3–55 Consider a hydraulic jack being used in a car repairshop, as in Fig. P3–55. The pistons have an area of A1 � 1cm2 and A2 � 0.04 m2. Hydraulic oil with a specific gravityof 0.870 is pumped in as the small piston on the left side ispushed up and down, slowly raising the larger piston on the

CHAPTER 3117

3–54 A multifluid container is connected to a U-tube, asshown in Fig. P3–54. For the given specific gravities andfluid column heights, determine the gage pressure at A. Alsodetermine the height of a mercury column that would createthe same pressure at A. Answers: 0.415 kPa, 0.311 cm

right side. A car that weighs 20,000 N is to be jacked up. (a)At the beginning, when both pistons are at the same elevation(h � 0), calculate the force F1 in newtons required to holdthe weight of the car. (b) Repeat the calculation after the carhas been lifted two meters (h � 2 m). Compare and discuss.

3–56 Consider a hydraulic jack as in Prob.3–55. The specific gravity of the hydraulic oil is 0.870. (a)For a given force F1 � 50 N and for the case in which h � 0,calculate force F2 as a function of area ratio for the range 50� A2/A1 � 1000. Discuss the relationship between F1 andA2/A1 (is the relationship linear, quadratic, etc.?) (b) RunFlowLab using template Hydraulic_press_area for the givenconditions and the same range of area ratio. Compare to yourmanual calculations.

3–57 Consider a hydraulic jack as in Prob.3–55. The area ratio is A2/A1 � 400. (a) For a given force F1� 50 N and for the case in which h � 0, calculate force F2 asa function of hydraulic fluid density for the range 780 kg/m3

(kerosene) � r � 998.2 kg/m3 (water). Discuss the relation-ship between F2 and r (is the relationship linear, quadratic,etc.?). (b) Run FlowLab using template Hydraulic_press_liq-uid for the given conditions and the same range of density.Compare to your manual calculations.

3–58 Consider a hydraulic jack as in Prob.3–55. The specific gravity of the hydraulic oil is 0.870, andthe area ratio is A2/A1 � 400 (A1 � 1 cm2 and A2 � 0.04 m2).(a) For a given elevation difference h � 2 m, calculate forceF2 as a function of applied force F1 for the range 10 N � F1� 500 N. Discuss the relationship between F2 and F1 (is therelationship linear, quadratic, etc.?). (b) Run FlowLab usingtemplate Hydraulic_press_force for the given conditions andthe same range of F1. Compare to your manual calculations.

3–59 Consider a hydraulic jack as in Prob.3–55. The specific gravity of the hydraulic oil is 0.870, andthe area ratio is A2/A1 � 400 (A1 � 1 cm2 and A2 � 0.04 m2).(a) For a given applied force F1 � 50 N, calculate force F2 asa function of elevation difference h for the range 0 m � h �4 m. Discuss the relationship between F2 and h (is the rela-tionship linear, quadratic, etc.?). (b) Run FlowLab using tem-plate Hydraulic_press_elevation for the given conditions andthe same range of h. Compare to your manual calculations.

Fluid Statics: Hydrostatic Forces on Plane and Curved Surfaces

3–60C Define the resultant hydrostatic force acting on asubmerged surface, and the center of pressure.

3–61C Someone claims that she can determine the magni-tude of the hydrostatic force acting on a plane surface sub-merged in water regardless of its shape and orientation if sheknew the vertical distance of the centroid of the surface fromthe free surface and the area of the surface. Is this a validclaim? Explain.

