The hemispherical dome in Fig. P2.91 weighs 30 kN and is fi lled with water and attached

For the two-dimensional stress fi eld shown in Fig. P2.1 it is found that xx 5 3000 lbf/ft2 yy 5 2000 lbf/ft2 xy 5 500 lbf/ft2 Find the shear and normal stresses (in lbf/ft 2 ) acting on plane AA cutting through the element at a 30 8 angle as shown.

For the two-dimensional stress fi eld shown in Fig. P2.1 suppose that xx 5 2000 lbf/ft2 yy 5 3000 lbf/ft2 n(AA) 5 2500 lbf/ft2 Compute ( a ) the shear stress xy and ( b) the shear stress on plane AA

A vertical, clean, glass piezometer tube has an inside diameter of 1 mm. When pressure is applied, water at 20 8 C rises into the tube to a height of 25 cm. After correcting for surface tension, estimate the applied pressure in Pa

Pressure gages, such as the bourdon gage in Fig. P2.4, are calibrated with a deadweight piston. If the bourdon gage is designed to rotate the pointer 10 degrees for every 2 psig of internal pressure, how many degrees does the pointer rotate if the piston and weight together total 44 newtons?

Quito, Ecuador, has an average altitude of 9350 ft. On a standard day, pressure gage A in a laboratory experiment reads 63 kPa and gage B reads 105 kPa. Express these readings in gage pressure or vacuum pressure, whichever is appropriate.

Any pressure reading can be expressed as a length or head, h 5 p / g . What is standard sea-level pressure expressed in ( a ) ft of glycerin, ( b ) inHg, ( c ) m of water, and ( d ) mm of ethanol? Assume all fl uids are at 20 8 C

La Paz, Bolivia, is at an altitude of approximately 12,000 ft. Assume a standard atmosphere. How high would the liquid rise in a methanol barometer, assumed at 20 8 C? Hint: Dont forget the vapor pressure.

Suppose, which is possible, that there is a half-mile deep lake of pure ethanol on the surface of Mars. Estimate the absolute pressure, in Pa, at the bottom of this speculative lake

A storage tank, 26 ft in diameter and 36 ft high, is fi lled with SAE 30W oil at 20 8 C. ( a ) What is the gage pressure, in lbf/in 2 , at the bottom of the tank? ( b ) How does your result in ( a ) change if the tank diameter is reduced to 15 ft? ( c ) Repeat ( a ) if leakage has caused a layer of 5 ft of water to rest at the bottom of the (full) tank.

A large open tank is open to sea-level atmosphere and fi lled with liquid, at 20 8 C, to a depth of 50 ft. The absolute pressure at the bottom of the tank is approximately 221.5 kPa. From Table A.3, what might this liquid be?

In Fig. P2.11, pressure gage A reads 1.5 kPa (gage). The fluids are at 20 8 C. Determine the elevations z , in meters, of the liquid levels in the open piezometer tubes B and C .

In Fig. P2.12 the tank contains water and immiscible oil at 20 8 C. What is h in cm if the density of the oil is 898 kg/m 3 ?

In Fig. P2.13 the 20 8 C water and gasoline surfaces are open to the atmosphere and at the same elevation. What is the height h of the third liquid in the right leg? 1.5 m 1 m h Water Gasoline Liquid, SG = 1.60

For the three-liquid system shown, compute h1 and h2 . Neglect the air density. h2 h1 Water Mercury 8 cm 27 cm 5 cm Oil, SG = 0.78

The airoilwater system in Fig. P2.15 is at 20 8 C. Knowing that gage A reads 15 lbf/in 2 absolute and gage B reads 1.25 lbf/in 2 less than gage C , compute ( a ) the specifi c weight of the oil in lbf/ft 3 and ( b ) the actual reading of gage C in lbf/in 2 absolute. P2.15 Air Oil Water 1 ft 1 ft 2 ft 2 ft A B C

If the absolute pressure at the interface between water and mercury in Fig. P2.16 is 93 kPa, what, in lbf/ft 2 , is ( a ) the pressure at the surface and ( b ) the pressure at the bottom of the container? P2.16 28 cm 8 cm Water Mercury 75 75 32 cm

The system in Fig. P2.17 is at 20 8 C. Determine the height h of the water in the left side. P2.17 Air, 200 Pa (gage) Oil, SG = 0.8 25 cm 20 c

The system in Fig. P2.18 is at 20 8 C. If atmospheric pressure is 101.33 kPa and the pressure at the bottom of the tank is 242 kPa, what is the specific gravity of fluid X ? P2.18 0.5 m

The U-tube in Fig. P2.19 has a 1-cm ID and contains mercury as shown. If 20 cm 3 of water is poured into the righthand leg, what will the free-surface height in each leg be after the sloshing has died down? P2.19 Mercury 10 cm 10 cm 10 cm

The hydraulic jack in Fig. P2.20 is fi lled with oil at 56 lbf/ft 3 . Neglecting the weight of the two pistons, what force F on the handle is required to support the 2000-lbf weight for this design?

