Fluid Mech Hw Problems Ch3

84

Chapter 3 Fluid Statics

Case Study The Falkirk Wheel

The Falkirk Wheel.

Hydrostatics, the study of fluids at rest, is an ancient discipline, so one might think there are no new or exciting applications still to be developed. The Falkirk wheel in Scotland is a dramatic demonstration that

this is not the case; it is a novel replacement for a lock, a device for moving a boat from one water level to another. The wheel, which has a diameter of 35 m, consists of two sets of axe-shaped opposing arms (which take the shape of a Celtic-inspired, doubleheaded axe). Sitting in bearings in the ends of these arms are two water-filled caissons, or tanks, each with a capacity of 80,000 gal. The hydrostatics concept of Archimedes’ principle, which we studied in this chapter, states that floating objects displace their own weight of water. Hence, the boat shown entering the lower caisson displaces water from the caisson weighing exactly the same as the boat itself. This means the entire wheel remains balanced at all times (both caissons always carry the same weight, whether containing boats or not), and so, despite its enormous mass, it rotates through 180 in less than four minutes while using very little power. The electric motors used for this use 22.5 kilowatts (kW) of power, so the energy used in four minutes is about 1.5 kilowatthours (kWh); even at current prices, this works out to be only a few cents worth of energy.

References 1. Burcher, R., and L. Rydill, Concepts in Submarine Design. Cambridge, UK: Cambridge University Press, 1994.

2. Marchaj, C. A., Aero-Hydrodynamics of Sailing, rev. ed. Camden, ME: International Marine Publishing, 1988.

Problems 3.1 Compressed nitrogen (140 lbm) is stored in a spherical tank of diameter D = 2.5 ft at a temperature of 77 F. What is the pressure inside the tank? If the maximum allowable stress in the tank is 30 ksi, find the minimum theoretical wall thickness of the tank.

3.4

When you are on a mountain face and boil water, you notice that the water temperature is 195 F. What is your approximate altitude? The next day, you are at a location where it boils at 185 F. How high did you climb between the two days? Assume a U.S. Standard Atmosphere.

Standard Atmosphere

3.2

Because the pressure falls, water boils at a lower temperature with increasing altitude. Consequently, cake mixes and boiled eggs, among other foods, must be cooked different lengths of time. Determine the boiling temperature of water at 1000 and 2000 m elevation on a standard day, and compare with the sea-level value.

Pressure Variation in a Static Fluid

3.5 A 125-mL cube of solid oak is held submerged by a tether as shown. Calculate the actual force of the water on the bottom surface of the cube and the tension in the tether. patm

3.3

Ear “popping” is an unpleasant phenomenon sometimes experienced when a change in pressure occurs, for example in a fast-moving elevator or in an airplane. If you are in a two-seater airplane at 3000 m and a descent of 100 m causes your ears to “pop,” what is the pressure change that your ears “pop” at, in millimeters of mercury? If the airplane now rises to 8000 m and again begins descending, how far will the airplane descend before your ears “pop” again? Assume a U.S. Standard Atmosphere.

d = 10 mm

Oil 0.5 m

0.3 m

SG = 0.8

H = 200 mm

Water

P3.5

3.6

F

Diameter, D = 50 mm

h = 25 mm

P3.6

The tube shown is filled with mercury at 20 C. Calculate the force applied to the piston.

Problems

3.7

The following pressure and temperature measurements were taken by a meteorological balloon rising through the lower atmosphere:

p (psia) 14.71 14.62 14.53 14.45 14.36 14.27 14.18 14.1 14.01 13.92 13.84 T ( F)

53.6

52

50.9

50.4

50.2

50

50.5 51.4 52.9

54

53.8

The initial values (top of table) correspond to ground level. Using the ideal gas law (p = ρRT with R = 53.3 ft ! lbf/ lbm ! R), compute and plot the variation of air density (in lbm/ft3) with height.

3.8

A hollow metal cube with sides 100 mm floats at the interface between a layer of water and a layer of SAE 10W oil such that 10% of the cube is exposed to the oil. What is the pressure difference between the upper and lower horizontal surfaces? What is the average density of the cube?

3.9

Your pressure gage indicates that the pressure in your cold tires is 0.25 MPa (gage) on a mountain at an elevation of 3500 m. What is the absolute pressure? After you drive down to sea level, your tires have warmed to 25 C. What pressure does your gage now indicate? Assume a U.S. Standard Atmosphere.

85

straw is in the bottom 15 cm. What is the pressure in the straw just below your thumb? Ignore any surface tension effects.

3.16 A water tank filled with water to a depth of 16 ft has in inspection cover (1 in. 3 1 in.) at its base, held in place by a plastic bracket. The bracket can hold a load of 9 lbf. Is the bracket strong enough? If it is, what would the water depth have to be to cause the bracket to break?

3.17 A container with two circular vertical tubes of diameters d1 5 39.5 mm and d2 5 12.7 mm is partially filled with mercury. The equilibrium level of the liquid is shown in the left diagram. A cylindrical object made from solid brass is placed in the larger tube so that it floats, as shown in the right diagram. The object is D 5 37.5 mm in diameter and H 5 76.2 mm high. Calculate the pressure at the lower surface needed to float the object. Determine the new equilibrium level, h, of the mercury with the brass cylinder in place. d1

Brass

d2

h

3.10

An air bubble, 0.3 in. in diameter, is released from the regulator of a scuba diver swimming 100 ft below the sea surface. (The water temperature is 86 F.) Estimate the diameter of the bubble just before it reaches the water surface.

