When brake is applied, a uniformly retarded wheel reduces its velocity from 300 rpm to 180 rpm during one minute. Find the angular acceleration of the wheel and the number of revolutions it completes in this time.

Solution

Given,
Initial speed, $straight omega subscript straight o equals 300 space rpm space equals space 10 space straight pi space rad divided by straight s$
Attained speed, $straight omega equals 180 space rpm space equals space 6 straight pi space rad divided by straight s$
Time of travel, $straight t space equals space 1 space min space equals space 60 straight s$
Now, using the formula,
   $straight omega minus straight omega subscript straight o equals alpha t$
or    $straight alpha equals fraction numerator straight omega minus straight omega subscript straight o over denominator straight t end fraction$
∴ $straight alpha equals fraction numerator 6 straight pi minus 10 straight pi over denominator 60 end fraction equals negative fraction numerator 4 straight pi over denominator 60 end fraction space rad divided by straight s squared$
   $equals negative straight pi over 15 rad divided by straight s squared$
Also, we have $straight omega squared minus straight omega subscript straight o superscript 2 equals 2 αθ$
That is,
   $straight theta equals fraction numerator straight omega squared minus straight omega subscript straight o superscript 2 over denominator 2 straight alpha end fraction space space equals fraction numerator left parenthesis 6 straight pi right parenthesis squared minus left parenthesis 10 straight pi right parenthesis squared over denominator 2 open parentheses begin display style fraction numerator negative straight pi over denominator 15 end fraction end style close parentheses end fraction$
   $equals fraction numerator negative 64 straight pi squared over denominator negative 2 straight pi divided by 15 end fraction equals 480 straight pi space rad$