Solve and explain ---**Other Relevant Text (Header)**
FLUID MECHANICS ASSIGNMENT 3
**Question Stem**
The rigid gate, AOB, of the Figure 1 below is hinged at O and rests against a rigid support at B. What minimum horizontal force, P, is required to hold the gate closed if its width is 3 m? Neglect the weight of the gate and friction in the hinge. The back of the gate is exposed to the atmosphere.
**Chart/Diagram Description**
* **Type:** Engineering diagram / Cross-sectional view illustrating a hydraulic gate system.
* **Main Elements:**
* A body of water is depicted on the left side, labeled "Water". Its free surface is indicated by an inverted triangle symbol and is annotated with "Open to atmosphere".
* A rigid, L-shaped gate, implicitly labeled "AOB" from the problem statement, is shown. It consists of a vertical section and a horizontal section.
* Point O is marked near the top of the vertical section of the gate and is labeled "Hinge", indicating the pivot point.
* Point A is the corner where the vertical and horizontal sections of the gate meet.
* Point B is the far end of the horizontal section of the gate.
* A textured, grey block represents a rigid support structure, against which point B of the gate rests, and part of which extends underneath the horizontal section of the gate.
* A horizontal arrow labeled "P" points to the left, indicating a horizontal force applied to the right side of the vertical section of the gate, near its bottom.
* **Dimensions:**
* A vertical dimension of "3 m" is indicated from the water surface down to the hinge point O.
* A vertical dimension of "4.4 m" is indicated from the hinge point O down to the corner A of the gate.
* A horizontal dimension of "2 m" is indicated for the length of the horizontal section of the gate (from A to B).
* **Labels and Annotations:** "Open to atmosphere", "Water", "Hinge", "P", "3 m", "4.4 m", "2 m", "O", "A", "B".
* **Relative Position and Direction:** The water is contained on the left side of the gate. The gate is hinged at point O, allowing rotation. Point B is supported by the rigid structure. The force P is applied horizontally from the right to hold the gate closed against the water pressure. The region to the right of the gate is open to the atmosphere.
**Other Relevant Text (Figure Title)**
Figure 1
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答案文本
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We have a hydraulic gate problem involving an L-shaped rigid gate AOB. The gate is hinged at point O and rests against a support at point B. Water is on the left side, creating hydrostatic pressure. The gate extends 3 meters from the water surface down to the hinge O, then 4.4 meters further down to corner A, with a 2-meter horizontal section from A to B. We need to find the minimum horizontal force P required to keep this gate closed against the water pressure. The gate width is 3 meters, and we'll neglect the gate's weight and friction at the hinge.
Now let's analyze the hydrostatic pressure distribution on the vertical section of the gate. Hydrostatic pressure increases linearly with depth according to the formula p equals rho g h, where rho is water density, g is gravitational acceleration, and h is depth below the surface. At the water surface, pressure is zero. At hinge O, which is 3 meters below the surface, pressure equals 1000 times 9.81 times 3, which is 29,430 pascals or 29.4 kilopascals. At point A, which is 7.4 meters below the surface, pressure reaches 72,594 pascals or 72.6 kilopascals. This creates a triangular pressure distribution on the gate, with maximum pressure at the bottom.
Now we calculate the resultant hydrostatic force acting on the vertical gate section. The triangular pressure distribution can be replaced by a single resultant force. The magnitude is calculated using the formula F_R equals one half times rho times g times h total squared times width. With h total equal to 7.4 meters and width equal to 3 meters, we get F_R equals one half times 1000 times 9.81 times 7.4 squared times 3, which equals 803,847 newtons or 803.8 kilonewtons. This resultant force acts at the center of pressure, located at two-thirds of the total height from the water surface. This puts the center of pressure 4.93 meters below the water surface, or 1.47 meters below hinge O.
Now we set up the force analysis using a free body diagram of the gate. The forces acting on the system include the hydrostatic force F_R of 803.8 kilonewtons acting horizontally at the center of pressure, the applied force P acting horizontally, and reaction forces at the supports. At hinge O, we have horizontal and vertical reaction components R_Ox and R_Oy. At support B, we have a vertical reaction R_By. For equilibrium, the sum of forces in x and y directions must be zero, and the sum of moments about any point must be zero. We'll use moment equilibrium about hinge O, where the hydrostatic force creates a moment of F_R times 1.47 meters, and the applied force P creates a moment of P times 4.4 meters.
Now we solve for the required force P using moment equilibrium about hinge O. The hydrostatic force creates a clockwise moment equal to F_R times its moment arm of 1.47 meters. This gives us 803,847 newtons times 1.47 meters, which equals 1,181,655 newton-meters. The applied force P creates a counterclockwise moment equal to P times 4.4 meters. For equilibrium, these moments must be equal, so P times 4.4 equals 1,181,655. Solving for P, we get P equals 1,181,655 divided by 4.4, which equals 268,558 newtons, or 268.6 kilonewtons. This is the minimum horizontal force required to hold the gate closed against the water pressure.