Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_172

⚡ Mecademy AIENG정역학 · ch5  Problem Statement The hydraulic cylinder operates the toggle which closes the vertical gate against the pressure of fresh water on the opposite side. The gate is rectangular with a horizontal width of 2 m perpendicular to the paper. For a depth h = 3 m of water, calculate the required oil pressure p which acts on the 150-mm-diameter piston of the hydraulic cylinder. Problem 5/172 (a) Determination of required oil pressure 1. Formula: Hydrostatic force on the gate: Moment equilibrium about hinge : Toggle force relationship: (where ) Required pressure: 2. Substitution: Water density , gravity , width , depth Total gate height Distance of toggle from , Distance of resultant water force from : Toggle link geometry: horizontal distance , vertical distance , so Piston diameter 3. Calculation: From : R=ρgA=h ˉ ρghw 2 12 AM =∑ A 0⟹F d = LL Rd R F = cyl 2F tanθ L tanθ= horizontal distance vertical distance p= = A p F cyl πd/4 2 F cyl ρ= 1000 kg/m 3 g= 9.81 m/s 2 w=2 m h=3 m H=2 m+2 m=4 m Ad = L 2 m RAd = R H− = 3 h 4− = 3 3 3 m =1 m=2 m tanθ= = 1 2 2 d=150 mm=0.15 m R= (1000)(9.81)(3)(2)= 2 1 2 88,290 N=88.29 kN M =∑ A 0F (2)− L R(3)=0⟹F = L 1.5R= 1.5(88.29)=132.435 kN Required cylinder force: Piston area: Required pressure: 4. Result: ● Final Conclusion: The required oil pressure to keep the gate closed against a 3-m water depth is 29.98 MPa. ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/172 F = cyl 2F (2)= L 4F = L 4(132.435)= 529.74 kN A