At 20 8 C gage A reads 350 kPa absolute. What is the height h of the water in cm? What should gage B read in kPa absolute? See Fig. P2.21. Oil 3-in diameter 1 in 15 in 1-in diameter F 2000 lbf P2.20 P2.21

The fuel gage for a gasoline tank in a car reads proportional to the bottom gage pressure as in Fig. P2.22. If the tank is 30 cm deep and accidentally contains 2 cm of water plus gasoline, how many centimeters of air remain at the top when the gage erroneously reads full? P2.22 Gasoline SG = 0.68 30 cm Water h? 2 cm

In Fig. P2.23 both fluids are at 20 8 C. If surface tension effects are negligible, what is the density of the oil, in kg/m 3 ? Problems 107 P2.23 8 cm 6 cm Water Oil 10 cm

In Prob. 1.2 we made a crude integration of the density distribution ( z ) in Table A.6 and estimated the mass of the earths atmosphere to be m < 6 E18 kg. Can this result be used to estimate sea-level pressure on the earth? Conversely, can the actual sea-level pressure of 101.35 kPa be used to make a more accurate estimate of the atmospheric mass?

As measured by NASAs Viking landers, the atmosphere of Mars, where g < 3.71 m/s 2 , is almost entirely carbon dioxide, and the surface pressure averages 700 Pa. The temperature is cold and drops off exponentially: T < To e 2Cz , where C 5 1.3E-5 m 21 and To = 250 K. For example, at 20,000 m altitude, T < 193 K. ( a ) Find an analytic formula for the variation of pressure with altitude. ( b ) Find the altitude where pressure on Mars has dropped to 1 pascal.

For gases that undergo large changes in height, the linear approximation, Eq. (2.14), is inaccurate. Expand the troposphere power-law, Eq. (2.20), into a power series, and show that the linear approximation p < pa 2 a gz is adequate when z ! 2T0 (n 2 1)B where n 5 g RB

Conduct an experiment to illustrate atmospheric pressure. Note: Do this over a sink or you may get wet! Find a drinking glass with a very smooth, uniform rim at the top. Fill the glass nearly full with water. Place a smooth, light, fl at plate on top of the glass such that the entire rim of the glass is covered. A glossy postcard works best. A small index card or one fl ap of a greeting card will also work. See Fig. P2.27 a . ( a ) Hold the card against the rim of the glass and turn the glass upside down. Slowly release pressure on the card. Does the water fall out of the glass? Record your experimental observations. ( b ) Find an expression for the pressure at points 1 and 2 in Fig. P2.27 b . Note that the glass is now inverted, so the original top rim of the glass is at the bottom of the picture, and the original bottom of the glass is at the top of the picture. The weight of the card can be neglected. ( c ) Estimate the theoretical maximum glass height at which this experiment could still work, such that the water would not fall out of the glass. P2.27 a Card Top of glass Bottom of glass P2.27b Card Original top of glass

A correlation of computational fl uid dynamics results indicates that, all other things being equal, the distance traveled by a well-hit baseball varies inversely as the 0.36 power of the air density. If a home-run ball hit in Citi Field in New York travels 400 ft, estimate the distance it would travel in ( a ) Quito, Ecuador, and ( b ) Colorado Springs, CO.

Follow up on Prob. P2.8 by estimating the altitude on Mars where the pressure has dropped to 20 percent of its surface value. Assume an isothermal atmosphere, not the exponential variation of P2.25.

For the traditional equal-level manometer measurement in Fig. E2.3, water at 20 8 C fl ows through the plug device from a to b . The manometer fl uid is mercury. If L 5 12 cm and h 5 24 cm, ( a ) what is the pressure drop through the device? ( b ) If the water fl ows through the pipe at a velocity V 5 18 ft/s, what is the dimensionless loss coeffi cient of the device, defi ned by K 5 Dp /( V2 )? We will study loss coeffi cients in Chap. 6.

In Fig. P2.31 all fl uids are at 20 8 C. Determine the pressure difference (Pa) between points A and B . 20 cm 40 cm 8 cm 9 cm 14 cm A B Kerosene Air Mercury Water Benzene

For the inverted manometer of Fig. P2.32, all fluids are at 20 8 C. If pB 2 pA 5 97 kPa, what must the height H be in cm? P2.32 18 cm H 35 cm Mercury Water Meriam red oil, SG = 0.827 A B

In Fig. P2.33 the pressure at point A is 25 lbf/in 2 . All fl uids are at 20 8 C. What is the air pressure in the closed chamber B , in Pa? A 4 cm 3 cm 6 cm 8 cm 5 cm 3

Sometimes manometer dimensions have a signifi cant effect. In Fig. P2.34 containers ( a ) and ( b ) are cylindrical and conditions are such that pa 5 pb . Derive a formula for the pressure difference pa 2 pb when the oilwater interface on the right rises a distance Dh , h , for ( a ) d ! D and ( b ) d 5 0.15 D . What is the percentage change in the value of Dp ? P2.34 (a) (b) d L h

Water fl ows upward in a pipe slanted at 30 8 , as in Fig. P2.35. The mercury manometer reads h 5 12 cm. Both fl uids are at 20 8 C. What is the pressure difference p1 2 p2 in the pipe? P2.35 h (1) (

In Fig. P2.36 both the tank and the tube are open to the atmosphere. If L 5 2.13 m, what is the angle of tilt of the tube?

The inclined manometer in Fig. P2.37 contains Meriam red manometer oil, SG 5 0.827. Assume that the reservoir is very large. If the inclined arm is fi tted with graduations 1 in apart, what should the angle be if each graduation corresponds to 1 lbf/ft 2 gage pressure for pA ?

If the pressure in container A in Fig. P2.38 is 200 kPa, compute the pressure in container B . P2.38 Water Oil, SG = 0.8 Mercury A B 18 cm 22 cm 16 cm 8 cm

In Fig. P2.39 the right leg of the manometer is open to the atmosphere. Find the gage pressure, in Pa, in the air gap in the tank.

