Question

The angular velocity of a wheel increases from 100 rps to 300 rps in 10 s. What is the number of revolutions made during this time?
(A) 600
(B) 1500
(C) 1000
(D) 2000

Answer 1

Hint
Angular acceleration is the change in angular velocity over a period of time. When used with correct units this parameter can give us information about the distance travelled or revolutions covered in a certain period of time.
Formula used: $ \dfrac{{\Delta \omega }}{t} $ , where $ \Delta \omega $ is the change in angular velocity and $ t $ is time covered.

Complete step by step answer
As the name suggests, angular velocity is the change in the angular position of an object per unit time. For full rotations in a circular motion, the units of angular velocity are generally taken as rotation per second or rps. Multiplying this velocity with a factor of radius gives us the linear velocity of the object.
In this question, we are provided with a wheel undergoing circular motion. We are required to find the revolutions (or rotations) made by it after some time. The data given to us is:
Initial angular velocity $ {\omega _1} = 100{\text{ rps}} $
Final angular velocity $ {\omega _2} = 300{\text{ rps}} $
Time taken $ t = 10s $
We keep the units in rps because we are asked to find the rotations. We know that the angular acceleration is given as:
 $ a = \dfrac{{\Delta \omega }}{t} = \dfrac{{{\omega _2} - {\omega _1}}}{t} $
Putting the values in this gives us:
 $ a = \dfrac{{300 - 100}}{{10}} = \dfrac{{200}}{{10}} = 20 $
The unit for this acceleration is $ {\text{rotations/}}{s^2} $ . As we need the number of rotations, we multiply this acceleration with $ {t^2} $ so that the units of numerator and denominator cancel out and we get the revolutions as
 $ a \times {t^2} = 20 \times {10^2} $
This implies that the number of rotations is 2000.
Hence, the correct answer is option (D).

Note
Simple examples of angular velocity include the hands of a clock. Be it the minutes hand or the hours hand, these cover small fixed angular distance as time passes by, and hence possess an angular speed. Matching the angular speed with the standard clocks is what provides the same time to all of us.