Pressure on a Dam As the reservoir behind a dam is filled with water, the pressu

Question

Pressure on a Dam As the reservoir behind a dam is filled with water, the pressure that the water exerts on the dam increases. Eventually, the force on the dam becomes substantial, and it could cause the dam to collapse. There are two significant issues to be considered: First, the base of the dam should be able to withstand the pressure , where is the density of the water behind the dam, is its depth, and is the magnitude of the acceleration due to gravity. This means that the material of which the dam is made needs to be strong enough so that it doesn't crack (compressive strength). The second issue has to do with the strength of the foundation of the dam. The water pressure exerts a clockwise torque on the dam, as shown in the figure. The foundation of the dam should be strong enough so that the dam does not topple. The material has to be strong enough that the dam does not snap (shear strength). To study this phenomenon, consider the simple model of a dam depicted in the diagram. (Figure 1) A reservoir of water (density ) behind the dam is filled to a height . Assume that the width of the dam (the dimension pointing into the screen) is .

Explanation / Answer

Since x = 0 is at the floor, the depth of the water is H - x and

force = area * pressure = dx * width * density * g * depth   

or

dF = (density) (width) g (H-x) dx

Note: in your problem description, it says "density" (near the end of the big paragraph) with a blank after it, as if there is supposed to be a letter that represents density. It was lost when you copied the problem. Same thing for "width". Whatever those letters are, put them in the expression in place of (density) and (width).

Now... the torque...

dt = x dF so total torque is integral of x dF or

total torque = (density) (width) g integral of x(H-x) dx =

= (density) (width) g (1/2) Hx^2 - (1/3) x^3 from 0 to H or

= (density) (width) g [ (1/2) H^3 - (1/3) H^3 ]

= (density) (width) g (1/6) H^3

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