In Fig. P2.40, if pressure gage A reads 20 lbf/in 2 absolute, fi nd the pressure in the closed air space B . The manometer fl uid is Meriam red oil, SG 5 0.827.

1 The system in Fig. P2.41 is at 20 8 C. Compute the pressure at point A in lbf/ft 2 absolute. Water Water 5 in 10 in 6 in Mercury A Oil, SG = 0.85 pa = 14.7 lbf/in2 P

Very small pressure differences pA 2 pB can be measured accurately by the two-fl uid differential manometer in Fig. P2.42. Density 2 is only slightly larger than that of the upper fl uid 1 . Derive an expression for the proportionality between h and pA 2 pB if the reservoirs are very large. 110 Chapter 2 Pressure Distribution in a Fluid P2.42 h1 pA 1 pB 1 h 2

The traditional method of measuring blood pressure uses a sphygmomanometer , fi rst recording the highest ( systolic ) and then the lowest ( diastolic ) pressure from which fl owing Korotkoff sounds can be heard. Patients with dangerous hypertension can exhibit systolic pressures as high as 5 lbf/in 2 . Normal levels, however, are 2.7 and 1.7 lbf/in 2 , respectively, for systolic and diastolic pressures. The manometer uses mercury and air as fl uids. ( a ) How high in cm should the manometer tube be? ( b ) Express normal systolic and diastolic blood pressure in millimeters of mercury

Water fl ows downward in a pipe at 45 8 , as shown in Fig. P2.44. The pressure drop p1 2 p2 is partly due to gravity and partly due to friction. The mercury manometer reads a 6-in height difference. What is the total pressure drop p1 2 p2 in lbf/in 2 ? What is the pressure drop due to friction only between 1 and 2 in lbf/in 2 ? Does the manometer reading correspond only to friction drop? Why? P2.44 5 ft Flow 1 2 45 6 in Mercury

In Fig. P2.45, determine the gage pressure at point A in Pa. Is it higher or lower than atmospheric? P2.45 45 cm 30 cm 15 cm 40 cm patm Air Oil, SG = 0.85 Wat

In Fig. P2.46 both ends of the manometer are open to the atmosphere. Estimate the specifi c gravity of fl uid X . P2.46 7 cm 4 cm 6 cm 9 cm 5 cm 12 cm

The cylindrical tank in Fig. P2.47 is being fi lled with water at 20 8 C by a pump developing an exit pressure of 175 kPa. At the instant shown, the air pressure is 110 kPa and H 5 35 cm. The pump stops when it can no longer raise the water pressure. For isothermal air compression, estimate H at that time. Problems 111 P2.47 75 cm H 50 cm Air 20 C Water Pump

The system in Fig. P2.48 is open to 1 atm on the right side. ( a ) If L 5 120 cm, what is the air pressure in container A ? ( b ) Conversely, if pA 5 135 kPa, what is the length L ? 15 cm 32 cm Air 18 cm 35 Mercury Water L

Conduct the following experiment to illustrate air pressure. Find a thin wooden ruler (approximately 1 ft in length) or a thin wooden paint stirrer. Place it on the edge of a desk or table with a little less than half of it hanging over the edge lengthwise. Get two full-size sheets of newspaper; open them up and place them on top of the ruler, covering only the portion of the ruler resting on the desk as illustrated in Fig. P2.49. ( a ) Estimate the total force on top of the newspaper due to air pressure in the room. ( b ) Careful! To avoid potential injury, make sure nobody is standing directly in front of the desk. Perform a karate chop on the portion of the ruler sticking out over the edge of the desk. Record your results. ( c ) Explain your results.

A small submarine, with a hatch door 30 in in diameter, is submerged in seawater. ( a ) If the water hydrostatic force on the hatch is 69,000 lbf, how deep is the sub? ( b ) If the sub is 350 ft deep, what is the hydrostatic force on the hatch?

Gate AB in Fig. P2.51 is 1.2 m long and 0.8 m into the paper. Neglecting atmospheric pressure, compute the force F on the gate and its center-of-pressure position X . 8 m 6 m 1.2 m F 40

Example 2.5 calculated the force on plate AB and its line of action, using the moment-of-inertia approach. Some teachers say it is more instructive to calculate these by direct integration of the pressure forces. Using Figs. P2.52 and E2.5 a , ( a ) fi nd an expression for the pressure variation p ( ) along the plate; ( b ) integrate this expression to fi nd the total force F ; ( c ) integrate the moments about point A to fi nd the position of the center of pressure.

The Hoover Dam, in Arizona, encloses Lake Mead, which contains 10 trillion gallons of water. The dam is 1200 ft wide and the lake is 500 ft deep. ( a ) Estimate the hydrostatic force on the dam, in MN. ( b ) Explain how you might analyze the stress in the dam due to this hydrostatic force

In Fig. P2.54, the hydrostatic force F is the same on the bottom of all three containers, even though the weights of liquid above are quite different. The three bottom shapes and the fl uids are the same. This is called the hydrostatic paradox . Explain why it is true and sketch a free body of each of the liquid columns. P2.54 (a) F (b) (c)

Gate AB in Fig. P2.55 is 5 ft wide into the paper, hinged at A , and restrained by a stop at B . The water is at 20 8 C. Compute ( a ) the force on stop B and ( b ) the reactions at A if the water depth h 5 9.5 ft. P2.55 pa Water pa 4 f

In Fig. P2.55, gate AB is 5 ft wide into the paper, and stop B will break if the water force on it equals 9200 lbf. For what water depth h is this condition reached?