3.11 A cube with 6 in. sides is suspended in a fluid by a wire. The top of the cube is horizontal and 8 in. below the free surface. If the cube has a mass of 2 slugs and the tension in the wire is T 5 50.7 lbf, compute the fluid specific gravity, and from this determine the fluid. What are the gage pressures on the upper and lower surfaces?

x Mercury

P3.17 3.18 A partitioned tank as shown contains water and mercury. What is the gage pressure in the air trapped in the left chamber? What pressure would the air on the left need to be pumped to in order to bring the water and mercury free surfaces level?

3.12

0.75 m

Assuming the bulk modulus is constant for seawater, derive an expression for the density variation with depth, h, below the surface. Show that the result may be written

3.75 m

Water

3m

1m

ρ " ρ0 þ bh where ρ0 is the density at the surface. Evaluate the constant b. Then, using the approximation, obtain an equation for the variation of pressure with depth below the surface. Determine the depth in feet at which the error in pressure predicted by the approximate solution is 0.01 percent.

3.13

Oceanographic research vessels have descended to 6.5 mi below sea level. At these extreme depths, the compressibility of seawater can be significant. One may model the behavior of seawater by assuming that its bulk modulus remains constant. Using this assumption, evaluate the deviations in density and pressure compared with values computed using the incompressible assumption at a depth, h, of 6.5 mi in seawater. Express your answers as a percentage. Plot the results over the range 0 # h # 7 mi.

Mercury

2.9 m

P3.18, P3.19

3.19

In the tank of Problem 3.18, if the opening to atmosphere on the right chamber is first sealed, what pressure would the air on the left now need to be pumped to in order to bring the water and mercury free surfaces level? (Assume the air trapped in the right chamber behaves isothermally.)

3.20 Consider the two-fluid manometer shown. Calculate the applied pressure difference. p1

3.14

p2

Water

An inverted cylindrical container is lowered slowly beneath the surface of a pool of water. Air trapped in the container is compressed isothermally as the hydrostatic pressure increases. Develop an expression for the water height, y, inside the container in terms of the container height, H, and depth of submersion, h. Plot y/H versus h/H.

l= 10.2 mm

3.15 You close the top of your straw with your thumb and lift the straw out of your glass containing Coke. Holding it vertically, the total length of the straw is 45 cm, but the Coke held in the

3m

P3.20

Carbon tetrachloride

Chapter 3 Fluid Statics

86

3.21 A manometer is formed from glass tubing with uniform inside diameter, D 5 6.35 mm, as shown. The U-tube is --- 5 3.25 cm3 of Meriam red partially filled with water. Then V oil is added to the left side. Calculate the equilibrium height, H, when both legs of the U-tube are open to the atmosphere.

Meriam red oil

Water (Tank 1)

Water (Tank 2)

H

Oil

Equilibrium level Water

P3.25

D

3.26 Water flows downward along a pipe that is inclined at 30 P3.21

3.22

The manometer shown contains water and kerosene. With both tubes open to the atmosphere, the free-surface elevations differ by H0 5 20.0 mm. Determine the elevation difference when a pressure of 98.0 Pa (gage) is applied to the right tube. p1

H0 = 20 mm

below the horizontal, as shown. Pressure difference pA 2 pB is due partly to gravity and partly to friction. Derive an algebraic expression for the pressure difference. Evaluate the pressure difference if L 5 5 ft and h 5 6 in.

L

p2

A Water

Liquid A Kerosene

30°

B

a z

h

Water

Liquid B

h __ 2

Mercury

g

h __ 2

P3.26 P3.22

P3.23

3.23 The manometer shown contains two liquids. Liquid A has SG 5 0.88 and liquid B has SG 5 2.95. Calculate the deflection, h, when the applied pressure difference is p1 2 p2 5 18 lbf/ft2.

3.27

Consider a tank containing mercury, water, benzene, and air as shown. Find the air pressure (gage). If an opening is made in the top of the tank, find the equilibrium level of the mercury in the manometer. D = 0.25 m

3.24 Determine the gage pressure in kPa at point a, if liquid A has SG = 1.20 and liquid B has SG = 0.75. The liquid surrounding point a is water, and the tank on the left is open to the atmosphere.

d = 0.025 m Air 0.1 m

Benzene

0.1 m

Water

0.1 m

Mercury

0.3 m

Liquid A 0.9 m

0.25 m Water a 0.125 m 0.4 m Liquid B

P3.24

3.25

An engineering research company is evaluating using a sophisticated $80,000 laser system between two large water storage tanks. You suggest that the job can be done with a $200 manometer arrangement. Oil less dense than water can be used to give a significant amplification of meniscus movement; a small difference in level between the tanks will cause a much larger deflection in the oil levels in the manometer. If you set up a rig using Meriam red oil as the manometer fluid, determine the amplification factor that will be seen in the rig.

P3.27

3.28

A reservoir manometer has vertical tubes of diameter D 5 18 mm and d 5 6 mm. The manometer liquid is Meriam red oil. Develop an algebraic expression for liquid deflection L in the small tube when gage pressure ∆p is applied to the reservoir. Evaluate the liquid deflection when the applied pressure is equivalent to 25 mm of water (gage). ∆p d = 6 mm

D = 18 mm

L Equilibrium level

x

P3.28

Problems

3.29 A rectangular tank, open to the atmosphere, is filled with water to a depth of 2.5 m as shown. A U-tube manometer is connected to the tank at a location 0.7 m above the tank bottom. If the zero level of the Meriam blue manometer fluid is 0.2 m below the connection, determine the deflection l after the manometer is connected and all air has been removed from the connecting leg.

