Fluid Pressure and Fluid Force

Transcription

1 0_0707.q //0 : PM Page 07 SECTION 7.7 Section 7.7 Flui Pressure an Flui Force 07 Flui Pressure an Flui Force Fin flui pressure an flui force. Flui Pressure an Flui Force Swimmers know that the eeper an object is submerge in a flui, the greater the pressure on the object. Pressure is efine as the force per unit of area over the surface of a bo. For eample, because a column of water that is 0 feet in height an inch square weighs. pouns, the flui pressure at a epth of 0 feet of water is. pouns per square inch.* At 0 feet, this woul increase to 8. pouns per square inch, an in general the pressure is proportional to the epth of the object in the flui. Definition of Flui Pressure The pressure on an object at epth h in a liqui is Pressure P wh where w is the weight-ensit of the liqui per unit of volume. The Granger Collection Below are some common weight-ensities of fluis in pouns per cubic foot. BLAISE PASCAL ( ) Pascal is well known for his work in man areas of mathematics an phsics, an also for his influence on Leibniz. Although much of Pascal s work in calculus was intuitive an lacke the rigor of moern mathematics, he nevertheless anticipate man important results. Ethl alcohol Gasoline Glcerin Kerosene Mercur Seawater Water When calculating flui pressure, ou can use an important (an rather surprising) phsical law calle Pascal s Principle, name after the French mathematician Blaise Pascal. Pascal s Principle states that the pressure eerte b a flui at a epth h is transmitte equall in all irections. For eample, in Figure 7.8, the pressure at the inicate epth is the same for all three objects. Because flui pressure is given in terms of force per unit area P F A, the flui force on a submerge horizontal surface of area A is Flui force F PA (pressure)(area). h The pressure at h is the same for all three objects. Figure 7.8 * The total pressure on an object in 0 feet of water woul also inclue the pressure ue to Earth s atmosphere. At sea level, atmospheric pressure is approimatel.7 pouns per square inch.

2 0_0707.q //0 : PM Page CHAPTER 7 Applications of Integration EXAMPLE Flui Force on a Submerge Sheet Fin the flui force on a rectangular metal sheet measuring feet b feet that is submerge in feet of water, as shown in Figure 7.9. The flui force on a horizontal metal sheet is equal to the flui pressure times the area. Figure 7.9 c L( i ) h( i ) Calculus methos must be use to fin the flui force on a vertical metal plate. Figure 7.70 Solution Because the weight-ensit of water is. pouns per cubic foot an the sheet is submerge in feet of water, the flui pressure is P. 7. pouns per square foot. Because the total area of the sheet is A square feet, the flui force is F PA pouns. This result is inepenent of the size of the bo of water. The flui force woul be the same in a swimming pool or lake. In Eample, the fact that the sheet is rectangular an horizontal means that ou o not nee the methos of calculus to solve the problem. Consier a surface that is submerge verticall in a flui. This problem is more ifficult because the pressure is not constant over the surface. Suppose a vertical plate is submerge in a flui of weight-ensit w (per unit of volume), as shown in Figure To etermine the total force against one sie of the region from epth c to epth, ou can subivie the interval c, into n subintervals, each of with. Net, consier the representative rectangle of with an length L i, where i is in the ith subinterval. The force against this representative rectangle is F i w epth area wh i L i. P wh The force against n such rectangles is n F i w n h i L i. i i pouns square feet square foot Note that w is consiere to be constant an is factore out of the summation. Therefore, taking the limit as 0 n suggests the following efinition. Definition of Force Eerte b a Flui The force F eerte b a flui of constant weight-ensit w (per unit of volume) against a submerge vertical plane region from c to is F w lim 0 n h i L i i w c h L where h is the epth of the flui at an L is the horizontal length of the region at.

3 0_0707.q //0 : PM Page 09 SECTION 7.7 Flui Pressure an Flui Force 09 EXAMPLE Flui Force on a Vertical Surface ft 8 ft ft (a) Water gate in a am ft h() = (, ) 0 (, 9) (b) The flui force against the gate Figure 7.7 A vertical gate in a am has the shape of an isosceles trapezoi 8 feet across the top an feet across the bottom, with a height of feet, as shown in Figure 7.7(a). What is the flui force on the gate when the top of the gate is feet below the surface of the water? Solution In setting up a mathematical moel for this problem, ou are at libert to locate the - an -aes in several ifferent was. A convenient approach is to let the -ais bisect the gate an place the -ais at the surface of the water, as shown in Figure 7.7(b). So, the epth of the water at in feet is Depth h. To fin the length L of the region at, fin the equation of the line forming the right sie of the gate. Because this line passes through the points, 9 an,, its equation is In Figure 7.7(b) ou can see that the length of the region at is Length L. Finall, b integrating from 9 to, ou can calculate the flui force to be F w h L c ,9 pouns. NOTE In Eample, the -ais coincie with the surface of the water. This was convenient, but arbitrar. In choosing a coorinate sstem to represent a phsical situation, ou shoul consier various possibilities. Often ou can simplif the calculations in a problem b locating the coorinate sstem to take avantage of special characteristics of the problem, such as smmetr.

