Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_189

⚡ Mecademy AIENG정역학 · ch5  Problem Statement The figure shows the cross section of a rectangular gate 4 m high and 6 m long (perpendicular to the paper) which blocks a fresh-water channel. The gate has a mass of 8.5 Mg and is hinged about a horizontal axis through C. Compute the vertical force P exerted by the foundation on the lower edge A of the gate. Neglect the mass of the frame to which the gate is attached. Problem 5/189 (a) Calculation of the vertical force P ● Calculation Process 1. Present Final Formula: We utilize the principle of static equilibrium by summing moments about the hinge at point . Taking clockwise moments as positive: Where: is the upward vertical force from the foundation at . is the weight of the rectangular gate. is the resultant hydrostatic force from the fresh water. are the horizontal or vertical moment arms relative to point . 2. Substitute Values: First, we calculate the known forces and their respective moment arms based on the provided geometry (, , , water depth ): Weight of the gate (): The weight acts through the geometric center of the high plate, which is located horizontally from hinge . Hydrostatic force (): The water is on the left side of the gate with a depth of (from level down to ). C M = ∑ C 0⟹d ⋅ P P−d ⋅ W W−d ⋅ Fw F = w 0 PA W F w d ,d ,d PWFw C AB=4 mw=6 mm=8.5 Mg h = w 3 m W W=m⋅g=(8.5×1000 kg)⋅ 9.81 m/s= 2 83,385 N 4 m 3 mC M = W W⋅d = W 83,385 N⋅3 m=250,155 N⋅m (Clockwise) F w 3 mCA The resultant acts at of the height from