Show that if a vertical dam in the shape of a rectangle is divided in half by means of a diagona

Show that if a vertical dam in the shape of a rectangle is...

Key Concepts

Hydrostatic Pressure Distribution

This concept explains that within a fluid at rest, the pressure increases linearly with depth due to the weight of the fluid above. In problems involving dams or submerged surfaces, the pressure at any point on the surface is dependent on the vertical distance from the free surface, which is typically expressed as ?gh (density × gravity × depth). This non-uniform pressure distribution across the surface is fundamental to calculating the net force acting on the structure.

Integration of Pressure Over a Submerged Surface

To determine the total force on a submerged surface, one must integrate the pressure over the entire area of the surface. Since the pressure varies with depth, the integration accounts for the changing pressure values across the surface. This technique is critical in fluid mechanics problems where the force distribution is not uniform, and it leads to an accurate calculation of the resultant hydrostatic force.

Centroid and Center of Pressure

The centroid of an area is a geometric center that can be useful in simplifying force calculations. However, for submerged surfaces, the center of pressure, which is the point where the resultant force acts, does not generally coincide with the centroid because the pressure distribution is not uniform. Understanding the relationship between these centers is important when analyzing the moments and stability of structures like dams.

Geometric Influence on Hydrostatic Force Distribution

The shape and segmentation of a structure, such as dividing a rectangular dam along a diagonal, have a significant impact on how the hydrostatic force is distributed over its surface. By splitting the dam into two regions with different depths, the integration of pressure over each segment yields different total forces, illustrating that even though the areas might be equal geometrically, the force magnitudes can vary due to the linear pressure gradient with depth.

Transcript

00:01 What we want to do is we are going to start with this represents a vertical dam.

00:07 It's in the shape of a rectangle.

00:10 And we are going to divide this vertical dam by a diagonal.

00:19 And we want to show that the total force exerted by the water on one half of the dam is twice that exerted on the other half.

00:32 Of the dam.

00:33 And so we're going to assume that this top edge of the dam is even with the surface of the water.

00:38 Okay.

00:39 So basically what we want to do is we kind of want to start with a coordinate plane.

00:44 And so we're going to go ahead and put our coordinate plane.

00:52 Oops, right here.

00:54 And so this is going to be x and this will be y.

00:58 And so this right here will be at zero zero.

01:03 We're going to note this at some x value that we're going to call a and zero.

01:10 And then this up here is going to be some y value.

01:16 We're just going to denote as zero b.

01:18 And then that means this fourth coordinate is a comma b.

01:23 Okay.

01:24 So that's the first thing.

01:26 So now what we're going to do is we are actually going to start with, with let's go ahead and start with this is going to be f1 and this will be f2.

01:55 Let's just go ahead and start with that.

01:58 Okay, so now what we want to do is we want to actually look at each of these forces separately.

02:04 And so we know that a fluid force is equal to the density of the fluid times the depth of the fluid times the area.

02:18 And so what we're going to do is we're going to start with a representative rectangle.

02:25 And the depth of that rectangle is going to be b minus y.

02:32 So that is going to be the depth of that rectangle that is h.

02:37 The thickness is delta y.

02:39 And now we've got to find the length of these representative rectangles.

02:43 And so that's the length is going to be this line minus zero.

02:49 And so let's go ahead and find that slope of that line...