The square vertical panel ABCD in Fig. P2.57 is submerged in water at 20 8 C. Side AB is at least 1.7 m below the surface. Determine the difference between the hydrostatic forces on subpanels ABD and BCD . P2.57 A D B C 60

In Fig. P2.58, the cover gate AB closes a circular opening 80 cm in diameter. The gate is held closed by a 200 - kg mass as shown. Assume standard gravity at 20 8 C. At what water level h will the gate be dislodged? Neglect the weight of the gate. P2.58 Water 30 cm 3

Gate AB has length L and width b into the paper, is hinged at B , and has negligible weight. The liquid level h remains at the top of the gate for any angle . Find an analytic expression for the force P , perpendicular to AB , required to keep the gate in equilibrium in Fig. P2.59. P2.59 B h Hinge A P L

In Fig. P2.60, vertical, unsymmetrical trapezoidal panel ABCD is submerged in fresh water with side AB 12 ft below the surface. Since trapezoid formulas are complicated, ( a ) estimate, reasonably, the water force on the panel, in lbf, neglecting atmospheric pressure. For extra credit, ( b ) look up the formula and compute the exact force on the panel. Problems 113 P2.60 C A B D

Gate AB in Fig. P2.61 is a homogeneous mass of 180 kg, 1.2 m wide into the paper, hinged at A , and resting on a smooth bottom at B . All fl uids are at 20 8 C. For what water depth h will the force at point B be zero? P2.61

Gate AB in Fig. P2.62 is 15 ft long and 8 ft wide into the paper and is hinged at B with a stop at A . The water is at 20 8 C. The gate is 1-in-thick steel, SG 5 7.85. Compute the water level h for which the gate will start to fall. P2.62 Water h B 15 ft 60 Pulley A 1

The tank in Fig. P2.63 has a 4-cm-diameter plug at the bottom on the right. All fl uids are at 20 8 C. The plug will pop out if the hydrostatic force on it is 25 N. For this condition, what will be the reading h on the mercury manometer on the left side? h 50 2 cm H Water Plug, D = 4 cm Mercury P

Gate ABC in Fig. P2.64 has a fi xed hinge line at B and is 2 m wide into the paper. The gate will open at A to release water if the water depth is high enough. Compute the depth h for which the gate will begin to open. Water at 20C A 20 cm B C 1 m h

Gate AB in Fig. P2.65 is semicircular, hinged at B , and held by a horizontal force P at A . What force P is required for equilibrium? P2.65 P A B 5 m W

Dam ABC in Fig. P2.66 is 30 m wide into the paper and made of concrete (SG 5 2.4). Find the hydrostatic force on surface AB and its moment about C . Assuming no seepage of water under the dam, could this force tip the dam over? How does your argument change if there is seepage under the dam? P2.66 60 m C

Generalize Prob. P2.66 as follows. Denote length AB as H , length BC as L , and angle ABC as . Let the dam material have specifi c gravity SG. The width of the dam is b . Assume no seepage of water under the dam. Find an analytic relation between SG and the critical angle c for which the dam will just tip over to the right. Use your relation to compute c for the special case SG 5 2.4 (concrete)

8 Isosceles triangle gate AB in Fig. P2.68 is hinged at A and weighs 1500 N. What horizontal force P is required at point B for equilibrium? A P 3 m Gate 50 B Oil, SG = 0.83 1 m 2 m

Consider the slanted plate AB of length L in Fig. P2.69. ( a ) Is the hydrostatic force F on the plate equal to the weight of the missing water above the plate? If not, correct this hypothesis. Neglect the atmosphere. ( b ) Can a missing water theory be generalized to curved surfaces of this type? P2.69 A B F Water specific weight

The swing-check valve in Fig. P2.70 covers a 22.86-cm diameter opening in the slanted wall. The hinge is 15 cm from the centerline, as shown. The valve will open when the hinge moment is 50 N m. Find the value of h for the water to cause this condition. P2.70 Water at 20C 15 cm Hinge Air 60

In Fig. P2.71 gate AB is 3 m wide into the paper and is connected by a rod and pulley to a concrete sphere (SG 5 2.40). What diameter of the sphere is just suffi cient to keep the gate closed? P2.71 A 4 m B Concrete sphere, SG = 2.4 8 m 6 m Water

In Fig. P2.72, gate AB is circular. Find the moment of the hydrostatic force on this gate about axis A . P2.72 A B Water 3 m 2

Gate AB is 5 ft wide into the paper and opens to let fresh water out when the ocean tide is dropping. The hinge at A is 2 ft above the freshwater level. At what ocean level h will the gate fi rst open? Neglect the gate weight. P2.73 A B h Stop 10 ft Tide range Seawater, SG = 1.025

Find the height H in Fig. P2.74 for which the hydrostatic force on the rectangular panel is the same as the force on the semicircular panel below. P2.74 2R H

The cap at point B on the 5-cm-diameter tube in Fig. P2.75 will be dislodged when the hydrostatic force on its base reaches 22 lbf. For what water depth h does this occur? P2.75 1 m Water Oil, SG = 0.8 B 2 m

Panel BC in Fig. P2.76 is circular. Compute ( a ) the hydrostatic force of the water on the panel, ( b ) its center of pressure, and ( c ) the moment of this force about point B .

The circular gate ABC in Fig. P2.77 has a 1-m radius and is hinged at B . Compute the force P just suffi cient to keep the gate from opening when h 5 8 m. Neglect atmospheric pressure.

