1. **Find the angular velocity of a wheel in radians per second spinning at 350 revolutions per minute.** - Given: $\text{Revolutions per minute} = 350$ rpm. - We want angular velocity in radians per second ($\omega$). - Recall: One revolution equals $2\pi$ radians. - Convert rpm to radians per second: $$\omega = 350 \times \frac{2\pi}{60}$$ - Calculate: $$\omega = 350 \times \frac{2\pi}{60} = \frac{700\pi}{60} = \frac{35\pi}{3} \approx 36.65 \text{ rad/s}$$ 2. **Calculate the rotational inertia of a thick ring.** - Given: - Mass, $m = 30$ kg - Outer radius, $R = 22.5$ cm = 0.225 m - Inner radius, $r = 7.5$ cm = 0.075 m - For a thick ring (cylindrical shell with inner and outer radii), rotational inertia is: $$I = \frac{1}{2} m (R^2 + r^2)$$ - Substitute values: $$I = \frac{1}{2} \times 30 \times (0.225^2 + 0.075^2)$$ $$I = 15 \times (0.050625 + 0.005625) = 15 \times 0.05625 = 0.84375 \text{ kg} \cdot \text{m}^2$$ **Final answers:** - Angular velocity $\approx 36.65$ rad/s - Rotational inertia $= 0.84375$ kg·m$^2$