Question

A fan is running at $3000\hspace{0.17em}\text{Rpm}$. It is switched off. It comes to rest by uniformly decreasing its angular speed in $10\hspace{0.17em}\text{Sec}$. The total number of revolutions in this period.

Answer 1

Step 1: Given:

Initial angular velocity ${\omega }_i=3000\text{Rpm}$

Final angular velocity ${\omega }_f=0$

Time $t=10\text{sec}$

Step 2: To Find:

Total number of revolutions

Step 3: Calculate angular acceleration $\left(\alpha \right)$:

.$$\begin{gathered} {\omega }_f={\omega }_i+\alpha t \\ 0=\frac{3000}{60}+\alpha \times 10 \\ \alpha =-5\frac{\text{rev.}}{{\text{sec}}^2} \end{gathered}$$

Step 4: Calculate the number of revolutions $\left(\theta \right)$:

$$\begin{gathered} {{\omega }_f}^2={{\omega }_i}^2+2\alpha \theta \\ {0}^2={\left(\frac{3000}{60}\right)}^2+2\times \left(-5\frac{\text{rev}}{{\text{sec}}^2}\right)\times \theta \\ \theta =250\text{rev} \end{gathered}$$

Hence, the number of revolutions is $250\text{rev}$.