Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_166

⚡ Mecademy AIENG정역학 · ch5  Problem Statement The gate is held in the vertical position against the action of the body of fresh water by a counterweight of mass . If the width of the gate is and the mass of the gate is , determine the required value of and the magnitude of the pin reaction at . Problem 5/166 (a) Required value of the counterweight mass 1. Formula: The resultant hydrostatic force on a rectangular vertical gate of width submerged to a depth is: The line of action of this force (center of pressure) is at a distance from the base (point ). Moment equilibrium about the hinge at : where is the cable tension and is the total height of the gate. 2. Substitution: Given values: , , , , and . First, calculate : Calculate : Substitute into the moment equation: 3. Calculation: m5 m2500 kg mA m F w bh F = w ρ g b h 2 1 2 y= p 3 h AA M = ∑ A 0⟹T⋅H−F ⋅ w y = p 0 T=mgH ρ= 1000 kg/m 3 g= 9.81 m/s 2 b=5 m h=3.5 mH=3.5+1.5=5.0 mF w F = w (1000)(9.81)(5)(3.5) 2 1 2 y p y = p m 3 3.5 (m⋅9.81)⋅5.0=F ⋅ w ( 3 3.5 ) F = w 0.5⋅9.81⋅5000⋅12.25=300431.25 N y ≈ p 1.1667 m 4. Result: or ● Final Conclusion: The required mass of the counterweight to hold the gate in the vertical position is . (b) Magnitude of the pin reaction at 1. Formula: From force equilibrium in the horizontal () and vertical () directions: The magnitude of the total reaction at is: 2. Substitution: Substitute into equilibrium equations: 3. Calculation: 4. Result: ● Final Conclusion: The magnitude of the pin reaction at