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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

Consider a flat metal plate to be placed vertically underwater with its top 2 m below the surface of the water. Determine a shape for the plate so that if the plate is divided into any number of horizontal strips of equal height, the hydrostatic force on each strip is the same.

To determine

A shape for the metal plate.

Explanation

Given information:

The flat metal plate is 2 m below the surface of the water.

The hydrostatic force on each strip is same.

Calculation:

Let the height of the plate be h and it is symmetric about x-axis.

Let f(x) be the width of the plate from vertical axis to the end.

Therefore, the total width of the plate be 2f(x).

Sketch the shape of the plate with the orientation of positive x-axis as shown in Figure 1.

Each of the n horizontal strips has the height hn and the ith strip goes from x=2+(i1n)h to x=2+(inh).

The expression to find the hydrostatic force on the ith strip is shown below:

F(i)=2+(i1n)h2+(inh)γx[2f(x)]dx (1)

Let the unit weight of water is γ=62.5lb/ft3.

Substitute 62.5lb/ft3 for γ in Equation (1).

F(i)=2+(i1n)h2+(inh)62

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