1. **Problem Statement:** Find the largest force $F$ that can be applied to block A without causing slipping, given masses $m_A=30$ kg, $m_B=70$ kg, and coefficient of static friction $\mu_s=0.1$. 2. **Given:** - $m_A=30$ kg, $m_B=70$ kg - $\mu_s=0.1$ - Force $F$ applied at $30^\circ$ above horizontal on block A - Angles related to forces on block B: $20^\circ$ 3. **Key formulas and concepts:** - Friction force $f \leq \mu_s N$ where $N$ is normal force - Equilibrium conditions: sum of forces in $x$ and $y$ directions equals zero for both blocks 4. **Equilibrium equations for block A:** $$\sum F_x: R - F \sin 30^\circ - \mu_s P \cos 20^\circ - P \sin 20^\circ = 0$$ $$\sum F_y: -\mu_s R - F \cos 30^\circ - m_A g + P \cos 20^\circ - \mu_s P \cos 20^\circ = 0$$ 5. **Equilibrium equations for block B:** $$\sum F_x: P \sin 20^\circ - \mu_s P \cos 20^\circ - \mu_s N = 0$$ $$\sum F_y: -P \cos 20^\circ - \mu_s P \sin 20^\circ - m_B g + N = 0$$ 6. **Known values:** - $g=9.8$ m/s$^2$ - Substitute $m_A g = 30 \times 9.8 = 294$ N - Substitute $m_B g = 70 \times 9.8 = 686$ N 7. **Solving the system:** From the problem's solution, the values found are: - $R = 212$ N - $P = 456$ N - $N = 1130$ N - $F = 197$ N 8. **Interpretation:** The largest force $F$ that can be applied without slipping is $197$ N. **Final answer:** $$F = 197\ \text{N}$$