1. **Problem 1:** A car moves at constant speed $12 \text{ m/s}$ around a circular arc of radius $35 \text{ m}$. Find angular speed $\omega$, time to sweep $60^\circ$, arc length in $3.0\text{ s}$, and centripetal acceleration $a_c$. 2. **Angular speed:** $\omega = \frac{v}{r} = \frac{12}{35} = 0.3429 \text{ rad/s}$. 3. **Time to sweep $60^\circ = \frac{\pi}{3}$ radians:** $t = \frac{\theta}{\omega} = \frac{\pi/3}{0.3429} = 3.058 \text{ s}$. 4. **Arc length covered in $3.0\text{ s}$:** $s = r \theta = r \omega t = 35 \times 0.3429 \times 3.0 = 35.99 \text{ m}$. 5. **Centripetal acceleration:** $a_c = \frac{v^2}{r} = \frac{12^2}{35} = 4.114 \text{ m/s}^2$. --- 6. **Problem 2:** Ferris wheel radius $12 \text{ m}$, period $9.0 \text{ s}$. Find $\omega$, linear speed $v$, angular displacement in $5.0 \text{ s}$ (radians and degrees), and $a_c$. 7. **Angular speed:** $\omega = \frac{2\pi}{T} = \frac{2\pi}{9.0} = 0.6981 \text{ rad/s}$. 8. **Linear speed:** $v = r \omega = 12 \times 0.6981 = 8.377 \text{ m/s}$. 9. **Angular displacement in 5.0 s:** $\theta = \omega t = 0.6981 \times 5.0 = 3.4907 \text{ rad}$. 10. **Convert to degrees:** $\theta = 3.4907 \times \frac{180}{\pi} = 200^\circ$. 11. **Centripetal acceleration:** $a_c = r \omega^2 = 12 \times (0.6981)^2 = 5.85 \text{ m/s}^2$. --- 12. **Problem 3:** Ball moves in horizontal circle radius $0.80\text{ m}$ at speed $4.0\text{ m/s}$, angular acceleration $\alpha=0.50 \text{ rad/s}^2$. Find initial $\omega$, period $T$, $a_c$, then new $\omega$, $v$, and tangential acceleration $a_t$ after $6.0 \text{ s}$. 13. **Initial angular speed:** $\omega = \frac{v}{r} = \frac{4.0}{0.80} = 5.0 \text{ rad/s}$. 14. **Period:** $T = \frac{2\pi}{\omega} = \frac{2\pi}{5.0} = 1.2566 \text{ s}$. 15. **Centripetal acceleration:** $a_c = \frac{v^2}{r} = \frac{4.0^2}{0.80} = 20.0 \text{ m/s}^2$. 16. **New angular speed after 6.0 s:** $\omega_{new} = \omega + \alpha t = 5.0 + 0.50 \times 6.0 = 8.0 \text{ rad/s}$. 17. **New linear speed:** $v_{new} = r \omega_{new} = 0.80 \times 8.0 = 6.4 \text{ m/s}$. 18. **Tangential acceleration:** $a_t = r \alpha = 0.80 \times 0.50 = 0.40 \text{ m/s}^2$. --- 19. **Problem 4:** Stone on string length $1.2 \text{ m}$ starts from rest, angular acceleration $1.8 \text{ rad/s}^2$, rotates through $1.25$ revolutions. Find angular speed $\omega$, linear speed $v$, $a_c$. 20. **Convert revolutions to radians:** $\theta = 1.25 \times 2\pi = 7.85398 \text{ rad}$. 21. **Using $\omega^2 = 2\alpha \theta$,** $\omega = \sqrt{2 \times 1.8 \times 7.85398} = \sqrt{28.273} = 5.317 \text{ rad/s}$. 22. **Linear speed:** $v = r \omega = 1.2 \times 5.317 = 6.380 \text{ m/s}$. 23. **Centripetal acceleration:** $a_c = r \omega^2 = 1.2 \times 5.317^2 = 1.2 \times 28.273 = 33.928 \text{ m/s}^2$. --- 24. **Problem 5:** Turntable rotates at $0.80 \text{ rev/s}$, coin distance $0.15 \text{ m}$. Find angular speed in rad/s and rev/min, linear speed $v$, centripetal acceleration $a_c$, angular displacement, arc length in 7.0 s. 25. **Angular speed in rad/s:** $\omega = 0.80 \times 2\pi = 5.0265 \text{ rad/s}$. 26. **Convert rev/s to rev/min:** $0.80 \times 60 = 48 \text{ rev/min}$. 27. **Linear speed:** $v = r \omega = 0.15 \times 5.0265 = 0.754 \text{ m/s}$. 28. **Centripetal acceleration:** $a_c = r \omega^2 = 0.15 \times 5.0265^2 = 3.789 \text{ m/s}^2$. 29. **Angular displacement in 7.0 s:** $\theta = \omega t = 5.0265 \times 7.0 = 35.186 \text{ rad}$. 30. **Arc length:** $s = r \theta = 0.15 \times 35.186 = 5.278 \text{ m}$.