*Before you read this, I suggest you read* posts 17.4 and 17.11.

In the picture above, the point P is moving in a circular path around the point O. In a time-interval, Δ*t*, it moves from P to P’, and sweeps out an angle Δ*θ*. (In this post the symbol Δ, the Greek letter capital “delta”, is used to mean a change in something, here angle or time, as in post 16.12). We define the *average angular speed* of P, in this time-interval, as

*ω _{av}* = Δ

*θ*/Δ

*t*.

Since we measure Δ*θ* in radians (abbreviated to rad – see post 17.11) and time in seconds (abbreviated to s – post 16.12), we measure *ω** _{av}* in rad.s

^{-1}(rad/s).

At an instant in time, we can define the *true angular speed* as

*ω*= *dθ*/*dt*

where *d/dt* denotes the rate of change of something (in this case *θ*) with time, as explained in post 17.4. If *ω* doesn’t change with time, we say that P has a *constant angular speed*.

The time *T*, for P to perform a complete revolution of the circle, is sometimes called its *time-period*. In this time, it sweeps out 2π radians, so that, if P has a constant angular speed, its value is *ω*= 2π/*T*. In post 16.14, we saw that the number of times P makes a complete revolution in unit time is called the frequency of revolution and is given by *f* = 1/*T*; if *T* is measured in seconds, *f* is measured in Hz (post 16.14). Since *ω*= 2π*f*, it’s sometimes called *angular frequency*; it’s still measured in rad.s^{-1}. The speed, *v*, of P is the distance it travels divided by the time it takes (see post 16.12). So, thinking about a complete revolution of the circle, the distance travelled in time *T* is 2π*r* so that *v* = 2π*r*/*T* = *ωr*. This relationship is useful because it enables us to convert between angular and ordinary (translational) speed.

If *ω* changes with time, P is undergoing an *angular acceleration*. But to understand angular acceleration properly, we first have to use the idea of angular displacement (post 17.11).

To define angulation acceleration, we first define *angular velocity* as

** ω**=

*d*/

**θ***dt.*

Here the direction of the vector ** θ** is the direction of the axis around which P rotates in an anticlockwise direction – as explained in post 17.11; its modulus (see post 17.2) is the angle

*θ*. So,

**changes if**

*ω**θ*changes with time or if the axis, about which the rotation occurs, moves.

We can then define *angular acceleration* by

** α**=

*d*/

**ω***dt.*

In post 16.13, acceleration, ** a**, was measured in m.s

^{-2}(m/s

^{2}); now we measure angular acceleration,

**, in rad.s**

*α*^{-2}(rad/s

^{2}). Be careful to notice that

**and**

*a***are not the same;**

*α***is the Greek letter “alpha”.**

*α*The purpose of this post was to define a series of concepts that will be used, in later posts, to explain more about angular motion. When we start to use these concepts, their purpose should become much clearer.

*Related posts*

17.11 Measuring angles

17.10 Torque