A 2.5-m -diameter gate is connected to a water reservoir as...

A $2.5-\mathrm{m}$ -diameter gate is connected to a water reservoir as shown in Figure 2.61 . The gate is oriented at $35^{\circ}$ to the horizontal, the centroid of the gate is $1.5 \mathrm{~m}$ vertically below the water surface, and the weight of the gate is $500 \mathrm{kN}$. The gate is pin-connected to the reservoir at the top of the gate (at $\mathrm{P}$ ) and is opened by applying a vertical force at the bottom of the gate (at Q). (a) What is the magnitude of the resultant hydrostatic force and its location relative to the top of the gate? (b) What is the magnitude of the force required to open the gate?

A $2.5-\mathrm{m}$ -diameter gate is connected to a water reservoir as shown in Figure 2.61 . The gate is oriented at $35^{\circ}$ to the horizontal, the centroid of the gate is $1.5 \mathrm{~m}$ vertically below the water surface, and the weight of the gate is $500 \mathrm{kN}$. The gate is pin-connected to the reservoir at the top of the gate (at $\mathrm{P}$ ) and is opened by applying a vertical force at the bottom of the gate (at Q). (a) What is the magnitude of the resultant hydrostatic force and its location relative to the top of the gate?
(b) What is the magnitude of the force required to open the gate?

Key Concepts

Hydrostatic Pressure Distribution

This concept describes how pressure in a fluid increases with depth due to the weight of the overlying fluid. The pressure at any point on a submerged surface is a function of the depth below the free surface, typically expressed as P = ?gh, where ? is the density of the fluid, g is gravity, and h is the depth below the surface. This variation in pressure is fundamental to determining the forces exerted by the fluid on a structure.

Resultant Force on a Submerged Surface

For a submerged surface, the total or resultant hydrostatic force is calculated by integrating the varying pressure over the entire surface area. This integration accounts for the continuous change in pressure with depth, allowing for the determination of the overall magnitude of the force that the fluid exerts on the structure.

Center of Pressure

The center of pressure is the point on a submerged surface where the resultant hydrostatic force can be considered to act. Because the pressure increases with depth, the center of pressure is typically located below the centroid of the area. Determining its location is crucial for analyzing the stability and rotational effects on the structure.

Moment Equilibrium

This principle involves balancing the moments caused by the hydrostatic force and any applied forces about a pivot or hinge point. In problems involving fluid-structure interaction, such as opening a gate, moment equilibrium is used to determine the magnitude of an applied force needed to counteract the moment created by the hydrostatic pressure acting at the center of pressure.

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