87

Determine the sensitivity of this manometer. Plot the manometer sensitivity as a function of the diameter ratio d2/d1. patm

patm

patm + ∆ p

patm

g

h Oil (SG = 0.85) Zero level 3m

2.5 m

d1 = 10 mm

d2 = 15 mm

0.2 m Water l

0.7 m

3.36

P3.29, P3.31, P3.37

3.30

A reservoir manometer is calibrated for use with a liquid of specific gravity 0.827. The reservoir diameter is 5/8 in. and the (vertical) tube diameter is 3/16 in. Calculate the required distance between marks on the vertical scale for 1 in. of water pressure difference.

3.31

The manometer fluid of Problem 3.29 is replaced with mercury (same zero level). The tank is sealed and the air pressure is increased to a gage pressure of 0.5 atm. Determine the deflection l.

3.32

The inclined-tube manometer shown has D 5 96 mm and d 5 8 mm. Determine the angle, θ, required to provide a 5 : 1 increase in liquid deflection, L, compared with the total deflection in a regular U-tube manometer. Evaluate the sensitivity of this inclined-tube manometer. ∆p d D

L θ

P3.32, P3.33

3.33

The inclined-tube manometer shown has D 5 76 mm and d 5 8 mm, and is filled with Meriam red oil. Compute the angle, θ, that will give a 15-cm oil deflection along the inclined tube for an applied pressure of 25 mm of water (gage). Determine the sensitivity of this manometer.

3.34

A barometer accidentally contains 6.5 inches of water on top of the mercury column (so there is also water vapor instead of a vacuum at the top of the barometer). On a day when the temperature is 70 F, the mercury column height is 28.35 inches (corrected for thermal expansion). Determine the barometric pressure in psia. If the ambient temperature increased to 85 F and the barometric pressure did not change, would the mercury column be longer, be shorter, or remain the same length? Justify your answer.

3.35

P3.35

A student wishes to design a manometer with better sensitivity than a water-filled U-tube of constant diameter. The student’s concept involves using tubes with different diameters and two liquids, as shown. Evaluate the deflection h of this manometer, if the applied pressure difference is ∆p 5 250 N/m2.

A water column stands 50 mm high in a 2.5-mm diameter glass tube. What would be the column height if the surface tension were zero? What would be the column height in a 1.0-mm diameter tube?

3.37

If the tank of Problem 3.29 is sealed tightly and water drains slowly from the bottom of the tank, determine the deflection, l, after the system has attained equilibrium.

3.38

Consider a small-diameter open-ended tube inserted at the interface between two immiscible fluids of different densities. Derive an expression for the height difference ∆h between the interface level inside and outside the tube in terms of tube diameter D, the two fluid densities ρ1 and ρ2, and the surface tension σ and angle θ for the two fluids’ interface. If the two fluids are water and mercury, find the height difference if the tube diameter is 40 mils (1 mil = 0.001 in.).

3.39

You have a manometer consisting of a tube that is 0.5 in. inner diameter (ID). On one side, the manometer leg contains mercury, 0.6 in.3 of an oil (SG = 1.4), and 0.2 in.3 of air as a bubble in the oil. The other leg contains only mercury. Both legs are open to the atmosphere and are in a static condition. An accident occurs in which 0.2 in.3 of the oil and the air bubble are removed from one leg. How much do the mercury height levels change?

3.40

Compare the height due to capillary action of water exposed to air in a circular tube of diameter D 5 0.5 mm, and between two infinite vertical parallel plates of gap a 5 0.5 mm.

3.41 Two vertical glass plates 12 in. 3 12 in. are placed in an open tank containing water. At one end the gap between the plates is 0.004 in., and at the other it is 0.080 in. Plot the curve of water height between the plates from one end of the pair to the other.

3.42

Based on the atmospheric temperature data of the U.S. Standard Atmosphere of Fig. 3.3, compute and plot the pressure variation with altitude, and compare with the pressure data of Table A.3.

3.43

On a certain calm day, a mild inversion causes the atmospheric temperature to remain constant at 30 C between sea level and 5000-m altitude. Under these conditions, (a) calculate the elevation change for which a 3 percent reduction in air pressure occurs, (b) determine the change of elevation necessary to effect a 5 percent reduction in density, and (c) plot p2/p1 and ρ2/ρ1 as a function of ∆z.

88

Chapter 3 Fluid Statics

3.44

At ground level in Denver, Colorado, the atmospheric pressure and temperature are 83.2 kPa and 25 C. Calculate the pressure on Pike’s Peak at an elevation of 2690 m above the city assuming (a) an incompressible and (b) an adiabatic atmosphere. Plot the ratio of pressure to ground level pressure in Denver as a function of elevation for both cases.

Air 4 in

H 2O A Meriam Blue

Meriam Blue

3.45

The Martian atmosphere behaves as an ideal gas with mean molecular mass of 32.0 and constant temperature of 200 K. The atmospheric density at the planet surface is ρ 5 0.015 kg/m3 and Martian gravity is 3.92 m/s2. Calculate the density of the Martian atmosphere at height z 5 20 km above the surface. Plot the ratio of density to surface density as a function of elevation. Compare with that for data on the Earth’s atmosphere.

