Question

A ceiling fan is rotating about its own axis with uniform angular velocity $\omega$. The electric current is switched off then due to constant opposing torque its angular velocity is reduced to $\frac{2\omega }{3}$ as it completes $30$ rotations. The number of rotations further it makes before coming to rest is :

• A. $18$
• B. $12$
• C. $9$
• D. $24$

Answer 1

Answer: (D)

$24$

${\omega }_1^2-{\omega }_2^2=2\alpha \theta$ (governing equation as per question)
$\therefore {\omega }^2-{\left(\frac{2\omega }{3}\right)}^2=2\times \alpha \times 30$
$\frac{5{\omega }^2}{9}=60\alpha$
similarly let the no of rotations further made be $n$
${\left(\frac{2\omega }{3}\right)}^2=2\alpha n$
$\frac{4{\omega }^2}{9}=2n\alpha$

$\Rightarrow \frac{5}{4}=\frac{30}{n}$

$n=24$