The tank in Figure P 14.13 is filled with water of depth d....

The tank in Figure $P 14.13$ is filled with water of depth $d$. At the bottom of one sidewall is a rectangular hatch of height $h$ and width $w$ that is hinged at the top of the hatch. (a) Determine the magnitude of the force the water exerts on the hatch. (b) Find the magnitude of the torque exerted by the water about the hinges.

Key Concepts

Center of Pressure

The center of pressure is the point on a submerged surface through which the total resultant force can be considered to act. It is found by considering the moments produced by the differential forces about an axis, and typically lies below the geometric centroid for vertically oriented surfaces due to the increasing pressure with depth. This concept is essential when calculating moments or torques about a pivot point, such as a hinge.

Torque Calculation Due to Fluid Pressure

Torque, or moment, about a pivot point is calculated by taking the product of the force and the perpendicular distance from the line of action of the force to the pivot. In the context of a submerged surface, this involves integrating the pressure distribution, while accounting for the varying moment arms across the surface. The total torque is then determined by the contribution of the force acting at the center of pressure relative to the pivot, which is critical in evaluating the rotational effects of the fluid force on structures.

Force on a Submerged Surface

The total force exerted by the fluid on a submerged surface is obtained by integrating the pressure distribution over the area of the surface. Since pressure varies linearly with depth, the integration takes into account the variation of pressure along the surface, leading to a force that is not uniformly distributed. This method is crucial for determining the resultant force acting on any slice or element of a structure submerged in a fluid.

Hydrostatic Pressure Distribution

In a static fluid, the pressure at any point increases linearly with depth due to the weight of the fluid above. This relationship is typically described by the equation p = ?gh, where ? is the fluid density, g is the acceleration due to gravity, and h is the depth below the free surface. This concept is central to evaluating forces on submerged structures, as the force at each point on the surface depends on the local pressure.

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