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Question

(A) Let the initial angular velocity be $\omega_0$ and the final angular velocity be $\omega$. The equation of rotational motion is $\omega^2 = \omega

Vedclass · Accepted Answer

$3$

Vedclass · Answer

(A) Let the initial angular velocity be $\omega_0$ and the final angular velocity be $\omega$. The equation of rotational motion is $\omega^2 = \omega_0^2 + 2\alpha	heta$. For the first part,the velocity becomes $\frac{\omega_0}{3}$ after $	heta_1 = 24$ rotations. $(\frac{\omega_0}{3})^2 = \omega_0^2 + 2\alpha(24 	imes 2\pi) \implies \frac{\omega_0^2}{9} - \omega_0^2 = 48\pi\alpha \implies -\frac{8}{9}\omega_0^2 = 48\pi\alpha \implies \alpha = -\frac{\omega_0^2}{54\pi}$. Now,for the motion from $\frac{\omega_0}{3}$ to $0$ (rest),let the additional rotations be $	heta_2$. $0^2 = (\frac{\omega_0}{3})^2 + 2\alpha(	heta_2 	imes 2\pi) \implies 0 = \frac{\omega_0^2}{9} + 2(-\frac{\omega_0^2}{54\pi})(2\pi	heta_2)$. $0 = \frac{\omega_0^2}{9} - \frac{\omega_0^2}{13.5}	heta_2 \implies \frac{\omega_0^2}{9} = \frac{\omega_0^2}{13.5}	heta_2 \implies 	heta_2 = \frac{13.5}{9} = 3$. Thus,the fan makes $3$ more rotations.

When a ceiling fan is switched off,its angular velocity falls to $\frac{1}{3}$ of its initial value while it makes $24$ rotations. How many more rotations will it make before coming to rest?

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