 * $$\Delta$$, read as Delta, an operator shorthand for "difference" or "change in."
 * Velocity in one dimension.

The Delta Operator
The capital (Greek letter) 'Delta', $$\Delta$$, represents "difference". When used in this way, $$\Delta$$ is not a number. Nor is it a variable that represents a number. For example, $$x$$ is a variable that often is used to represent the distance from zero on a number line. If we have two different positions on a number line, $$x_1$$ and $$x_2$$, then we might, for example, define:


 * $$\Delta x = x_2 - x_1$$

It is convention to put the larger subscript first. One way to define $$\Delta$$ is to say it is shorthand for 'take the difference of'. Such entities are sometimes called 'operators'. As a trivial example, one can use this operator to define energy conservation. Let us take E to be the sum of kinetic and potential energy, given by


 * $$E=K+U$$.

The expression,


 * $$E_2 = E_1$$

could be instead written,


 * $$\Delta E = 0$$, where $$\Delta E \equiv E_2 - E_1$$.

We take the 'triple' equal sign to mean 'defined as', as in let us agree to use this variable to signify that expression. Used in this way, the symbol is not a statement of fact, but a convenient way to label an expression. Essentially, the Delta symbol allows us to more quickly and succinctly write our expressions.

Velocity and Acceleration in One Dimension
As another example of using the Delta operator, consider velocity. Velocity is the rate at which position changes. Acceleration is the rate at which velocity changes. If the time interval is not infinitesimally small, we refer to these as "average" rates. The average velocity or acceleration is often denoted by a bar above. Alternatives to $$\bar{v}$$ to are the brakcet $$$$ and the subscript $$v_\text{ave}$$.


 * $$\bar{v} = \frac{\Delta x}{\Delta t} = \frac{x_f-x_i}{t_f-t_i}$$,        $$\bar a  = \frac{\Delta v}{\Delta t}=\frac{v_f-v_i}{t_f-t_i}$$.

Instantaneous velocity and acceleration are derivatives, $$ v(t) = dx/dt$$,  $$ a(t)=dv/dt=d^2x/dt^2$$, and occur in the limit that $$\Delta x$$ and $$\Delta x$$ are small.

Example
A driver gets on mile 25 of a freeway at 3:30 pm and exits at mile 150 at 5:30 pm. If the road is straight, what is the velocity and is it average or instantaneous?

Solution:


 * $$ v_{\text{average}} = \frac{\Delta x}{\Delta t}=\frac{x_f-x_i}{t_f-t_i}=\frac{150-25}{5.5-3.5}=\frac{125}{2}= 62.5\frac{\text{miles}}{\text{hour}}$$

It is an average velocity because the time interval is not infinitesimally small. (In physics we like to be precise and call it velocity and not speed because if the person was going in the opposite direction, the result would have been negative. Velocity has direction as a property, speed does not.

Rotational Motion About a Fixed Axis


Angular displacement may be measured in radians or degrees. If using radians, it provides a very simple relationship between distance traveled around the circle and the distance r from the center.

A particle moves in a circle of radius $$r$$. Having moved an arc length $$s$$, its angular position is $$\theta$$ relative to its original position, where $$\theta=\frac{s}{r}$$.

In mathematics and physics it is usual to use the natural unit radians rather than degrees or revolutions. Units are converted as follows:


 * $$1 \mathrm{\ rev} = 360^{\circ} = 2\pi \mathrm{\ rad}$$

An angular displacement is described as


 * $$ \Delta \theta = \theta_{2} - \theta_{1}, \!$$

The relation between radius, r, at speed, v, and period, T, can be understood by assuming constant angular velocity.


 * $$vT=2\pi r$$.

Angular frequency, ω, can now be defined by the relation,


 * $$\omega T=2\pi$$,

so that,


 * $$v=\omega r$$.

Angular Velocity
Angular velocity is the change in angular displacement per unit time. The symbol for angular velocity is $$\omega$$ and the units are typically rad s−1. Angular speed is the magnitude of angular velocity.


 * $$\overline{\omega} = \frac{\Delta \theta}{\Delta t} = \frac{\theta_2 - \theta_1}{t_2 - t_1}.$$.