Fluid Pressure and Fluid Force

Transcription

1 SECTION 7.7 Flui Pressure an Flui Force 07 Section 7.7 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.. 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. MathBio Below are some common weight-ensities of fluis in pouns per cubic foot. Ethl alcohol 9. Gasoline.0.0 Glcerin 78. Kerosene. Mercur 89.0 Seawater.0 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 FA, 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 08 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 The flui force on a horizontal metal sheet is equal. to the flui pressure times the area. Figure 7.9 h( i ) c L( 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. Tr It 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 weptharea 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 Eploration A 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 wc hl where h is the epth of the flui at an L is the horizontal length of the region at.

3 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 9to, ou can calculate the flui force to be F w hl c ,9 pouns. Tr It Eploration A Open Eploration 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 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 hl 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 L 08. pouns. 0 So, the flui force on the winow is 08. pouns. Tr It Eploration A 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 SECTION 7.7 Flui Pressure an Flui Force Eercises for Section 7.7 The smbol inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem. to view the complete solution of the eercise. to print an enlarge cop of the graph. 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 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.. Rectangle. Triangle 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. 7. Trapezoi 8. Semicircle. Rectangle. Semiellipse, 0 ft ft ft ft 9. Parabola, 0. Semiellipse, 7. Rectangle 8. Triangle 9 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 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

7 REVIEW EXERCISES Review Eercises for Chapter 7 The smbol inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem. to view the complete solution of the eercise. to print an enlarge cop of the graph. In Eercises 0, sketch the region boune b the graphs of the equations, an etermine the area of the region...,,., 0,,.,, 0.,., 7. e, e, 0 8. csc, (one region) In Eercises, use a graphing utilit to graph the region boune b the graphs of the functions, an use the integration capabilities of the graphing utilit to fin the area of the region.. 8, 8.,, 0., 0, 0., In Eercises 8, use vertical an horizontal representative rectangles to set up integrals for fining the area of the region boune b the graphs of the equations. Fin the area of the region b evaluating the easier of the two integrals , 0,, sin, cos, cos,,, 0 9. Think About It A person has two job offers. The starting salar for each is $0,000, an after 0 ears of service each will pa $,000. The salar increases for each offer are shown in the figure. From a strictl monetar viewpoint, which is the better offer? Eplain. Salar (in ollars) 0,000 0,000 0,000 S, 7,,,, 0, 0 Job Job Year t 0. Moeling Data The table shows the annual service revenue in billions of ollars for the cellular telephone inustr for the ears 99 through 00. (Source: Cellular Telecommunications & Internet Association) (a) Use the regression capabilities of a graphing utilit to fin an eponential moel for the ata. Let t represent the ear, with t corresponing to 99. Use the graphing utilit to plot the ata an graph the moel in the same viewing winow. (b) A financial consultant believes that a moel for service revenue for the ears 00 through 00 is What is the ifference in total service revenue between the two moels for the ears 00 through 00? In Eercises 8, fin the volume of the soli generate b revolving the plane region boune b the equations about the inicate line(s).., 0, (a) the -ais (b) the -ais (c) the line () the line.,, 0 (a) the -ais (b) the line (c) the -ais () the line. (a) the -ais (oblate spheroi) 9 (b) the -ais (prolate spheroi) Year R R.8e 0.t.. (a) the -ais (oblate spheroi) a b (b) the -ais (prolate spheroi)., 0, 0, revolve about the -ais., 0,, revolve about the -ais 7., 0,, revolve about the -ais 8. e, 0, 0, revolve about the -ais In Eercises 9 an 0, consier the region boune b the graphs of the equations an Area Fin the area of the region. 0. Volume Fin the volume of the soli generate b revolving the region about (a) the -ais an (b) the -ais. R

8 CHAPTER 7 Applications of Integration. Depth of Gasoline in a Tank A gasoline tank is an oblate spheroi generate b revolving the region boune b the graph of 9 about the -ais, where an are measure in feet. Fin the epth of the gasoline in the tank when it is fille to one-fourth its capacit.. Magnitue of a Base The base of a soli is a circle of raius a, an its vertical cross sections are equilateral triangles. The volume of the soli is 0 cubic meters. Fin the raius of the circle. In Eercises an, fin the arc length of the graph of the function over the given interval.. f., 0,,. Length of a Catenar A cable of a suspension brige forms a catenar moele b the equation 00 cosh 80, where an are measure in feet. Use a graphing utilit to approimate the length of the cable.. Approimation Determine which value best approimates the length of the arc represente b the integral sec. 0 (Make our selection on the basis of a sketch of the arc an not b performing an calculations.) (a) (b) (c) () (e) 7. Surface Area Use integration to fin the lateral surface area of a right circular cone of height an raius. 8. Surface Area The region boune b the graphs of, 0, an is revolve about the -ais. Fin the surface area of the soli generate. 9. Work A force of pouns is neee to stretch a spring inch from its natural position. Fin the work one in stretching the spring from its natural length of 0 inches to a length of inches. 0. Work The force require to stretch a spring is 0 pouns. Fin the work one in stretching the spring from its natural length of 9 inches to ouble that length.. Work A water well has an eight-inch casing (iameter) an is 7 feet eep. The water is feet from the top of the well. Determine the amount of work one in pumping the well r, assuming that no water enters it while it is being pumpe.. Work Repeat Eercise, assuming that water enters the well at a rate of gallons per minute an the pump works at a rate of gallons per minute. How man gallons are pumpe in this case?. Work A chain 0 feet long weighs pouns per foot an is hung from a platform 0 feet above the groun. How much work is require to raise the entire chain to the 0-foot level?,. Work A winlass, 00 feet above groun level on the top of a builing, uses a cable weighing pouns per foot. Fin the work one in wining up the cable if (a) one en is at groun level. (b) there is a 00-poun loa attache to the en of the cable.. Work The work one b a variable force in a press is 80 footpouns. The press moves a istance of feet an the force is a quaratic of the form F a. Fin a.. Work Fin the work one b the force F shown in the figure. Pouns In Eercises 7 0, fin the centroi of the region boune b the graphs of the equations. 7. a, 0, 0 8., 9. a, 0 0.,. Centroi A blae on an inustrial fan has the configuration of a semicircle attache to a trapezoi (see figure). Fin the centroi of the blae. 0 8 F 8 0 Feet 7 (9, ). Flui Force A swimming pool is feet eep at one en an 0 feet eep at the other, an the bottom is an incline plane. The length an with of the pool are 0 feet an 0 feet. If the pool is full of water, what is the flui force on each of the vertical walls?. Flui Force Show that the flui force against an vertical region in a liqui is the prouct of the weight per cubic volume of the liqui, the area of the region, an the epth of the centroi of the region.. Flui Force Using the result of Eercise, fin the flui force on one sie of a vertical circular plate of raius feet that is submerge in water so that its center is feet below the surface.