Panels AB and CD in Fig. P2.78 are each 120 cm wide into the paper. ( a ) Can you deduce, by inspection, which panel has the larger water force? ( b ) Even if your deduction is brilliant, calculate the panel forces anyway.

Gate ABC in Fig. P2.79 is 1 m square and is hinged at B . It will open automatically when the water level h becomes high enough. Determine the lowest height for which the gate will open. Neglect atmospheric pressure. Is this result independent of the liquid density?

A concrete dam (SG 5 2.5) is made in the shape of an isosceles triangle, as in Fig. P2.80. Analyze this geometry to fi nd the range of angles for which the hydrostatic force will tend to tip the dam over at point B . The width into the paper is b . P2.80 B h

For the semicircular cylinder CDE in Example 2.9, fi nd the vertical hydrostatic force by integrating the vertical component of pressure around the surface from 5 0 to 5 .

The dam in Fig. P2.82 is a quarter circle 50 m wide into the paper. Determine the horizontal and vertical components of the hydrostatic force against the dam and the point CP where the resultant strikes the dam. P2.82 Water 20 m 20 m CP pa = 0

Gate AB in Fig. P2.83 is a quarter circle 10 ft wide into the paper and hinged at B . Find the force F just suffi cient to keep the gate from opening. The gate is uniform and weighs 3000 lbf. P2.83 Water A B F r = 8 ft

Panel AB in Fig. P2.84 is a parabola with its maximum at point A . It is 150 cm wide into the paper. Neglect atmospheric pressure. Find ( a ) the vertical and ( b ) the horizontal water forces on the panel. P2.84 25 cm Water Parabola 75 cm 40 cm B

Compute the horizontal and vertical components of the hydrostatic force on the quarter-circle panel at the bottom of the water tank in Fig. P2.85. P2.85 Water 6 m 2 m 5 m 2 m

The quarter circle gate BC in Fig. P2.86 is hinged at C. Find the horizontal force P required to hold the gate stationary. Neglect the weight of the gate. P2.86 P B C

The bottle of champagne (SG 5 0.96) in Fig. P2.87 is under pressure, as shown by the mercury-manometer reading. Compute the net force on the 2-in-radius hemispherical end cap at the bottom of the bottle. P2.87 4 in 2 in 6 in Mercury

Gate ABC is a circular arc, sometimes called a Tainter gate , which can be raised and lowered by pivoting about point O . See Fig. P2.88. For the position shown, determine ( a ) the hydrostatic force of the water on the gate and ( b ) its line of action. Does the force pass through point O ? Water B C O R = 6 m 6 m 6 m A

The tank in Fig. P2.89 contains benzene and is pressurized to 200 kPa (gage) in the air gap. Determine the vertical hydrostatic force on circular-arc section AB and its line of action.

The tank in Fig. P2.90 is 120 cm long into the paper. Determine the horizontal and vertical hydrostatic forces on the quarter-circle panel AB . The fl uid is water at 20 8 C. Neglect atmospheric pressure.

The hemispherical dome in Fig. P2.91 weighs 30 kN and is fi lled with water and attached to the fl oor by six equally spaced bolts. What is the force in each bolt required to hold down the dome?

A 4-m-diameter water tank consists of two half cylinders, each weighing 4.5 kN/m, bolted together as shown in Fig. P2.92. If the support of the end caps is neglected, determine the force induced in each bolt. 118 Chapter 2 Pressure Distribution in a Fluid P2.92 2 m Water Bolt spacing 25 cm 2

In Fig. P2.93, a one-quadrant spherical shell of radius R is submerged in liquid of specifi c weight and depth h . R. Find an analytic expression for the resultant hydrostatic force, and its line of action, on the shell surface. P2.93 z , R R x

4 Find an analytic formula for the vertical and horizontal forces on each of the semicircular panels AB in Fig. P2.94. The width into the paper is b. Which force is larger? Why? h A B d + h A B d + P2.94

The uniform body A in Fig. P2.95 has width b into the paper and is in static equilibrium when pivoted about hinge O . What is the specifi c gravity of this body if ( a ) h 5 0 and ( b ) h 5 R ?

In Fig. P2.96, curved section AB is 5 m wide into the paper and is a 60 8 circular arc of radius 2 m. Neglecting atmospheric pressure, calculate the vertical and horizontal hydrostatic forces on arc AB . P2.96 A C O B Water 4 m 60

The contractor ran out of gunite mixture and fi nished the deep corner of a 5-m-wide swimming pool with a quartercircle piece of PVC pipe, labeled AB in Fig. P2.97. Compute the horizontal and vertical water forces on the curved panel AB . P2.97 2 m 1 m Water A B

The curved surface in Fig. P2.98 consists of two quarterspheres and a half cylinder. A side view and front view are shown. Calculate the horizontal and vertical forces on the surface. Problems 119 P2.98 Water Side 2 m F

The mega-magnum cylinder in Fig. P2.99 has a hemispherical bottom and is pressurized with air to 75 kPa (gage) at the top. Determine ( a ) the horizontal and ( b ) the vertical hydrostatic forces on the hemisphere, in lbf. P2.99 Water Air 20 ft 12

Pressurized water fi lls the tank in Fig. P2.100. Compute the net hydrostatic force on the conical surface ABC . P2.100 2 m A C B 150 kPa gage 4 m 7 m Water

The closed layered box in Fig. P2.101 has square horizontal cross sections everywhere. All fl uids are at 20 8 C. Estimate the gage pressure of the air if ( a ) the hydrostatic force on panel AB is 48 kN or ( b ) the hydrostatic force on the bottom panel BC is 97 kN. P2.101 A C B 30 cm Air SAE 30W oil Water 160 cm 90 cm 80 cm 60 cm

A cubical tank is 3 m 3 3 m 3 3 m and is layered with 1 meter of fl uid of specifi c gravity 1.0, 1 meter of fl uid with SG 5 0.9, and 1 meter of fl uid with SG 5 0.8. Neglect atmospheric pressure. Find ( a ) the hydrostatic force on the bottom and ( b ) the force on a side panel.

A solid block, of specifi c gravity 0.9, fl oats such that 75 percent of its volume is in water and 25 percent of its volume is in fl uid X , which is layered above the water. What is the specifi c gravity of fl uid X ?

The can in Fig. P2.104 fl oats in the position shown. What is its weight in N? P2.104 Water 3 cm 8 cm D

It is said that Archimedes discovered the buoyancy laws when asked by King Hiero of Syracuse to determine whether his new crown was pure gold (SG 5 19.3). Archimedes measured the weight of the crown in air to be 11.8 N and its weight in water to be 10.9 N. Was it pure gold?

A spherical helium balloon has a total mass of 3 kg. It settles in a calm standard atmosphere at an altitude of 5500 m. Estimate the diameter of the balloon.

Repeat Prob. 2.62, assuming that the 10,000-lbf weight is aluminum (SG 5 2.71) and is hanging submerged in the water.

A 7-cm-diameter solid aluminum ball (SG 5 2.7) and a solid brass ball (SG 5 8.5) balance nicely when submerged in a liquid, as in Fig. P2.108. ( a ) If the fl uid is water at 20 8 C, what is the diameter of the brass ball? ( b ) If the brass ball has a diameter of 3.8 cm, what is the density of the fl uid? P2.108 + Brass +

A hydrometer fl oats at a level that is a measure of the specifi c gravity of the liquid. The stem is of constant diameter D , and a weight in the bottom stabilizes the body to fl oat vertically, as shown in Fig. P2.109. If the position h 5 0 is pure water (SG 5 1.0), derive a formula for h as a function of total weight W , D , SG, and the specifi c weight 0 of water. P2.109 h Fluid, SG > 1 W

0 A solid sphere, of diameter 18 cm, fl oats in 20 8 C water with 1527 cubic centimeters exposed above the surface. ( a ) What are the weight and specifi c gravity of this sphere? ( b ) Will it fl oat in 20 8 C gasoline? If so, how many cubic centimeters will be exposed?

A solid wooden cone (SG = 0.729) fl oats in water. The cone is 30 cm high, its vertex angle is 90 8 , and it fl oats with vertex down. How much of the cone protrudes above the water?

The uniform 5-m-long round wooden rod in Fig. P2.112 is tied to the bottom by a string. Determine ( a ) the tension in the string and ( b ) the specifi c gravity of the wood. Is it possible for the given information to determine the inclination angle ? Explain.

3 A spar buoy is a buoyant rod weighted to fl oat and protrude vertically, as in Fig. P2.113. It can be used for measurements or markers. Suppose that the buoy is maple wood (SG 5 0.6), 2 in by 2 in by 12 ft, fl oating in seawater (SG 5 1.025). How many pounds of steel (SG 5 7.85) should be added to the bottom end so that h 5 18 in? P2.113 Wsteel

The uniform rod in Fig. P2.114 is hinged at point B on the waterline and is in static equilibrium as shown when 2 kg of lead (SG 5 11.4) are attached to its end. What is the specifi c gravity of the rod material? What is peculiar about the rest angle 5 30 8 ? P2.114 8 m Hinge = 30

The 2-in by 2-in by 12-ft spar buoy from Fig. P2.113 has 5 lbm of steel attached and has gone aground on a rock, as in Fig. P2.115. Compute the angle at which the buoy will lean, assuming that the rock exerts no moments on the spar. P2.115 Rock Seawater

The bathysphere of the chapter-opener photo is steel, SG < 7.85, with inside diameter 54 inches and wall thickness 1.5 inches. Will the empty sphere fl oat in seawater?

The solid sphere in Fig. P2.117 is iron (SG < 7.9). The tension in the cable is 600 lbf. Estimate the diameter of the sphere, in cm. P2.117 Water

An intrepid treasure-salvage group has discovered a steel box, containing gold doubloons and other valuables, resting in 80 ft of seawater. They estimate the weight of the box and treasure (in air) at 7000 lbf. Their plan is to attach the box to a sturdy balloon, infl ated with air to 3 atm pressure. The empty balloon weighs 250 lbf. The box is 2 ft wide, 5 ft long, and 18 in high. What is the proper diameter of the balloon to ensure an upward lift force on the box that is 20 percent more than required?

When a 5-lbf weight is placed on the end of the uniform fl oating wooden beam in Fig. P2.119, the beam tilts at an angle with its upper right corner at the surface, as shown. Determine ( a ) the angle and ( b ) the specifi c gravity of the wood. Hint: Both the vertical forces and the moments about the beam centroid must be balanced.