Water A

MercurySG = 13.6

2a

u

26.8 cma

a

Water B

FIGURE P3–53

80 cmOil

SG = 0.90

Water

GlycerinSG = 1.26

35 cm

18 cm

15 cm

90 cm

A

FIGURE P3–54

Hydraulic oilSG = 0.870

h

A1

F1 A2

F2

FIGURE P3–55

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3–83 Repeat Prob. 3–82 for a radius of 4 m for the gate.Answer: 314 kN

3–84 Consider a flat plate of thickness t, width w into thepage, and length b submerged in water, as in Fig. P3–84. Thedepth of water from the surface to the center of the plate is H,and angle u is defined relative to the center of the plate. (a)Generate an equation for the force F on the upper face of theplate as a function of (at most) H, b, t, w, g, r, and u. Ignoreatmospheric pressure. In other words, calculate the force thatis in addition to the force due to atmospheric pressure. (b) As atest of your equation, let H � 1.25 m, b � 1 m, t � 0.2 m, w �1 m, g � 9.807 m/s2, r � 998.3 kg/m3, and u � 30o. If yourequation is correct, you should get a force of 11.4 kN.

120PRESSURE AND FLUID STATICS

in water as in Fig. P3–87. The depth of water is h. (a) Gener-ate an equation for the force F acting on the cylinder as afunction of (at most) h, R, w, g, r, and L. Ignore atmosphericpressure since it acts on both sides of the cylinder. (b) As atest of your equation, let h � 5 m, R � 0.5 m, w � 1 m, g �9.807 m/s2, and r � 998.3 kg/m3. If your equation is correct,you should get a force of 11.4 kN.

3–88 Consider the cylindrical gate of Prob.3–87. (a) For the values given there, calculate force F as afunction of depth h in the range 0.6 � h � 5 m. Discuss therelationship between F and h (is the relationship linear, qua-dratic, etc.?) (b) Run FlowLab using templateCylindrical_gate_depth for the given conditions and the samerange of depth h. Compare to your manual calculations.

3–89 Consider the cylindrical gate of Prob.3–87. (a) For the values given there and with h � 5 m, calcu-late force F as a function of the width L of water in the reser-voir upstream of the gate in the range 0.2 � L � 2 m. Dis-cuss the relationship between F and L (is the relationshiplinear, quadratic, etc.?) (b) Run FlowLab using templateCylindrical_gate_width for the given conditions and the samerange of width L. Compare to your manual calculations.

Buoyancy

3–90C What is buoyant force? What causes it? What is themagnitude of the buoyant force acting on a submerged bodywhose volume is V? What are the direction and the line ofaction of the buoyant force?

3–91C Consider two identical spherical balls submerged inwater at different depths. Will the buoyant forces acting onthese two balls be the same or different? Explain.

3–92C Consider two 5-cm-diameter spherical balls—onemade of aluminum, the other of iron—submerged in water.Will the buoyant forces acting on these two balls be the sameor different? Explain.

3–93C Consider a 3-kg copper cube and a 3-kg copper ballsubmerged in a liquid. Will the buoyant forces acting onthese two bodies be the same or different? Explain.

3–85 Consider the submerged plate of Prob.3–84. (a) For the values given there, and for the plate alignedvertically (u � 90o), calculate force F as a function of depthH in the range 1 � H � 10 m. Discuss the relationshipbetween F and H (is the relationship linear, quadratic, etc.?)(b) Run FlowLab using template Submerged_plate_depth forthe given conditions and the same range of depth H. Com-pare to your manual calculations. Note: FlowLab usesabsolute pressure in its solution, so you need to subtract theforce due to atmospheric pressure from the FlowLab results.

3–86 Consider the submerged plate of Prob.3–84. (a) For the values given there, and for the plate centerfixed at H � 1.25 m), calculate force F as a function of angleu in the range 0 � u � 90o. Discuss the relationship betweenF and u (is the relationship linear, quadratic, etc.?) (b) RunFlowLab using template Submerged_plate_angle for thegiven conditions and the same range of angle u. Compare toyour manual calculations. Note: FlowLab uses absolute pres-sure in its solution, so you need to subtract the force due toatmospheric pressure from the FlowLab results.

3–87 Consider a two-dimensional hinged cylindrical gate ofradius R and width w into the page. The cylinder is resting atground level with one quarter of its circumference submerged

R

R

F

Hinge

h

L

FIGURE P3–87

H

F

t

b

FIGURE P3–84

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