4 0_0707.q //0 : PM Page 0 0 CHAPTER 7 Applications of Integration EXAMPLE Flui Force on a Vertical Surface 8 Observation winow The flui force on the winow Figure A circular observation winow on a marine science ship has a raius of foot, an the center of the winow is 8 feet below water level, as shown in Figure 7.7. What is the flui force on the winow? Solution To take avantage of smmetr, locate a coorinate sstem such that the origin coincies with the center of the winow, as shown in Figure 7.7. The epth at is then Depth h 8. The horizontal length of the winow is, an ou can use the equation for the circle,, to solve for as follows. Length Finall, because ranges from to, an using pouns per cubic foot as the weight-ensit of seawater, ou have F w L h L c 8. Initiall it looks as if this integral woul be ifficult to solve. However, if ou break the integral into two parts an appl smmetr, the solution is simple. F The secon integral is 0 (because the integran is o an the limits of integration are smmetric to the origin). Moreover, b recognizing that the first integral represents the area of a semicircle of raius, ou obtain F pouns. So, the flui force on the winow is 08. pouns. 0. f is not ifferentiable at ±. Figure 7.7. TECHNOLOGY To confirm the result obtaine in Eample, ou might have consiere using Simpson s Rule to approimate the value of 8 8. From the graph of f 8 however, ou can see that f is not ifferentiable when ± (see Figure 7.7). This means that ou cannot appl Theorem.9 from Section. to etermine the potential error in Simpson s Rule. Without knowing the potential error, the approimation is of little value. Use a graphing utilit to approimate the integral.

5 0_0707.q //0 : PM Page SECTION 7.7 Flui Pressure an Flui Force Eercises for Section 7.7 Force on a Submerge Sheet In Eercises an, the area of the top sie of a piece of sheet metal is given. The sheet metal is submerge horizontall in feet of water. Fin the flui force on the top sie.. square feet. square feet See for worke-out solutions to o-numbere eercises. Flui Force of Water In Eercises, fin the flui force on the vertical plate submerge in water, where the imensions are given in meters an the weight-ensit of water is 9800 newtons per cubic meter.. Square. Square Buoant Force In Eercises an, fin the buoant force of a rectangular soli of the given imensions submerge in water so that the top sie is parallel to the surface of the water. The buoant force is the ifference between the flui forces on the top an bottom sies of the soli... h h ft ft ft ft ft 8 ft. Triangle. Rectangle Flui Force on a Tank Wall In Eercises 0, fin the flui force on the vertical sie of the tank, where the imensions are given in feet. Assume that the tank is full of water. 9. Rectangle. Triangle 7. Trapezoi 8. Semicircle 9. Parabola, 0. Semiellipse, 9 Force on a Concrete Form In Eercises 8, the figure is the vertical sie of a form for poure concrete that weighs 0.7 pouns per cubic foot. Determine the force on this part of the concrete form.. Rectangle. Semiellipse, 0 ft ft 7. Rectangle 8. Triangle ft ft ft ft ft ft 9. Flui Force of Gasoline A clinrical gasoline tank is place so that the ais of the cliner is horizontal. Fin the flui force on a circular en of the tank if the tank is half full, assuming that the iameter is feet an the gasoline weighs pouns per cubic foot.

6 0_0707.q //0 : PM Page CHAPTER 7 Applications of Integration 0. Flui Force of Gasoline Repeat Eercise 9 for a tank that is full. (Evaluate one integral b a geometric formula an the other b observing that the integran is an o function.). Flui Force on a Circular Plate A circular plate of raius r feet is submerge verticall in a tank of flui that weighs w pouns per cubic foot. The center of the circle is k k > r feet below the surface of the flui. Show that the flui force on the surface of the plate is F wk r. (Evaluate one integral b a geometric formula an the other b observing that the integran is an o function.). Flui Force on a Circular Plate Use the result of Eercise to fin the flui force on the circular plate shown in each figure. Assume the plates are in the wall of a tank fille with water an the measurements are given in feet. (a) (b). Flui Force on a Rectangular Plate A rectangular plate of height h feet an base b feet is submerge verticall in a tank of flui that weighs w pouns per cubic foot. The center is k feet below the surface of the flui, where h k. Show that the flui force on the surface of the plate is F wkhb.. Flui Force on a Rectangular Plate Use the result of Eercise to fin the flui force on the rectangular plate shown in each figure. Assume the plates are in the wall of a tank fille with water an the measurements are given in feet. (a) (b). Submarine Porthole A porthole on a vertical sie of a submarine (submerge in seawater) is square foot. Fin the flui force on the porthole, assuming that the center of the square is feet below the surface.. Submarine Porthole Repeat Eercise for a circular porthole that has a iameter of foot. The center is feet below the surface Moeling Data The vertical stern of a boat with a superimpose coorinate sstem is shown in the figure. The table shows the with w of the stern at inicate values of. Fin the flui force against the stern if the measurements are given in feet. w Water level 8. Irrigation Canal Gate The vertical cross section of an irrigation canal is moele b f Stern where is measure in feet an 0 correspons to the center of the canal. Use the integration capabilities of a graphing utilit to approimate the flui force against a vertical gate use to stop the flow of water if the water is feet eep. In Eercises 9 an 0, use the integration capabilities of a graphing utilit to approimate the flui force on the vertical plate boune b the -ais an the top half of the graph of the equation. Assume that the base of the plate is feet beneath the surface of the water Think About It (a) Approimate the epth of the water in the tank in Eercise if the flui force is one-half as great as when the tank is full. (b) Eplain wh the answer in part (a) is not. Writing About Concepts. Define flui pressure.. Define flui force against a submerge vertical plane region.. Two ientical semicircular winows are place at the same epth in the vertical wall of an aquarium (see figure). Which has the greater flui force? Eplain. w 7