Question

When a ceiling fan is switched off, its angular velocity reduces by $50$% while it makes $36$ rotations. How many more rotations will it make before coming to rest? (Assume uniforms angular retardation)

• A. $48$
• B. $36$
• C. $12$
• D. $18$

Answer 1

The correct option is C $12$

$\begin{array}{c}Youhavetousetheequation,\\ {\omega }^2={{\omega }_0}^2+2\alpha \theta forfindingtheangularacceleration\alpha and\\ hencethenumberoffurtherrotations.\\ Notethatthisequationistherotationalanalogueoftheequation\\ v^2={v_0}^2+2as(or,v^2=u^2+2as)inlinearmotion.\\ Sincetheangularvelocityhasreducetohalfoftheinitialvalue{\omega }_{0}after36rotations,wehave\\ {{\left({\omega }_0/2\right)}_{}}^2={{\omega }_0}^2+2\alpha \times 36fromwhich\alpha =--{{\omega }_0}^2/96\\ \left[Wehaveexpressedtheangulardisplacement\theta inrotationsitselfforconvenience\right]\\ Iftheadditionalnumberofrotationsisx,wehave\\ 0={{\left({\omega }_0/2\right)}_{}}^2+2\alpha x={{\left({\omega }_0/2\right)}_{}}^2+2\times (--{{\omega }_0}^2/96)x\\ Thisgives,\\ x=12\end{array}$

Hence,

option $\left(C\right)$ is correct answer.