When a ceiling fan is switched off, its angular velocity falls to half while it makes 36 rotations. How many more rotations will it make before coming to rest?

From third equation of angular motion,
\( \begin{array}{l} \quad \omega^{2}=\omega_{0}^{2}-2 \alpha \theta \\ {\left[\text {Here, } \omega=\frac{\omega_{0}}{2}, \theta=36 \times 2 \pi\right]} \\ \therefore\left(\frac{\omega_{0}}{2}\right)^{2}=\omega_{0}^{2}-2 \alpha \times 36 \times 2 \pi \\ \text { or } \quad 4 \times 36 \pi \alpha=\omega_{0}^{2}-\frac{\omega_{0}^{2}}{4} \end{array} \)
or
\(4 \times 36 \pi \alpha=\frac{3 \omega_{0}^{2}}{4}\)
or \(\alpha=\frac{\omega_{0}^{2}}{16 \times 12 \pi}\) ...(i)
According to question again applying the third equation of angular motion
\(\omega^{2}=\omega_{0}^{2}-2 \alpha \theta\)
[Here, \(\omega=0\) ]
\(0=\left(\frac{\omega_{0}}{2}\right)^{2}-2 \times \frac{\omega_{0}^{2} \cdot \theta}{16 \times 12 \pi}\)
or \(\theta=24 \pi \quad\) or \(\quad \theta=12 \times 2 \pi\)
\(\begin{array}{lc}\text { But } & 2 \pi=1 \text { cycle } \\ \text { So, } & \theta=12 \text { cycle }\end{array}\)