Question

I’ve been racking my brain this one. My book showed the answer on how the total force was derived. But I don’t get how it got to that answer. Can someone help me please?

ethe ratio of the input work to the output work wi-F:aa-EN )-(20 )-1 t verifies that the work input and output are the same, as they must be to conserve energy. This result ,Example 15.2 The Force on a Dam lled to a height H behind a dam of width to (Fig. 15.6). Determine the ant force exerted by the water on the dam SOLUTION concetualize Because pressure varies with depth, we cannot calculate the force sim- by multiplying the area by the pressure. As the pressure in the water increases with depth, the force on the adjacent portion of the dam also increases. Categorize Because of the variation of pressure with depth, we must use integration to solve this example, so we categorize it as an analysis problem. Analyze Let's imagine a vertical y axis, with y = 0 at the bottorn of the dam, we divide he face of the dam into narrow horizontal strips at a distance yabove the bottom, such asthe red strip in Figure 15.6. The pressure on each such strip is due only to the water; atmospheric pressure acts on both sides of the dam. Figure 15.6 (Exam exerts a force on a da Use Equation 15.4 to calculate the pressure due to the water at the depth h: Use Equation 15.1 to find the force exerted on the shaded strip of area da wdy Integrate to find the total force on the dam: inalize Notice that the thickness of the dam shown in Figure 15.6 increases with depth. This design ac greater force the water exerts on the dam at greater depths. What if? What if you were asked to find this force without using calculus? How could you determine its val Answer We know from Equation 15.4 that pressure varies linearly with depth. Therefore, the average pres at the top and the pressure at the bottom: top 2PgH avg force on the dam is equal to the product of the average pressure and the area of the face of the d hich is the same result we obtained using calculus.

Answer 1

This is simply integration of area with height.

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