1. **State the problem:** Find the period $T$ of oscillation of a mass-spring system with amplitude $A=0.1225$ m, maximum speed $V_{max}=5.13$ m/s, and spring constant $k=5.03$ N/m. 2. **Recall the formulas:** - Maximum speed formula: $$V_{max} = \sqrt{\frac{k A^2}{m}}$$ - Period formula: $$T = 2\pi \sqrt{\frac{m}{k}}$$ 3. **Find the mass $m$ using the maximum speed formula:** $$V_{max} = \sqrt{\frac{k A^2}{m}} \implies V_{max}^2 = \frac{k A^2}{m} \implies m = \frac{k A^2}{V_{max}^2}$$ 4. **Substitute known values:** $$m = \frac{5.03 \times (0.1225)^2}{(5.13)^2} = \frac{5.03 \times 0.01500625}{26.3169}$$ 5. **Calculate numerator and denominator:** $$5.03 \times 0.01500625 = 0.07553$$ 6. **Calculate mass:** $$m = \frac{0.07553}{26.3169} \approx 0.00287 \text{ kg}$$ 7. **Calculate the period $T$ using the period formula:** $$T = 2\pi \sqrt{\frac{m}{k}} = 2\pi \sqrt{\frac{0.00287}{5.03}}$$ 8. **Calculate inside the square root:** $$\frac{0.00287}{5.03} \approx 0.000570$$ 9. **Calculate the square root:** $$\sqrt{0.000570} \approx 0.02387$$ 10. **Calculate the period:** $$T = 2\pi \times 0.02387 \approx 0.150 \text{ seconds}$$ **Final answer:** The period of oscillation is approximately $0.15$ seconds.