0 A uniform wooden beam (SG 5 0.65) is 10 cm by 10 cm by 3 m and is hinged at A, as in Fig. P2.120. At what angle will the beam fl oat in the 20 8 C water? P2.120 Water 1 m

Show that this can happen only ( a ) when b 5 /3 and ( b ) when the sphere has size D 5 c Lhb (SG 2 1) d 1/3

A uniform block of steel (SG 5 7.85) will fl oat at a mercurywater interface as in Fig. P2.122. What is the ratio of the distances a and b for this condition? P2.122 Steel block Mercury: SG = 13.56 a

A barge has the trapezoidal shape shown in Fig. P2.123 and is 22 m long into the paper. If the total weight of barge and cargo is 350 tons, what is the draft H of the barge when fl oating in seawater? P2.123 H 2.5 m 8 m 60 60

4 A balloon weighing 3.5 lbf is 6 ft in diameter. It is fi lled with hydrogen at 18 lbf/in 2 absolute and 60 8 F and is released. At what altitude in the U.S. standard atmosphere will this balloon be neutrally buoyant?

A uniform cylindrical white oak log, = 710 kg/m 3 , fl oats lengthwise in fresh water at 20 8 C. Its diameter is 24 inches. What height of the log is visible above the surface?

A block of wood (SG 5 0.6) fl oats in fl uid X in Fig. P2.126 such that 75 percent of its volume is submerged in fl uid X . Estimate the vacuum pressure of the air in the tank. P2.126 Air = 0 kPa gage 40 cm 70 cm Air pressure? Wood Fluid X

Consider a cylinder of specifi c gravity S , 1 fl oating vertically in water ( S 5 1), as in Fig. P2.127. Derive a formula for the stable values of D / L as a function of S and apply it to the case D / L 5 1.2. P2.127 h D

An iceberg can be idealized as a cube of side length L , as in Fig. P2.128. If seawater is denoted by S 5 1.0, then glacier ice (which forms icebergs) has S 5 0.88. Determine if this cubic iceberg is stable for the position shown in Fig. P2.128. P2.128 Water S = 1.0 M? G B Specific gravity

9 The iceberg idealization in Prob. P2.128 may become unstable if its sides melt and its height exceeds its width. In Fig. P2.128 suppose that the height is L and the depth into the paper is L , but the width in the plane of the paper is H , L. Assuming S 5 0.88 for the iceberg, fi nd the ratio H / L for which it becomes neutrally stable (about to overturn).

Consider a wooden cylinder (SG 5 0.6) 1 m in diameter and 0.8 m long. Would this cylinder be stable if placed to fl oat with its axis vertical in oil (SG 5 0.8)?

A barge is 15 ft wide and 40 ft long and fl oats with a draft of 4 ft. It is piled so high with gravel that its center of gravity is 3 ft above the waterline. Is it stable?

A solid right circular cone has SG 5 0.99 and fl oats vertically as in Fig. P2.132. Is this a stable position for the cone? P2.132 Water : SG = 1.0 SG = 0.99

Consider a uniform right circular cone of specifi c gravity S , 1, fl oating with its vertex down in water ( S 5 1). The base radius is R and the cone height is H. Calculate and plot the stability MG of this cone, in dimensionless form, versus H / R for a range of S , 1.

When fl oating in water (SG 5 1.0), an equilateral triangular body (SG 5 0.9) might take one of the two positions shown in Fig. P2.134. Which is the more stable position? Assume large width into the paper. P2.134 (a) (b)

Consider a homogeneous right circular cylinder of length L , radius R , and specifi c gravity SG, fl oating in water (SG 5 1). Show that the body will be stable with its axis vertical if R L . 32SG(1 2 SG)4 1/2

6 Consider a homogeneous right circular cylinder of length L , radius R , and specifi c gravity SG 5 0.5, fl oating in water (SG 5 1). Show that the body will be stable with its axis horizontal if L / R . 2.0.

A tank of water 4 m deep receives a constant upward acceleration az . Determine ( a ) the gage pressure at the tank bottom if az 5 5 m 2 /s and ( b ) the value of az that causes the gage pressure at the tank bottom to be 1 atm.

A 12-fl -oz glass, of 3-in diameter, partly full of water, is attached to the edge of an 8-ft-diameter merry-go-round, which is rotated at 12 r/min. How full can the glass be before water spills? Hint: Assume that the glass is much smaller than the radius of the merry-go-round.

The tank of liquid in Fig. P2.139 accelerates to the right with the fl uid in rigid-body motion. ( a ) Compute ax in m/s 2 . ( b ) Why doesnt the solution to part ( a ) depend on the density of the fl uid? ( c ) Determine the gage pressure at point A if the fl uid is glycerin at 20 8 C. P2.139 A 28 cm 100 cm ax 15 c

The U-tube in Fig. P2.140 is moving to the right with variable velocity. The water level in the left tube is 6 cm, and the level in the right tube is 16 cm. Determine the acceleration and its direction. P2.140 20 cm

1 The same tank from Prob. P2.139 is now moving with constant acceleration up a 30 8 inclined plane, as in Fig. P2.141. Assuming rigid-body motion, compute ( a ) the value of the acceleration a , ( b ) whether the acceleration is up or down, and ( c ) the gage pressure at point A if the fl uid is mercury at 20 8 C. P2.141 28 cm 100 cm 15 cm V A a? 30 z x

The tank of water in Fig. P2.142 is 12 cm wide into the paper. If the tank is accelerated to the right in rigid-body motion at 6.0 m/s 2 , compute ( a ) the water depth on side AB and ( b ) the water-pressure force on panel AB . Assume no spilling. Water at 20C 24 cm 9 cm A B

The tank of water in Fig. P2.143 is full and open to the atmosphere at point A. For what acceleration ax in ft/s 2 will the pressure at point B be ( a ) atmospheric and ( b ) zero absolute? P2.143 Water 2 ft 2 ft 1 ft 1 ft A B ax

Consider a hollow cube of side length 22 cm, fi lled completely with water at 20 8 C. The top surface of the cube is horizontal. One top corner, point A , is open through a small hole to a pressure of 1 atm. Diagonally opposite to point A is top corner B . Determine and discuss the various rigidbody accelerations for which the water at point B begins to cavitate, for ( a ) horizontal motion and ( b ) vertical motion.

