Answer
171.41 rotations per minute
Work Step by Step
We know from Newton's second law that $\tau=I\alpha$, and we know that the moment of inertia of a wheel is given by $I=Mr^2$. Thus, we find:
$\alpha=\frac{\tau}{Mr^2}$
$\alpha=\frac{-\mu_k F_n r}{Mr^2}$
$\alpha=\frac{-\mu_k F_n }{Mr}$
$\alpha=\frac{-(.46)(2.7)}{(1.9)(.33)}=-1.98 \ rads/s^2 = 18.9\ rpm/s$
Thus, it follows:
$\omega_f = \omega_0 +\alpha t$
$\omega_f = 230-(18.9)(3.1)=\fbox{171.41 rotations per minute} $
