A wheel turns through 1800rev in 1.0min. Find; a.) Its average angular velocity in rad/s. b.) Through how many radians does the wheel turn in 15 seconds.

We know that $1 \text{ revolution} = 2\pi \text{ radians}$.

$$\Delta \theta = 1800 \text{ rev} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} = 3600\pi \text{ rad}$$

We know that $1 \text{ min} = 60 \text{ s}$.

$$\Delta t = 1.0 \text{ min} \times \frac{60 \text{ s}}{1 \text{ min}} = 60 \text{ s}$$

The formula for average angular velocity is $\omega = \frac{\Delta \theta}{\Delta t}$.

$$\omega = \frac{3600\pi \text{ rad}}{60 \text{ s}} = 60\pi \text{ rad/s}$$ $$\omega \approx 60 \times 3.14159 \text{ rad/s} \approx 188.495 \text{ rad/s}$$

The average angular velocity is $\boxed{60\pi \text{ rad/s}}$ or approximately $\boxed{188.5 \text{ rad/s}}$.

We have $\omega = 60\pi \text{ rad/s}$ and $\Delta t' = 15 \text{ s}$.
The formula for angular displacement is $\Delta \theta = \omega \times \Delta t'$.

$$\Delta \theta = (60\pi \text{ rad/s}) \times (15 \text{ s})$$

$$\Delta \theta = 900\pi \text{ rad}$$ $$\Delta \theta \approx 900 \times 3.14159 \text{ rad} \approx 2827.43 \text{ rad}$$

The wheel turns through $\boxed{900\pi \text{ rad}}$ or approximately $\boxed{2827 \text{ rad}}$ in 15 seconds.