It is straightforward to determine the fluid force on a horizontal metal sheet s

Question

It is straightforward to determine the fluid force on a horizontal metal sheet submerged in water (See Figure 7.69 on page 510). We can use the formula shown on page 509 to determine the fluid pressure, P = wh, where w is the weight-density of the liquid and h is the depth of the metal sheet (the weight-density for water is 62.4 pounds per cubic foot). Then the fluid force can be calculated as F = PA where A is the total area of the metal sheet (see Example #1 in page 510). In your own words, explain why calculus-based methods must be used to determine the fluid force on a vertical metal plate.

Explanation / Answer

Since the metal plate is submerged, its submerged area would be needed to calculate vis-a-vis calculating the pressure on it ( since the pressure varies vertically with the depth and so the area ), it becomes neceassary to

devise a mechanism to calculate to that.

Either we can deivide the plate into finite no. of sections and then assuming the pressure exerted on the center of mass of respective sections. But that would give the approximate answer.

Here, Calculus provides a way out , If we divide the sections into n no. of sections and then calculate the pressure on them and hence the force exerted on them and then take sum of all those to calculate the exact value. This is possible when we take those n no. of sections to infinity, taking limit ( n tends to infinity) , in effect calculating the integral.

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