A coin lies 0.12 m from the axis of a turntable spinning at 45 revolutions per minute…

Step 1. Angular velocity. $\omega = 45 \times \dfrac{2\pi\ \text{rad}}{60\ \text{s}} = 4.71\ \text{rad/s}$.

Step 2. (a) $\mu_s \ge \dfrac{\omega^2r}{g} = \dfrac{22.2\ \text{rad}^2/\text{s}^2 \times 0.12\ \text{m}}{9.8\ \text{m/s}^2} = 0.272$.

Step 3. (b) $r \le \dfrac{\mu_s g}{\omega^2} = \dfrac{0.40 \times 9.8\ \text{m/s}^2}{22.2\ \text{rad}^2/\text{s}^2} = 0.177\ \text{m}$.