9 P.S. Problem Solving P.S. Problem Solving The smbol inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem. to view the complete solution of the eercise. to print an enlarge cop of the graph.. Let R be the area of the region in the first quarant boune b the parabola an the line c, c > 0. Let T be the area of the triangle AOB. Calculate the limit lim T c 0 R.. A hole is cut through the center of a sphere of raius r (see figure). The height of the remaining spherical ring is h. Fin the volume of the ring an show that it is inepenent of the raius of the sphere. c A B(c, c ) h r T = R O c. Let R be the region boune b the parabola an the -ais. Fin the equation of the line m that ivies this region into two regions of equal area.. (a) A torus is forme b revolving the region boune b the circle about the -ais (see figure). Use the isk metho to calculate the volume of the torus. (b) Use the isk metho to fin the volume of the general torus if the circle has raius r an its center is R units from the ais of rotation.. Graph the curve 8. = r = ( ) + = (, 0) Use a computer algebra sstem to fin the surface area of the soli of revolution obtaine b revolving the curve about the -ais. Centroi = m. A rectangle R of length l an with w is revolve about the line L (see figure). Fin the volume of the resulting soli of revolution. L 8 = S R A(, ) w B R Figure for Figure for 7 7. (a) The tangent line to the curve at the point A, intersects the curve at another point B. Let R be the area of the region boune b the curve an the tangent line. The tangent line at B intersects the curve at another point C (see figure). Let S be the area of the region boune b the curve an this secon tangent line. How are the areas R an S relate? (b) Repeat the construction in part (a) b selecting an arbitrar point A on the curve. Show that the two areas R an S are alwas relate in the same wa. 8. The graph of f passes through the origin. The arc length of the curve from 0, 0 to, f is given b s e t t. 0 Ientif the function f. 9. Let f be rectifiable on the interval a, b, an let s ft t. a s (a) Fin. (b) Fin s an s. (c) If f t t, fin s on,. () Calculate s an escribe what it signifies. C

10 CHAPTER 7 Applications of Integration 0. The Archimees Principle states that the upwar or buoant force on an object within a flui is equal to the weight of the flui that the object isplaces. For a partiall submerge object, ou can obtain information about the relative ensities of the floating object an the flui b observing how much of the object is above an below the surface. You can also etermine the size of a floating object if ou know the amount that is above the surface an the relative ensities. You can see the top of a floating iceberg (see figure). The ensit of ocean water is.0 0 kilograms per cubic meter, an that of ice is kilograms per cubic meter. What percent of the total iceberg is below the surface?. Sketch the region boune on the left b, boune above b, an boune below b. (a) Fin the centroi of the region for. (b) Fin the centroi of the region for b. (c) Where is the centroi as b?. Sketch the region to the right of the -ais, boune above b an boune below b. (a) Fin the centroi of the region for. (b) Fin the centroi of the region for b. (c) Where is the centroi as b?. Fin the work one b each force F. (a) (b). Estimate the surface area of the pon using (a) the Trapezoial Rule an (b) Simpson s Rule. 0 ft L F 8 ft 7 ft 80 ft ft 8 ft 7 ft 0 ft h = L h = 0 = h F In Eercises an, fin the consumer surplus an proucer surplus for the given eman [ p ] an suppl [ p ] curves. The consumer surplus an proucer surplus are represente b the areas shown in the figure. P Consumer Suppl surplus curve Point of equilibrium P 0 ( 0, P 0 ) Proucer surplus. p 0 0., p 0.. p , p 7. A swimming pool is 0 feet wie, 0 feet long, feet eep at one en, an 8 feet eep at the other en (see figure). The bottom is an incline plane. Fin the flui force on each vertical wall. 8 ft 8 Δ 0 ft 0 0 Deman curve 0 ft (0, ) (a) Fin at least two continuous functions f that satisf each conition. (i) f 0 on 0, (ii) f 0 0 an f 0 (iii) The area boune b the graph of f an the -ais for 0 equals. (b) For each function foun in part (a), approimate the arc length of the graph of the function on the interval 0,. (Use a graphing utilit if necessar.) (c) Can ou fin a function f that satisfies the conitions in part (a) an whose graph has an arc length of less than on the interval 0,? ft