Fluid Mechanics: Fundamentals and Applications

Yunus Cengel

Chapter 3

PRESSURE AND FLUID STATICS - all with Video Answers

Educators

Chapter Questions

Problem 1

What is the difference between gage pressure and absolute pressure?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 2

Explain why some people experience nose bleeding and some others experience shortness of breath at high elevations.

Mayukh Banik

Numerade Educator

Problem 3

Someone claims that the absolute pressure in a liquid of constant density doubles when the depth is doubled. Do you agree? Explain.

Mayukh Banik

Numerade Educator

Problem 4

A tiny steel cube is suspended in water by a string. If the lengths of the sides of the cube are very small, how would you compare the magnitudes of the pressures on the top, bottom, and side surfaces of the cube?

Chris Trentman

Numerade Educator

Problem 5

Express Pascal's law, and give a real-world example of it.

Mayukh Banik

Numerade Educator

Problem 6

Consider two identical fans, one at sea level and the other on top of a high mountain, running at identical speeds. How would you compare (a) the volume flow rates and (b) the mass flow rates of these two fans?

Mayukh Banik

Numerade Educator

Problem 7

A vacuum gage connected to a chamber reads 24 kPa at a location where the atmospheric pressure is 92 kPa . Determine the absolute pressure in the chamber.

Mayukh Banik

Numerade Educator

Problem 8

A manometer is used to measure the air pressure in a tank. The fluid used has a specific gravity of 1.25 , and the differential height between the two arms of the manometer is 28 in . If the local atmospheric pressure is 12.7 psia , determine the absolute pressure in the tank for the cases of the manometer arm with the (a) higher and (b) lower fluid level being attached to the tank.

Hariprasad Annamalai

Hariprasad Annamalai

Numerade Educator

Problem 9

The water in a tank is pressurized by air, and the pressure is measured by a multifluid manometer as shown in Fig. P3-9. Determine the gage pressure of air in the tank if $h_1$ $=0.2 \mathrm{~m}, h_2=0.3 \mathrm{~m}$, and $h_3=0.46 \mathrm{~m}$. Take the densities of water, oil, and mercury to be $1000 \mathrm{~kg} / \mathrm{m}^3, 850 \mathrm{~kg} / \mathrm{m}^3$, and $13,600 \mathrm{~kg} / \mathrm{m}^3$, respectively.

Hariprasad Annamalai

Hariprasad Annamalai

Numerade Educator

Problem 10

Determine the atmospheric pressure at a location where the barometric reading is 750 mmHg . Take the density of mercury to be $13,600 \mathrm{~kg} / \mathrm{m}^3$.

Mayukh Banik

Numerade Educator

Problem 11

The gage pressure in a liquid at a depth of 3 m is read to be 28 kPa . Determine the gage pressure in the same liquid at a depth of 12 m .

Mayukh Banik

Numerade Educator

Problem 12

The absolute pressure in water at a depth of 5 m is read to be 145 kPa . Determine (a) the local atmospheric pressure, and (b) the absolute pressure at a depth of 5 m in a liquid whose specific gravity is 0.85 at the same location.

Nick Johnson

Numerade Educator

Problem 13

Show that $1 \mathrm{kgf} / \mathrm{cm}^2=14.223 \mathrm{psi}$.

Prabhu Ramji

Numerade Educator

Problem 14

A 200-lb man has a total foot imprint area of $72 \mathrm{in}^2$. Determine the pressure this man exerts on the ground if (a) he stands on both feet and (b) he stands on one foot.

Mayukh Banik

Numerade Educator

Problem 15

Consider a 70-kg woman who has a total foot imprint area of $400 \mathrm{~cm}^2$. She wishes to walk on the snow, but the snow cannot withstand pressures greater than 0.5 kPa . Determine the minimum size of the snowshoes needed (imprint area per shoe) to enable her to walk on the snow without sinking.

Mayukh Banik

Numerade Educator

Problem 16

A vacuum gage connected to a tank reads 30 kPa at a location where the barometric reading is 755 mmHg . Determine the absolute pressure in the tank. Take $\rho_{\mathrm{Hg}}=13,590 \mathrm{~kg} / \mathrm{m}^3$. 3-

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 17

A pressure gage connected to a tank reads 50 psi at a location where the barometric reading is 29.1 inHg . Determine the absolute pressure in the tank. Take $\rho_{\mathrm{Hg}}=848.4 \mathrm{lbm} / \mathrm{ft}^3$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 18

A pressure gage connected to a tank reads 500 kPa at a location where the atmospheric pressure is 94 kPa . Determine the absolute pressure in the tank.

