1. **Problem statement:** We have a block of mass 1500 kg on a 40° incline with a downward vertical force of 2500 N acting on it. The coefficient of friction is 0.25. We want to find: A) The minimum force $P$ applied up the incline to prevent the block from sliding down. B) The maximum force $P$ applied up the incline so the block does not move upward. 2. **Known values:** - Mass $m = 1500$ kg - Gravity $g = 9.8$ m/s² (assumed) - Weight $W = mg = 1500 \times 9.8 = 14700$ N - Vertical force $F_v = 2500$ N downward - Incline angle $\theta = 40^\circ$ - Coefficient of friction $\mu = 0.25$ 3. **Step 1: Resolve forces and find normal force $F_N$ and friction force $F_f$** - Weight components along and perpendicular to incline: $$W_x = W \sin \theta = 14700 \times \sin 40^\circ = 14700 \times 0.6428 = 9452.16\,N$$ $$W_y = W \cos \theta = 14700 \times \cos 40^\circ = 14700 \times 0.7660 = 11260.2\,N$$ - Vertical force $F_v$ components along and perpendicular to incline: $$F_{v_x} = F_v \sin \theta = 2500 \times 0.6428 = 1607\,N$$ $$F_{v_y} = F_v \cos \theta = 2500 \times 0.7660 = 1915\,N$$ - Total normal force $F_N$ is sum of perpendicular components: $$F_N = W_y + F_{v_y} = 11260.2 + 1915 = 13175.2\,N$$ - Friction force magnitude: $$F_f = \mu F_N = 0.25 \times 13175.2 = 3293.8\,N$$ 4. **Step 2: Forces along the incline** - Downhill forces (trying to slide down): $$F_{down} = W_x + F_{v_x} = 9452.16 + 1607 = 11059.16\,N$$ - Friction force acts uphill to resist sliding down, so friction force $F_f$ acts uphill. - Force $P$ acts uphill (unknown magnitude). 5. **Step 3: Equilibrium conditions** - To prevent sliding down (minimum $P$): $$P + F_f = F_{down}$$ $$P = F_{down} - F_f = 11059.16 - 3293.8 = 7715.36\,N$$ - To prevent moving upward (maximum $P$): friction acts downhill resisting upward motion, so friction force reverses direction. - Uphill forces: $$P + F_f$$ - Downhill forces: $$F_{down}$$ - For maximum $P$ without moving up: $$P - F_f = F_{down}$$ $$P = F_{down} + F_f = 11059.16 + 3293.8 = 14352.96\,N$$ 6. **Final answers:** A) Minimum force $P$ to avoid sliding down: $$\boxed{7715.36\,N}$$ B) Maximum force $P$ so block does not move upward: $$\boxed{14352.96\,N}$$