A fi sh tank 14 in deep by 16 by 27 in is to be carried in a car that may experience accelerations as high as 6 m/s 2 . What is the maximum water depth that will avoid spilling in rigid-body motion? What is the proper alignment of the tank with respect to the car motion?

The tank in Fig. P2.146 is fi lled with water and has a vent hole at point A. The tank is 1 m wide into the paper. Inside the tank, a 10-cm balloon, fi lled with helium at 130 kPa, is tethered centrally by a string. If the tank accelerates to the right at 5 m/s 2 in rigid-body motion, at what angle will the balloon lean? Will it lean to the right or to the left? Water at 20C 60 cm 40 cm 20 cm D = 10 cm String 1

7 The tank of water in Fig. P2.147 accelerates uniformly by freely rolling down a 30 8 incline. If the wheels are frictionless, what is the angle ? Can you explain this interesting result? P2.147 30

A child is holding a string onto which is attached a heliumfi lled balloon. ( a ) The child is standing still and suddenly accelerates forward. In a frame of reference moving with the child, which way will the balloon tilt, forward or backward? Explain. ( b ) The child is now sitting in a car that is stopped at a red light. The helium-fi lled balloon is not in contact with any part of the car (seats, ceiling, etc.) but is held in place by the string, which is in turn held by the child. All the windows in the car are closed. When the traffi c light turns green, the car accelerates forward. In a frame of reference moving with the car and child, which way will the balloon tilt, forward or backward? Explain. ( c ) Purchase or borrow a helium-fi lled balloon. Conduct a scientifi c experiment to see if your predictions in parts ( a ) and ( b ) above are correct. If not, explain

The 6-ft-radius waterwheel in Fig. P2.149 is being used to lift water with its 1-ft-diameter half-cylinder blades. If the wheel rotates at 10 r/min and rigid-body motion is assumed, what is the water surface angle at position A ?

A cheap accelerometer, probably worth the price, can be made from a U-tube as in Fig. P2.150. If L 5 18 cm and D 5 5 mm, what will h be if ax 5 6 m/s 2 ? Can the scale markings on the tube be linear multiples of ax ? P2.150 D ax L

The U-tube in Fig. P2.151 is open at A and closed at D. If accelerated to the right at uniform ax , what acceleration Problems 125 will cause the pressure at point C to be atmospheric? The fl uid is water (SG 5 1.0). P2.151 1 ft 1 ft 1 ft

A 16-cm-diameter open cylinder 27 cm high is full of water. Compute the rigid-body rotation rate about its central axis, in r/min, ( a ) for which one-third of the water will spill out and ( b ) for which the bottom will be barely exposed.

A tall cylindrical container, 14 in in diameter, is used to make a mold for forming 14-in salad bowls. The bowls are to be 8 in deep. The cylinder is half-fi lled with molten plastic, 5 1.6 kg/(m-s), rotated steadily about the central axis, then cooled while rotating. What is the appropriate rotation rate, in r/min?

A very tall 10-cm-diameter vase contains 1178 cm 3 of water. When spun steadily to achieve rigid-body rotation, a 4-cmdiameter dry spot appears at the bottom of the vase. What is the rotation rate, in r/min, for this condition?

For what uniform rotation rate in r/min about axis C will the U-tube in Fig. P2.155 take the confi guration shown? The fl uid is mercury at 20 8 C. P2.155 A B C 20 cm 10 cm 5 cm

Suppose that the U-tube of Fig. P2.151 is rotated about axis DC . If the fl uid is water at 122 8 F and atmospheric pressure is 2116 lbf/ft 2 absolute, at what rotation rate will the fl uid within the tube begin to vaporize? At what point will this occur?

The 45 8 V-tube in Fig. P2.157 contains water and is open at A and closed at C . What uniform rotation rate in r/min about axis AB will cause the pressure to be equal at points B and C ? For this condition, at what point in leg BC will the pressure be a minimum? P2.157 30 cm 45 A C B

It is desired to make a 3-m-diameter parabolic telescope mirror by rotating molten glass in rigid-body motion until the desired shape is achieved and then cooling the glass to a solid. The focus of the mirror is to be 4 m from the mirror, measured along the centerline. What is the proper mirror rotation rate, in r/min, for this task?

The three-legged manometer in Fig. P2.159 is fi lled with water to a depth of 20 cm. All tubes are long and have equal small diameters. If the system spins at angular velocity V about the central tube, ( a ) derive a formula to fi nd the change of height in the tubes; ( b ) fi nd the height in cm in each tube if V 5 120 r/min. Hint: The central tube must supply water to both the outer legs. P2.159 20 cm 10 cm 10 cm

Figure P2.160 shows a gage for very low pressures, invented in 1874 by Herbert McLeod. ( a ) Can you deduce, from the fi gure, how it works? ( b ) If not, read about it and explain it to the class.

Figure P2.161 shows a sketch of a commercial pressure gage. ( a ) Can you deduce, from the fi gure, how it works? P2.161 P2 P