4 in

Air Meriam Blue

B

6 in

C

P3.49

Hydrostatic Force on Submerged Surfaces

3.50 Semicircular plane gate AB is hinged along B and held by horizontal force FA applied at A. The liquid to the left of the gate is water. Calculate the force FA required for equilibrium.

3.46

A door 1 m wide and 1.5 m high is located in a plane vertical wall of a water tank. The door is hinged along its upper edge, which is 1 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/F0 and yu/yc for different values of the surface pressure ratio ps/patm. (F0 is the resultant force when ps 5 patm.)

3.47

A door 1 m wide and 1.5 m high is located in a plane vertical wall of a water tank. The door is hinged along its upper edge, which is 1 m below the water surface. Atmospheric pressure acts on the outer surface of the door. (a) If the pressure at the water surface is atmospheric, what force must be applied at the lower edge of the door in order to keep the door from opening? (b) If the water surface gage pressure is raised to 0.5 atm, what force must be applied at the lower edge of the door to keep the door from opening? (c) Find the ratio F/F0 as a function of the surface pressure ratio ps/patm. (F0 is the force required when ps 5 patm.)

3.48

A hydropneumatic elevator consists of a piston-cylinder assembly to lift the elevator cab. Hydraulic oil, stored in an accumulator tank pressurized by air, is valved to the piston as needed to lift the elevator. When the elevator descends, oil is returned to the accumulator. Design the least expensive accumulator that can satisfy the system requirements. Assume the lift is 3 floors, the maximum load is 10 passengers, and the maximum system pressure is 800 kPa (gage). For column bending strength, the piston diameter must be at least 150 mm. The elevator cab and piston have a combined mass of 3000 kg, and are to be purchased. Perform the analysis needed to define, as a function of system operating pressure, the piston diameter, the accumulator volume and diameter, and the wall thickness. Discuss safety features that your company should specify for the complete elevator system. Would it be preferable to use a completely pneumatic design or a completely hydraulic design? Why?

3.49 Find the pressures at points A, B, and C, as shown in the figure, and in the two air cavities.

H = 25 ft A

FA Gate: side view

R = 10 ft B

P3.50

3.51 A triangular access port must be provided in the side of a form containing liquid concrete. Using the coordinates and dimensions shown, determine the resultant force that acts on the port and its point of application.

Liquid a = 1.25 ft concrete

y Port

b = 1 ft

P3.51

3.52 A plane gate of uniform thickness holds back a depth of water as shown. Find the minimum weight needed to keep the gate closed.

θ = 30°

L=3m

Water w=2m

P3.52

3.53 Consider a semicylindrical trough of radius R and length L. Develop general expressions for the magnitude and line of action of the hydrostatic force on one end, if the trough is partially filled with water and open to atmosphere. Plot the results (in nondimensional form) over the range of water depth 0 # d/R # 1.

Problems

3.54

A rectangular gate (width w 5 2 m) is hinged as shown, with a stop on the lower edge. At what depth H will the gate tip?

89

3.60 A large open tank contains water and is connected to a 6-ft-diameter conduit as shown. A circular plug is used to seal the conduit. Determine the magnitude, direction, and location of the force of the water on the plug.

Water H

Hinge

0.55 m

9 ft

Stop

0.45 m

P3.54 For a mug of tea (65 mm diameter), imagine it cut symmetrically in half by a vertical plane. Find the force that each half experiences due to an 80-mm depth of tea.

Water

3.55

Plug

3.56 Gates in the Poe Lock at Sault Ste. Marie, Michigan, close a channel W 5 34 m wide, L 5 360 m long, and D 5 10 m deep. The geometry of one pair of gates is shown; each gate is hinged at the channel wall. When closed, the gate edges are forced together at the center of the channel by water pressure. Evaluate the force exerted by the water on gate A. Determine the magnitude and direction of the force components exerted by the gate on the hinge. (Neglect the weight of the gate.) Plan view:

y Hinge

P3.60

3.61

What holds up a car on its rubber tires? Most people would tell you that it is the air pressure inside the tires. However, the air pressure is the same all around the hub (inner wheel), and the air pressure inside the tire therefore pushes down from the top as much as it pushes up from below, having no net effect on the hub. Resolve this paradox by explaining where the force is that keeps the car off the ground.

3.62

x Gate A Water

D = 6 ft

W = 34 m 15°

The circular access port in the side of a water standpipe has a diameter of 0.6 m and is held in place by eight bolts evenly spaced around the circumference. If the standpipe diameter is 7 m and the center of the port is located 12 m below the free surface of the water, determine (a) the total force on the port and (b) the appropriate bolt diameter.

3.63 P3.56 3.57 A section of vertical wall is to be constructed from readymix concrete poured between forms. The wall is to be 3 m high, 0.25 m thick, and 5 m wide. Calculate the force exerted by the ready-mix concrete on each form. Determine the line of application of the force.

As water rises on the left side of the rectangular gate, the gate will open automatically. At what depth above the hinge will this occur? Neglect the mass of the gate. A Gate D

3.58

A window in the shape of an isosceles triangle and hinged at the top is placed in the vertical wall of a form that contains liquid concrete. Determine the minimum force that must be applied at point D to keep the window closed for the configuration of form and concrete shown. Plot the results over the range of concrete depth 0 # c # a b = 0.3 m

Hinge line

5 ft 12 ft

Hinge

Water O 8 ft

P3.63

a = 0.4 m

P3.64

Neglecting the weight of the gate, determine the force in bar AB. The gate is sealed at C. The gate shown is 3 m wide and for analysis can be considered massless. For what depth of water will this rectangular gate be in equilibrium as shown?

2500 kg

D

P3.58

3.59 Solve Example 3.6 again using the two separate pressures method. Consider the distributed force to be the sum of a force F1 caused by the uniform gage pressure and a force F2 caused by the liquid. Solve for these forces and their lines of action. Then sum moments about the hinge axis to calculate Ft.

C 6 ft

3.64 The gate AOC shown is 6 ft wide and is hinged along O. 3.65

c = 0.25 m

B

3 ft

d

5m 60°

P3.65

90

Chapter 3 Fluid Statics

3.66

The gate shown is hinged at H. The gate is 3 m wide normal to the plane of the diagram. Calculate the force required at A to hold the gate closed.

3.70

For the dam shown, what is the vertical force of the water on the dam? 3 ft 3 ft

1.5 m

3 ft

H

F A

3m

Water

Top

3 ft 3 ft

30°

6 ft

3 ft 3 ft

3 ft 3 ft 3 ft Water

P3.66

3 ft 3 ft

3.67 A long, square wooden block is pivoted along one edge.

3 ft

The block is in equilibrium when immersed in water to the depth shown. Evaluate the specific gravity of the wood, if friction in the pivot is negligible. L

3 ft Front

Side

P3.70

Air

3.71 d = 0.5 m Wood

L = 1.0 m

Water

The gate shown is 1.5 m wide and pivoted at O; a 5 1.0 m22, D 5 1.20 m, and H 5 1.40 m. Determine (a) the magnitude and moment of the vertical component of the force about O, and (b) the horizontal force that must be applied at point A to hold the gate in position.

Pivot, O

y

P3.67

3.68 A solid concrete dam is to be built to hold back a depth D of water. For ease of construction the walls of the dam must be planar. Your supervisor asks you to consider the following dam cross-sections: a rectangle, a right triangle with the hypotenuse in contact with the water, and a right triangle with the vertical in contact with the water. She wishes you to determine which of these would require the least amount of concrete. What will your report say? You decide to look at one more possibility: a nonright triangle, as shown. Develop and plot an expression for the cross-section area A as a function of a, and find the minimum cross-sectional area. Water

A

Gate Water

H

D

x = ay3

x

O

P3.71

3.72 The parabolic gate shown is 2 m wide and pivoted at O; c 5 0.25 m21, D 5 2 m, and H 5 3 m. Determine (a) the magnitude and line of action of the vertical force on the gate due to the water, (b) the horizontal force applied at A required to maintain the gate in equilibrium, and (c) the vertical force applied at A required to maintain the gate in equilibrium. y

D A

H

Gate

Water D

αb

y = cx2

b

x O

P3.68

P3.72

3.69 For the geometry shown, what is the vertical force on the dam? The steps are 0.5 m high, 0.5 m deep, and 3 m wide.

3.73

Liquid concrete is poured into the form (R = 2 ft). The form is w = 15 ft wide normal to the diagram. Compute the magnitude of the vertical force exerted on the form by the concrete, and specify its line of action. FV

Water

Dam

Concrete

dF h

R

θ

P3.69

P3.73

Problems

91

3.74 An open tank is filled with water to the depth indicated.

3.79 Consider the cylindrical weir of diameter 3 m and length

Atmospheric pressure acts on all outer surfaces of the tank. Determine the magnitude and line of action of the vertical component of the force of the water on the curved part of the tank bottom.

6 m. If the fluid on the left has a specific gravity of 1.6, and on the right has a specific gravity of 0.8, find the magnitude and direction of the resultant force.

D = 3.0 m

3.0 m

1.5 m Water

P3.79, P3.80

10 ft 4 ft

3.80 A cylindrical weir has a diameter of 3 m and a length of

10 ft 12 ft

6 m. Find the magnitude and direction of the resultant force acting on the weir from the water.

P3.74

3.75

A spillway gate formed in the shape of a circular arc is w m wide. Find the magnitude and line of action of the vertical component of the force due to all fluids acting on the gate. A = 1 ft B = 10 ft2 Water

2 ft

R

xy – Ay = B 10 ft

H = 9 ft

h=R y

Water

x 7 ft

P3.75

1.67 ft

3.81 A cylindrical log of diameter D rests against the top of a dam. The water is level with the top of the log and the center of the log is level with the top of the dam. Obtain expressions for (a) the mass of the log per unit length and (b) the contact force per unit length between the log and dam.

3.82

A curved surface is formed as a quarter of a circular cylinder with R 5 0.750 m as shown. The surface is w 5 3.55 m wide. Water stands to the right of the curved surface to depth H 5 0.650 m. Calculate the vertical hydrostatic force on the curved surface. Evaluate the line of action of this force. Find the magnitude and line of action of the horizontal force on the surface.

P3.76

3.76

A dam is to be constructed using the cross-section shown. Assume the dam width is w 5 160 ft. For water height H 5 9 ft, calculate the magnitude and line of action of the vertical force of water on the dam face. Is it possible for water forces to overturn this dam? Under what circumstances will this happen?

Water

θ

3.77

A Tainter gate used to control water flow from the Uniontown Dam on the Ohio River is shown; the gate width is w 5 35 m. Determine the magnitude, direction, and line of action of the force from the water acting on the gate.

H

R

P3.82

Buoyancy and Stability

3.83 If you throw an anchor out of your canoe but the rope is too short for the anchor to rest on the bottom of the pond, will your canoe float higher, lower, or stay the same? Prove your answer.

