Answer
(a) $v = 1.5~m/s$
(b) The fan turns through 12.5 revolutions while stopping.
Work Step by Step
(a) Initially, the fan rotates at 60 rpm which is 1 rev/s which is $2\pi~rad/s$. Therefore $\omega_0 = 2\pi~rad/s$.
We can find the rate of angular deceleration as the fan comes to a stop.
$\alpha = \frac{\omega_f-\omega_0}{t}$
$\alpha = \frac{0-2\pi~rad/s}{25~s}$
$\alpha = -0.251~rad/s^2$
We then find the angular velocity after 10 seconds.
$\omega = \omega_0+at$
$\omega = 2\pi~rad/s-(0.251~rad/s^2)(10~s)$
$\omega = 3.77~rad/s$
We then find the speed of the tip of the blade.
$v = \omega~r$
$v = (3.77~rad/s)(0.40~m)$
$v = 1.5~m/s$
(b) We first find $\theta$ while the fan is stopping;
$\theta = \omega_0~t+\frac{1}{2}\alpha~t^2$
$\theta = (2\pi~rad/s)(25~s)-\frac{1}{2}(0.251~rad/s^2)(25~s)^2$
$\theta = 78.6~rad$
We can use $\theta$ to find the number of revolutions $N$.
$N = (78.6~rad)(\frac{1~rev}{2\pi~rad})$
$N = 12.5~rev$
The fan turns through 12.5 revolutions while stopping.
