# Angular Speed Formula

## Angular Speed Formulas - Rotational Speed Definition & Problems

In circular motion, angular speed is the time rate with which the angular displacement occurs. It is also referred to as the rotational speed.

Angular speed is denoted by $\omega$.
If $\Delta \theta$is the angular displacement in a time $\Delta t$, the angular speed $\omega$is given by:
$\omega = \frac{{\Delta \theta }}{{\Delta t}}$ (This gives the average angular speed over a time $\Delta t$)
The instantaneous angular speed is given by: $\omega = \frac{{d\theta }}{{dt}}$
In the case of uniform circular motion, the average and instantaneous values are equal.
It is expressed in the units of radians per second (or rad/s).
When a particle is in circular motion, it has linear speed as well. The linear speed (v) is related to angular speed $\omega$as:
v = r$\omega$, where, r is the radius of the circular path.

Example:
A particle in uniform circular motion makes an angular displacement of 1.57 rad in half a second. Find its angular speed as well as linear speed if the radius of the circular path is $\frac{1}{\pi }$m.
Solution:
$\Delta \theta = 1.57\,rad,\,\,\Delta t = 0.5\,s,\,\,r = \frac{1}{\pi }\,m$
$\omega = \frac{{\Delta \theta }}{{\Delta t}} = \frac{{1.57}}{{0.5}} = 3.14\,rad/s$
This gives the value of the angular speed.
Now, linear speed: $v = r\omega = \frac{1}{\pi } \times 3.14 = 1\,m/s$

Question:
A vehicle travels at a steady speed on a straight road. Each of its wheel rotates 5 times per second. If each wheel has a diameter of 40 cm, then the angular speed of the wheel and the speed of the car are (approximate) respectively:
Options: