Question

A fan is running at $3000\,rpm$. It is switched off. It comes to rest by uniformly decreasing its angular speed in 10 seconds. Find the total number of revolutions in this period.
A. 150
B. 250
C. 350
D. 300

Answer 1

Hint- To solve this question, we can first use the equation of motion connecting the angular velocity, angular acceleration and time t. From this we will get the value of angular acceleration. Then we can use the second equation of motion connecting the angular velocity, angular acceleration and the number of revolutions. By substituting all the values in this relation, we can find the number of revolutions.

Complete step by step answer:
It is given that the angular velocity of the fan is initially $3000\,rpm$
Let this be denoted as ${\omega _i}$
That is, ${\omega _i} = \dfrac{{3000}}{{60}}revolutions/s$
When it is switched off it comes to rest in 10 seconds.
We need to find the total number of revolutions in this time period.
First of all, let us use the equation of motion connecting the angular velocity, angular acceleration and time t.
${\omega _f} = {\omega _i} + \alpha t$.................(1)
Where ${\omega _i}$ is the initial angular velocity, ${\omega _f}$ is the final angular velocity, $\alpha$ is the angular acceleration, $t$ is the time taken.
Since the fan comes to rest after 10 seconds, we can take the final velocity as zero.
On substituting the values, we get the above equation as
$0 = \dfrac{{3000}}{{60}} + \alpha \times 10$
From this we get the value of angular acceleration as,
$\alpha = - 5\,revolutions/{s^2}$
Now let us use another equation of motion connecting the angular velocity, angular acceleration and the number of revolutions.
${\omega _f}^2 = {\omega _i}^2 + 2\alpha \theta$.....................(2)
Where $\theta$ denotes the number of revolutions.
On substituting the values in the equation 2, we get
$0 = {\left( {\dfrac{{3000}}{{60}}} \right)^2} + 2 \times - 5 \times \theta$
On solving for $\theta$ ,we get
$\theta = 250\,revolutions$
This is the number of revolutions in 10 seconds.

So, the correct answer is option B.

Note: The equations of motion should be used in solving kinematic problems only when the acceleration is constant. In the question it is given that the angular speed decreases uniformly while coming to rest. This means there is a constant angular acceleration. When the acceleration keeps varying then the situation will be more complex so we cannot use these equations of motion.