R = 20 m

3.84

D = 10 m Water

P3.77

3.78

A gate, in the shape of a quarter-cylinder, hinged at A and sealed at B, is 3 m wide. The bottom of the gate is 4.5 m below the water surface. Determine the force on the stop at B if the gate is made of concrete; R 5 3 m.

A curved submerged surface, in the shape of a quarter cylinder with radius R 5 1.0 ft is shown. The form can withstand a maximum vertical load of 350 lbf before breaking. The width is w 5 4 ft. Find the maximum depth H to which the form may be filled. Find the line of action of the vertical force for this condition. Plot the results over the range of concrete depth 0 # H # R. w d R H

Water

A

y x

D B

R

P3.84

3.85 P3.78

y = ax 2

P3.85

The cross-sectional shape of a canoe is modeled by the curve y 5 ax2, where a 5 1.2 ft21 and the coordinates are in

92

Chapter 3 Fluid Statics

feet. Assume the width of the canoe is constant at w = 2 ft over its entire length L 5 18 ft. Set up a general algebraic expression relating the total mass of the canoe and its contents to distance d between the water surface and the gunwale of the floating canoe. Calculate the maximum total mass allowable without swamping the canoe.

3.86

The cylinder shown is supported by an incompressible liquid of density ρ, and is hinged along its length. The cylinder, of mass M, length L, and radius R, is immersed in liquid to depth H. Obtain a general expression for the cylinder specific gravity versus the ratio of liquid depth to cylinder radius, α 5 H/R, needed to hold the cylinder in equilibrium for 0 # α , 1. Plot the results. R Hinge

the net weight. Develop an expression for the specific gravity of a person in terms of their weight in air, net weight in water, and SG 5 f(T) for water.

*3.92

Quantify the statement, “Only the tip of an iceberg shows (in seawater).”

*3.93

An open tank is filled to the top with water. A steel cylindrical container, wall thickness δ 5 1 mm, outside diameter D 5 100 mm, and height H 5 1 m, with an open top, is gently placed in the water. What is the volume of water that overflows from the tank? How many 1 kg weights must be placed in the container to make it sink? Neglect surface tension effects.

*3.94 Quantify the experiment performed by Archimedes to identify the material content of King Hiero’s crown. Assume you can measure the weight of the king’s crown in air, Wa, and the weight in water, Ww. Express the specific gravity of the crown as a function of these measured values.

*3.95

Gas bubbles are released from the regulator of a submerged scuba diver. What happens to the bubbles as they rise through the seawater? Explain.

H

*3.96 P3.86

3.87 A canoe is represented by a right semicircular cylinder, with R 5 1.2 ft and L 5 17 ft. The canoe floats in water that is d 5 1 ft deep. Set up a general algebraic expression for the total mass (canoe and contents) that can be floated, as a function of depth. Evaluate for the given conditions. Plot the results over the range of water depth 0 # d # R.

3.88 A glass observation room is to be installed at the corner of the bottom of an aquarium. The aquarium is filled with seawater to a depth of 35 ft. The glass is a segment of a sphere, radius 5 ft, mounted symmetrically in the corner. Compute the magnitude and direction of the net force on the glass structure.

*3.89

A hydrometer is a specific gravity indicator, the value being indicated by the level at which the free surface intersects the stem when floating in a liquid. The 1.0 mark is the level when in distilled water. For the unit shown, the immersed volume in distilled water is 15 cm3. The stem is 6 mm in diameter. Find the distance, h, from the 1.0 mark to the surface when the hydrometer is placed in a nitric acid solution of specific gravity 1.5. 1.0 h 10 kg Nitric acid

P3.89

Water V = 0.025 m3

P3.90

*3.90

Find the specific weight of the sphere shown if its volume is 0.025m3. State all assumptions. What is the equilibrium position of the sphere if the weight is removed?

*3.91 The fat-to-muscle ratio of a person may be determined from a specific gravity measurement. The measurement is made by immersing the body in a tank of water and measuring

Hot-air ballooning is a popular sport. According to a recent article, “hot-air volumes must be large because air heated to 150 F over ambient lifts only 0.018 lbf/ft3 compared to 0.066 and 0.071 for helium and hydrogen, respectively.” Check these statements for sea-level conditions. Calculate the effect of increasing the hot-air maximum temperature to 250 F above ambient.

*3.97

Hydrogen bubbles are used to visualize water flow streaklines in the video, Flow Visualization. A typical hydrogen bubble diameter is d 5 0.001 in. The bubbles tend to rise slowly in water because of buoyancy; eventually they reach terminal speed relative to the water. The drag force of the water on a bubble is given by FD 5 3πµVd, where µ is the viscosity of water and V is the bubble speed relative to the water. Find the buoyancy force that acts on a hydrogen bubble immersed in water. Estimate the terminal speed of a bubble rising in water.

*3.98

It is desired to use a hot air balloon with a volume of 320,000 ft3 for rides planned in summer morning hours when the air temperature is about 48 F. The torch will warm the air inside the balloon to a temperature of 160 F. Both inside and outside pressures will be “standard” (14.7 psia). How much mass can be carried by the balloon (basket, fuel, passengers, personal items, and the component of the balloon itself) if neutral buoyancy is to be assured? What mass can be carried by the balloon to ensure vertical takeoff acceleration of 2.5 ft/s2? For this, consider that both balloon and inside air have to be accelerated, as well as some of the surrounding air (to make way for the balloon). The rule of thumb is that the total mass subject to acceleration is the mass of the balloon, all its appurtenances, and twice its volume of air. Given that the volume of hot air is fixed during the flight, what can the balloonists do when they want to go down?

