Answer to Question #270860 in Physics for Kai

Question

The fan of a CPU starts decelerating at a constant rate from an angular speed of 897 rpm. The fan has undergone 1856 revolutions before it comes to rest. How long (in s) did it take the CPU fan to stop rotating?

Answer 1

Let's first convert rev/min to rad/s:

$$\omega_i=897\ \dfrac{rev}{min}\times\dfrac{1\ min}{60\ s}\times\dfrac{2\pi\ rad}{1\ rev}=94\ \dfrac{rad}{s}.$$

Then, we can find the angular displacement of the CPU fan:

$$\theta=2\pi\ \dfrac{rad}{rev} n=2\pi\ \dfrac{rad}{rev}\times1856\ rev=11662\ rad.$$

Let's find the angular deceleration of the CPU fan from the kinematic equation:

$$\omega_f^2=\omega_i^2+2\alpha \theta,$$ $$0=\omega_i^2+2\alpha \theta,$$ $$\alpha=-\dfrac{\omega_i^2}{2\theta}=-\dfrac{(94\ \dfrac{rad}{s})^2}{2\times11662\ rad}=-0.38\ \dfrac{rad}{s^2}.$$

Finally, we can find the time that the CPU fan takes to stop rotating:

$$t=\dfrac{\omega_f-\omega_i}{\alpha}=\dfrac{0-94\ \dfrac{rad}{s}}{-0.38\ \dfrac{rad}{s^2}}=247\ s.$$