The upper edge of a gate in a dam runs along the water...

The upper edge of a gate in a dam runs along the water surface. The gate is $2.00 \mathrm{~m}$ high and $4.00 \mathrm{~m}$ wide and is hinged along a horizontal line through its center (Fig. $\mathbf{P 1 2 . 5 7}$ ). Calculate the torque about the hinge arising from the force due to the water. (Hint: Use a procedure similar to that used in Problem 12.55; calculate the torque on a thin, horizontal strip at a depth $h$ and integrate this over the gate.)

Key Concepts

Integration Over a Surface

Because the pressure varies with depth, the total force and torque on a submerged surface must be calculated by integrating the differential forces over the entire surface. In this context, small horizontal strips are considered, with their individual contributions to the force (or torque) summed by an integral over the range of depths corresponding to the size of the object.

Torque (Moment) Calculation

Torque or moment is the rotational effect produced by a force acting at a distance from a pivot point or axis. When calculating the torque on a submerged gate, each differential force due to pressure acts at a specific lever arm distance from the hinge. By integrating the products of these forces and their respective moment arms, one obtains the total torque about the hinge.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity. It increases linearly with depth, meaning that deeper parts of a fluid experience higher pressures. This concept is crucial for understanding forces on submerged surfaces, such as gates in dams, since the pressure varies continuously over the surface area.

Pressure Distribution on a Submerged Surface

The pressure distribution over a submerged surface is not uniform; it varies with depth. In problems like these, the pressure at any horizontal strip is given by the product of the fluid density, gravitational acceleration, and the depth below the surface. Understanding this distribution is essential for computing forces and resulting torques on structures in fluid environments.

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