/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 46 A door \(1 \mathrm{m}\) wide and... [FREE SOLUTION] | 91影视

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A door \(1 \mathrm{m}\) wide and \(1.5 \mathrm{m}\) high is located in a plane vertical wall of a water tank. The door is hinged along its upper edge, which is \(1 \mathrm{m}\) below the water surface. Atmospheric pressure acts on the outer surface of the door and at the water surface. (a) Determine the magnitude and line of action of the total resultant force from all fluids acting on the door. (b) If the water surface gage pressure is raised to 0.3 atm, what is the resultant force and where is its line of action? (c) Plot the ratios \(F / F_{9}\) and \(y^{\prime} / y_{c}\) for different values of the surface pressure ratio \(p_{s} / p_{\text {atm }}\). \(\left(F_{0}\right.\) is the resultant force when \(\left.p_{s}=p_{\text {atm }}\right)\).

Short Answer

Expert verified
The initial resultant force and its line of action on the door is \(F=25927.25N\) and \(h_c=2.375m\) below the water surface. When the water surface pressure is increased, the resultant force and its line of action changes. For any given changes in surface pressure, the ratios \(F/F_0\) and \(y'/y_c\) can be plotted.

Step by step solution

01

Calculate Initial resultant force and its line of action

Firstly, using the principle of hydrostatic pressure, the initial force acting on the door can be calculated using the formula \(F=蟻ghA\), where \(蟻\) is the water density, \(g\) is the gravitational constant, \(h\) is the height of water surface above the centroid of the door and \(A\) the area of the door. \nHere, \(蟻=1000kg/m^3\), \(g=9.81m/s^2\), \(h=1.75m\) and \(A=1*1.5m^2\). Substituting these values into the formula gives \(F=25927.25N\).\nThe line of action is calculated by finding the center of pressure, which is given by the formula \(h_c=\frac{(I_g+Ah^2)}{Ah}\), where in this problem the area moment of inertia (\(I_g\)) of a rectangular surface area about its own centroidal x axis is simply \(bh^3/12\), where \(b\) is the base width of the rectangle and \(h\) the height (of the door in this case). A is the area of the door and hs the distance of the centroid of the door from the water surface. Hence \(h_c = 2.375m\) below the water surface.
02

Determine resultant force and its line of action when surface pressure is increased

Next, the water pressure on the top surface of the door now becomes the sum of the atmospheric pressure and the gauge pressure i.e., 1 atm + 0.3 atm = 1.3 atm = 1.3 * 101325 Pa. If we consider the end of the door (consider it as \(h\)) to be at the 1m beneath the surface of water, then the pressure at that point is \(蟻gh + P_{atm}\) = 1000 * 9.81 * 1 + 1.3 * 101325 = 1.33 * 10^5 Pa. \nSince the pressure varies linearly along the door, the average pressure on the door is the average of the pressure at the top and bottom i.e. (pressure at top + pressure at bottom)/2. Hence the total force exerted is the average pressure times the area of the door. The line of action of the force i.e., center of pressure will still be at the previously calculated \(h_c=2.375m\).
03

Plot the ratios

Finally, ratios of the initial resultant force to the new resultant force and the initial line of action to the new line of action for different values of surface pressure can be plotted. \(F/F_0\) and \(y'/y_c\) will be the y and x coordinates respectively.

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