Question

An exhaust fan is rotating with an angular velocity of $216\text{rad/min}$. When it is switched off, it is observed that the angular retardation of the fan is $\alpha =\frac{3}{2}\sqrt{t}{\text{rad/min}}^2$. The time taken by the fan to stop completely is

• A. $36\text{min}$
• B. $6\text{min}$
• C. $17\text{min}$
• D. $22\text{min}$

Answer 1

The correct option is A $36\text{min}$

Initial angular velocity $\left({\omega }_0\right)=216\text{rad/min}$
When switched off,
Final angular velocity $\left(\omega \right)=0\text{rad/min}$
Given that $\alpha =\frac{3}{2}\sqrt{t}$
Since $\alpha$ is function of time and angular velocity of the fan is decreasing
$\alpha =-\frac{d\omega }{dt}$
$\Rightarrow -\frac{d\omega }{dt}=\frac{3}{2}\sqrt{t}$
$\Rightarrow d\omega =-\frac{3}{2}\sqrt{t}dt$
Integrating on both sides,
${\int }_{{\omega }_0}^{0}d\omega =-{\int }_0^t\frac{3}{2}\sqrt{t}dt$
$\Rightarrow {\omega }_0=\frac{3}{2}{\begin{array}{c}\left(\frac{t^{3/2}}{3/2}\right)\end{array}|}_0^t$
$\Rightarrow {\omega }_0=t^{3/2}$
$\Rightarrow t=({\omega }_0{)}^{2/3}=(216{)}^{2/3}=6^2=36\text{minutes}$