A fan rotates with a velocity corresponding to a frequency of 900 rev/min. When its motor is switched off, the fan uniformly slows down and performs 75 revolutions before it comes to a stop. How much time elapsed from the moment the fan was switched off to the moment it stopped?

Solution

Here,
$straight omega subscript straight o space equals space 900 space rpm space equals space 30 straight pi space rad divided by straight s$
$straight omega space equals space 0 space rad divided by straight s comma$
$straight theta equals 75 space rev space equals space 150 straight pi space rad$
We have the formula,
   $straight omega squared minus straight omega subscript straight o superscript 2 equals 2 αθ$
That is,
   $straight alpha equals fraction numerator straight omega squared minus straight omega subscript straight o superscript 2 over denominator 2 straight theta end fraction$
∴    $straight alpha equals fraction numerator left parenthesis 0 right parenthesis squared minus left parenthesis 30 straight pi right parenthesis squared over denominator 2 cross times 150 straight pi end fraction equals negative 3 straight pi space rad divided by straight s squared$
Now $straight omega minus straight omega subscript straight o space equals space αt$
$rightwards double arrow$    $space space space space space space space space space space space space space space space space straight t equals fraction numerator straight omega minus straight omega subscript straight o over denominator straight alpha end fraction$
∴ $straight t equals fraction numerator 0 minus 30 straight pi over denominator negative 3 straight pi end fraction equals 10 straight s$
So, 10 seconds elapsed from the moment the fan was switched off to the moment it stopeed.