*3.99

Scientific balloons operating at pressure equilibrium with the surroundings have been used to lift instrument packages to extremely high altitudes. One such balloon,

*These problems require material from sections that may be omitted without loss of continuity in the text material.

Problems filled with helium, constructed of polyester with a skin thickness of 0.013 mm and a diameter of 120 m, lifted a payload of 230 kg. The specific gravity of the skin material is 1.28. Determine the altitude to which the balloon would rise. Assume that the helium used in the balloon is in thermal equilibrium with the ambient air, and that the balloon is a perfect sphere.

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tank bottom. When the sphere is released, will it stay on the bottom of the tank or float to the surface?

R = 1 in.

H = 2.5 ft

*3.100

A helium balloon is to lift a payload to an altitude of 40 km, where the atmospheric pressure and temperature are 3.0 mbar and 225 C, respectively. The balloon skin is polyester with specific gravity of 1.28 and thickness of 0.015 mm. To maintain a spherical shape, the balloon is pressurized to a gage pressure of 0.45 mbar. Determine the maximum balloon diameter if the allowable tensile stress in the skin is limited to 62 MN/m2. What payload can be carried?

*3.101

A block of volume 0.025 m3 is allowed to sink in water as shown. A circular rod 5 m long and 20 cm2 in crosssection is attached to the weight and also to the wall. If the rod mass is 1.25 kg and the rod makes an angle of 12 degrees with the horizontal at equilibrium, what is the mass of the block? 0.25 m 5m

θ = 12°

a = 0.075 in.

P3.106

*3.107

A cylindrical timber, with D 5 1 ft and L 5 15 ft, is weighted on its lower end so that it floats vertically with 10 ft submerged in seawater. When displaced vertically from its equilibrium position, the timber oscillates or “heaves” in a vertical direction upon release. Estimate the frequency of oscillation in this heave mode. Neglect viscous effects and water motion.

*3.108

You are in the Bermuda Triangle when you see a bubble plume eruption (a large mass of air bubbles, similar to a foam) off to the side of the boat. Do you want to head toward it and be part of the action? What is the effective density of the water and air bubbles in the drawing on the right that will cause the boat to sink? Your boat is 10 ft long, and weight is the same in both cases.

M V = 0.025 m3

Water rushing in!

1ft

P3.101

*3.102

The stem of a glass hydrometer used to measure specific gravity is 5 mm in diameter. The distance between marks on the stem is 2 mm per 0.1 increment of specific gravity. Calculate the magnitude and direction of the error introduced by surface tension if the hydrometer floats in kerosene. (Assume the contact angle between kerosene and glass is 0 .)

*3.103 A sphere, of radius R, is partially immersed, to depth d, in a liquid of specific gravity SG. Obtain an algebraic expression for the buoyancy force acting on the sphere as a function of submersion depth d. Plot the results over the range of water depth 0 # d # 2R.

*3.104

If the mass M in Problem 3.101 is released from the rod, at equilibrium how much of the rod will remain submerged? What will be the minimum required upward force at the tip of the rod to just lift it out of the water?

7ft 60° Sea water

Sea water and air bubbles

Sinking

Floating

P3.108

*3.109 A bowl is inverted symmetrically and held in a dense fluid, SG 5 15.6, to a depth of 200 mm measured along the centerline of the bowl from the bowl rim. The bowl height is 80 mm, and the fluid rises 20 mm inside the bowl. The bowl is 100 mm inside diameter, and it is made from an old clay recipe, SG 5 6.1. The volume of the bowl itself is about 0.9 L. What is the force required to hold it in place?

*3.105 In a logging operation, timber floats downstream to a lumber mill. It is a dry year, and the river is running low, as low as 60 cm in some locations. What is the largest diameter log that may be transported in this fashion (leaving a minimum 5 cm clearance between the log and the bottom of the river)? For the wood, SG 5 0.8.

200 mm D = 100 mm

*3.106

A sphere of radius 1 in., made from material of specific gravity of SG 5 0.95, is submerged in a tank of water. The sphere is placed over a hole of radius 0.075 in., in the

20 mm

P3.109

*These problems require material from sections that may be omitted without loss of continuity in the text material.

80 mm

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Chapter 3 Fluid Statics

*3.110

In the “Cartesian diver” child’s toy, a miniature “diver” is immersed in a column of liquid. When a diaphragm at the top of the column is pushed down, the diver sinks to the bottom. When the diaphragm is released, the diver again rises. Explain how the toy might work.

*3.117

The U-tube shown is filled with water at T 5 68 F. It is sealed at A and open to the atmosphere at D. The tube is rotated about vertical axis AB at 1600 rpm. For the dimensions shown, would cavitation occur in the tube? A

*3.111

Consider a conical funnel held upside down and submerged slowly in a container of water. Discuss the force needed to submerge the funnel if the spout is open to the atmosphere. Compare with the force needed to submerge the funnel when the spout opening is blocked by a rubber stopper.

Water

H = 12 in.

ω

*3.112

Three steel balls (each about half an inch in diameter) lie at the bottom of a plastic shell floating on the water surface in a partially filled bucket. Someone removes the steel balls from the shell and carefully lets them fall to the bottom of the bucket, leaving the plastic shell to float empty. What happens to the water level in the bucket? Does it rise, go down, or remain unchanged? Explain.

