Give the definition of: (a) the average angular velocity

Give the definition of: (a) the average angular velocity of a point rotating about a fixed point, (b) the instantaneous angular velocity of a point.

Step 1: Setting up the notation

Let a point move along a circular arc about a fixed centre. Let

\[ \varphi = \varphi(t) \]

denote the angle, measured in radians from a fixed reference direction, that has been swept out by the time $t$.

Step 2: Part (a): Average angular velocity

Over the time interval from $t$ to $t + \Delta t$ the point sweeps through the angular increment

\[ \Delta\varphi = \varphi(t + \Delta t) - \varphi(t). \]

The average angular velocity over this interval is defined as the ratio of the angular displacement to the elapsed time:

\[ \omega_{\text{avg}} = \frac{\Delta\varphi}{\Delta t}. \]

It is measured in radians per second (rad/s) and describes how fast the point rotates on average over the interval $[t,\; t+\Delta t]$.

Step 3: Part (b): Instantaneous angular velocity

The instantaneous angular velocity (usually called simply the angular velocity) at the instant $t$ is the limit of the average angular velocity as the time increment tends to zero:

\[ \omega = \lim_{\Delta t \to 0} \frac{\Delta\varphi}{\Delta t} = \frac{d\varphi}{dt} = \dot{\varphi}. \]

In other words, $\omega$ is the derivative of the angular displacement with respect to time. It describes the rate of rotation of the point about the fixed centre at the single moment $t$.

Step 4: Illustration

For uniform rotation $\varphi(t) = \omega_0 t$ the two quantities coincide: $\omega_{\text{avg}} = \frac{\omega_0(t+\Delta t) - \omega_0 t}{\Delta t} = \omega_0$.

\[\omega = \frac{d}{dt}\bigl(\omega_0 t\bigr) = \omega_0 \;\;\checkmark\]