Physics - Rotation Question with Solution | TestHub

Let the initial angular velocity be ${\omega }_0$.

After $24$ rotations, the angular velocity becomes $\frac{{\omega }_0}{3}$.

Using the third equation of rotational kinematics, ${\omega }^2={\omega }_0^2+2\alpha \theta$:

${\left(\frac{{\omega }_0}{3}\right)}^2={\omega }_0^2+2\alpha (24)$

$\frac{{\omega }_0^2}{9}-{\omega }_0^2=48\alpha$

$-\frac{8{\omega }_0^2}{9}=48\alpha \Rightarrow \alpha =-\frac{{\omega }_0^2}{54}$

Let the fan make ${\theta }^{\prime }$ more rotations before coming to rest. The initial angular velocity for this part is $\frac{{\omega }_0}{3}$ and the final angular velocity is $0$.

$0^2={\left(\frac{{\omega }_0}{3}\right)}^2+2\alpha {\theta }^{\prime }$

$0=\frac{{\omega }_0^2}{9}+2\left(-\frac{{\omega }_0^2}{54}\right){\theta }^{\prime }$

$\frac{{\omega }_0^2}{9}=\frac{{\omega }_0^2}{27}{\theta }^{\prime }$

${\theta }^{\prime }=\frac{27}{9}=3$

The fan will make $3$ more rotations before coming to rest.

Answer: $3$