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Chapter 2: Problem 91

Find the center of pressure of an elliptical area of minor axis \(2 a\) and major axis \(2 b\) where axis \(2 a\) is vertical and axis \(2 b\) is horizontal. The center of the ellipse is a vertical distance \(h\) below the surface of the water \((h>a)\). The fluid density is constant. Will the center of pressure of the ellipse change if the fluid is replaced by another constant-density fluid? Will the center of pressure of the ellipse change if the vertical axis is tilted back an angle \(\alpha\) from the vertical about its horizontal axis? Explain.

Short Answer

Expert verified
The center of pressure \( Y_p \) is calculated using the moment of inertia, the location of the centroid, and the area of the ellipse. Replacing the fluid with another fluid of different constant density will not change the center of pressure. If the vertical axis of the ellipse is tilted back, it changes the location of the center of pressure.

Step by step solution

01

Compute the centroid of the elliptical area

The centroid of an ellipse is its geometric center, which in this case is a distance \(h\) below the surface of the water. So, the centroid \( \bar{Y} = h \).
02

Calculate the moment of inertia of the ellipse

The moment of inertia of an ellipse is given by the formula \( I = \frac{\pi}{4} a b^{3} \) where a is the minor axis and b is the major axis of the ellipse. Calculate the value of the moment of inertia using the provided values of a and b.
03

Compute the center of pressure

The center of pressure \( Y_p \) can be calculated using the formula: \( Y_p = \bar{Y} + \frac{I}{\bar{A} \bar{Y}} \) where \( \bar{A} \) is the area of the ellipse \( \bar{A} = \pi a b \), \( \bar{Y} \) is the centroid (from step 1) and \( I \) is the moment of inertia (from step 2). Substitute the values into the formula to calculate \( Y_p \).
04

Analyze the effect of fluid replacement

Looking at the formula from step 3, you can see that all components of the formula depend only on the geometric properties of the elliptical area and its location relative to the surface of the fluid. Since war density does not appear in the formula, replacing the fluid with another fluid of different density will not alter the center of pressure as long as the density of the new fluid is also constant.
05

Analyze the effect of tilting the axis

The formula for the center of pressure assumes that the fluid pressure varies linearly with depth and is therefore only accurate when the object is oriented such that one of its axes is perpendicular to the free surface of the fluid. If the vertical axis of the ellipse is tilted back, the pressure distribution across the surface of the ellipse becomes non-linear, and the center of pressure is no longer given by the formula used in step 3. So, yes, tilting the axis will change the location of the center of pressure in most cases.

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