*3.113

A proposed ocean salvage scheme involves pumping air into “bags” placed within and around a wrecked vessel on the sea bottom. Comment on the practicality of this plan, supporting your conclusions with analyses.

Fluids in Rigid-Body Motion

*3.114

A cylindrical container, similar to that analyzed in Example 3.10 (on the Web), is rotated at a constant rate of 2 Hz about its axis. The cylinder is 0.5 m in diameter and initially contains water that is 0.3 m deep. Determine the height of the liquid free surface at the center of the container. Does your answer depend on the density of the liquid? Explain.

*3.115

A crude accelerometer can be made from a liquidfilled U-tube as shown. Derive an expression for the liquid level difference h caused by an acceleration ~ a , in terms of the tube geometry and fluid properties. d

Liquid density, ρ

h a

y x

D

B

C L = 3 in.

P3.117, P3.118

*3.118

If the U-tube of Problem 3.117 is spun at 300 rpm, what will the pressure be at A? If a small leak appears at A, how much water will be lost at D?

*3.119 A centrifugal micromanometer can be used to create small and accurate differential pressures in air for precise measurement work. The device consists of a pair of parallel disks that rotate to develop a radial pressure difference. There is no flow between the disks. Obtain an expression for pressure difference in terms of rotation speed, radius, and air density. Evaluate the speed of rotation required to develop a differential pressure of 8 µm of water using a device with a 50 mm radius.

*3.120

A test tube is spun in a centrifuge. The tube support is mounted on a pivot so that the tube swings outward as rotation speed increases. At high speeds, the tube is nearly horizontal. Find (a) an expression for the radial component of acceleration of a liquid element located at radius r, (b) the radial pressure gradient dp/dr, and (c) the required angular velocity to generate a pressure of 250 MPa in the bottom of a test tube containing water. (The free surface and bottom radii are 50 and 130 mm, respectively.)

*3.121 A rectangular container, of base dimensions 0.4 m 3 L

P3.115

*3.116 A rectangular container of water undergoes constant acceleration down an incline as shown. Determine the slope of the free surface using the coordinate system shown.

0.2 m and height 0.4 m, is filled with water to a depth of 0.2 m; the mass of the empty container is 10 kg. The container is placed on a plane inclined at 30 to the horizontal. If the coefficient of sliding friction between the container and the plane is 0.3, determine the angle of the water surface relative to the horizontal.

*3.122

y g x

If the container of Problem 3.121 slides without friction, determine the angle of the water surface relative to the horizontal. What is the slope of the free surface for the same acceleration up the plane?

*3.123 ax = 3 m/s2

θ = 30°

A cubical box, 80 cm on a side, half-filled with oil (SG 5 0.80), is given a constant horizontal acceleration of 0.25 g parallel to one edge. Determine the slope of the free surface and the pressure along the horizontal bottom of the box.

*3.124 P3.116

Gas centrifuges are used in one process to produce enriched uranium for nuclear fuel rods. The maximum

*These problems require material from sections that may be omitted without loss of continuity in the text material.

Problems peripheral speed of a gas centrifuge is limited by stress considerations to about 950 ft/s. Assume a gas centrifuge containing uranium hexafluoride gas, with molecular gas Mm = 352, and ideal gas behavior. Develop an expression for the ratio of maximum pressure to pressure at the centrifuge axis. Evaluate the pressure ratio for a gas temperature of 620 F.

*3.127

*3.125 A pail, 400 mm in diameter and 400 mm deep, weighs

*3.128

15 N and contains 200 mm of water. The pail is swung in a vertical circle of 1-m radius at a speed of 5 m/s. Assume the water moves as a rigid body. At the instant when the pail is at the top of its trajectory, compute the tension in the string and the pressure on the bottom of the pail from the water.

*3.126 A partially full can of soda is placed at the outer edge of a child’s merry-go-round, located R 5 5 ft from the axis of rotation. The can diameter and height are 2.5 in. and 5 in., respectively. The can is half full, and the soda has specific gravity SG 5 1.05. Evaluate the slope of the liquid surface in the can if the merry-go-round spins at 20 rpm. Calculate the spin rate at which the can would spill, assuming no slippage between the can bottom and the merry-go-round. Would the can most likely spill or slide off the merry-go-round?

95

When a water polo ball is submerged below the surface in a swimming pool and released from rest, it is observed to pop out of the water. How would you expect the height to which it rises above the water to vary with depth of submersion below the surface? Would you expect the same results for a beach ball? For a table-tennis ball?

Cast iron or steel molds are used in a horizontalspindle machine to make tubular castings such as liners and tubes. A charge of molten metal is poured into the spinning mold. The radial acceleration permits nearly uniformly thick wall sections to form. A steel liner, of length L 5 6 ft, outer radius ro 5 6 in., and inner radius ri 5 4 in., is to be formed by this process. To attain nearly uniform thickness, the angular velocity should be at least 300 rpm. Determine (a) the resulting radial acceleration on the inside surface of the liner and (b) the maximum and minimum pressures on the surface of the mold.

*3.129 The analysis of Problem 3.121 suggests that it may be possible to determine the coefficient of sliding friction between two surfaces by measuring the slope of the free surface in a liquid-filled container sliding down an inclined surface. Investigate the feasibility of this idea.

*These problems require material from sections that may be omitted without loss of continuity in the text material.