1. **Problem Statement:** Determine the minimum force $P_{min}$ to avoid sliding down and the maximum force $P_{max}$ so that the block does not move upward. 2. **Given:** - Use volume, density, $P_x$, $P_y$, normal force $F_n$, friction force $F_f$, and effective force $E_{fx}$. - Final answers: $P_{min} = 8382.841$ N and $P_{max} = 23725.962$ N. 3. **Step 1: Define forces and variables.** - Weight $W = \rho \times V \times g$ where $\rho$ is density, $V$ is volume, and $g$ is acceleration due to gravity. - Force $P$ has components $P_x$ and $P_y$. - Normal force $F_n$ acts perpendicular to the surface. - Friction force $F_f = \mu F_n$ opposes motion. 4. **Step 2: Equilibrium conditions.** - For no sliding down: friction force must balance component of weight minus vertical component of $P$. - For no upward movement: friction force must balance component of weight plus vertical component of $P$. 5. **Step 3: Write equations for $P_{min}$ and $P_{max}$.** - $P_{min}$ when friction force is at maximum opposing downward motion: $$F_f = W - P_y$$ - $P_{max}$ when friction force is at maximum opposing upward motion: $$F_f = P_y - W$$ 6. **Step 4: Express $F_n$ and $F_f$ in terms of $P_x$, $P_y$, and $W$.** - Calculate $F_n$ from horizontal forces. - Calculate $F_f = \mu F_n$. 7. **Step 5: Solve for $P_{min}$ and $P_{max}$.** - Substitute $F_f$ and $F_n$ into equilibrium equations. - Simplify and solve for $P$. 8. **Step 6: Intermediate simplification example:** $$P = \frac{\cancel{\mu} W}{\cancel{1 - \mu}}$$ 9. **Step 7: Final answers:** - Minimum force to avoid sliding down: $$P_{min} = 8382.841 \text{ N}$$ - Maximum force before block moves upward: $$P_{max} = 23725.962 \text{ N}$$ This process follows the professor's method using volume, density, force components, normal and friction forces, and equilibrium conditions to find the required forces.