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: (a) 31.4 rad/s, 45.2 km/h (b) 31.4 rad/s, 22.6 km/h (c) 15.7 rad/s, 22.6 km/h (d) 15.7 rad/s, 11.3 km/h Answer: (b)