17.12 Going round in circles

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

17-12-picture-cropped

In the picture above, the point P is moving in a circular path around the point O. In a time-interval, Δt, it moves from P to P’, and sweeps out an angle Δθ. (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

ωav = Δθt.

Since we measure Δθ in radians (abbreviated to rad – see post 17.11) and time in seconds (abbreviated to s – post 16.12), we measure ωav in rad.s-1 (rad/s).

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

ω= /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= 2πf, 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 2πr so that v = 2πr/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= dθ/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, ω changes if θ changes with time or if the axis, about which the rotation occurs, moves.

We can then define angular acceleration by

α= dω/dt.

In post 16.13, acceleration, a, was measured in m.s-2 (m/s2); now we measure angular acceleration, α, in rad.s-2 (rad/s2). Be careful to notice that a and α are not the same; α is the Greek letter “alpha”.

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

 

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