Ma Ednelyn Lim

Numerade Educator

Problem 19

The barometer of a mountain hiker reads 930 mbars at the beginning of a hiking trip and 780 mbars at the end. Neglecting the effect of altitude on local gravitational acceleration, determine the vertical distance climbed. Assume an average air density of $1.20 \mathrm{~kg} / \mathrm{m}^3$.

Mayukh Banik

Numerade Educator

Problem 20

The basic barometer can be used to measure the height of a building. If the barometric readings at the top and at the bottom of a building are 730 and 755 mmHg , respectively, determine the height of the building. Assume an average air density of $1.18 \mathrm{~kg} / \mathrm{m}^3$.

Mayukh Banik

Numerade Educator

Problem 21

Solve Prob. 3-20 using EES (or other) software.
Print out the entire solution, including the numerical results with proper units, and take the density of mercury to be $13,600 \mathrm{~kg} / \mathrm{m}^3$.

Check back soon!

Problem 22

Determine the pressure exerted on a diver at 30 m below the free surface of the sea. Assume a barometric pressure of 101 kPa and a specific gravity of 1.03 for seawater.

Mayukh Banik

Numerade Educator

Problem 23

Determine the pressure exerted on the surface of a submarine cruising 300 ft below the free surface of the sea. Assume that the barometric pressure is 14.7 psia and the specific gravity of seawater is 1.03 .

Mayukh Banik

Numerade Educator

Problem 24

A gas is contained in a vertical, frictionless pistoncylinder device. The piston has a mass of 4 kg and a crosssectional area of $35 \mathrm{~cm}^2$. A compressed spring above the piston exerts a force of 60 N on the piston. If the atmospheric pressure is 95 kPa , determine the pressure inside the cylinder.

Ranjeet Singh

Numerade Educator

Problem 25

Reconsider Prob. 3-24. Using EES (or other) software, investigate the effect of the spring force in the range of 0 to 500 N on the pressure inside the cylinder. Plot the pressure against the spring force, and discuss the results.

Hariprasad Annamalai

Hariprasad Annamalai

Numerade Educator

Problem 26

Both a gage and a manometer are attached to a gas tank to measure its pressure. If the reading on the pressure gage is 80 kPa , determine the distance between the two fluid levels of the manometer if the fluid is (a) mercury ( $\rho=13,600 \mathrm{~kg} / \mathrm{m}^3$ ) or (b) water ( $\rho=1000 \mathrm{~kg} / \mathrm{m}^3$ ).

Mayukh Banik

Numerade Educator

Problem 27

Reconsider Prob. 3-26. Using EES (or other) software, investigate the effect of the manometer fluid density in the range of 800 to $13,000 \mathrm{~kg} / \mathrm{m}^3$ on the differential fluid height of the manometer. Plot the differential fluid height against the density, and discuss the results.

Hariprasad Annamalai

Hariprasad Annamalai

Numerade Educator

Problem 28

A manometer containing oil ( $\rho=850 \mathrm{~kg} / \mathrm{m}^3$ ) is attached to a tank filled with air. If the oil-level difference between the two columns is 45 cm and the atmospheric pressure is 98 kPa , determine the absolute pressure of the air in the tank.

Mayukh Banik

Numerade Educator

Problem 29

A mercury manometer ( $\rho=13,600 \mathrm{~kg} / \mathrm{m}^3$ ) is connected to an air duct to measure the pressure inside. The difference in the manometer levels is 15 mm , and the atmospheric pressure is 100 kPa . (a) Judging from Fig. P3-29, determine if the pressure in the duct is above or below the atmospheric pressure. (b) Determine the absolute pressure in the duct.

Mayukh Banik

Numerade Educator

Problem 30

Repeat Prob. 3-29 for a differential mercury height of 30 mm .

Mayukh Banik

Numerade Educator

Problem 31

Blood pressure is usually measured by wrapping a closed air-filled jacket equipped with a pressure gage around the upper arm of a person at the level of the heart. Using a mercury manometer and a stethoscope, the systolic pressure (the maximum pressure when the heart is pumping) and the diastolic pressure (the minimum pressure when the heart is resting) are measured in mmHg . The systolic and diastolic pressures of a healthy person are about 120 mmHg and 80 mmHg , respectively, and are indicated as $120 / 80$. Express both of these gage pressures in $\mathrm{kPa}, \mathrm{psi}$, and meter water column.

Mayukh Banik

Numerade Educator

Problem 32

The maximum blood pressure in the upper arm of a healthy person is about 120 mmHg . If a vertical tube open to the atmosphere is connected to the vein in the arm of the person, determine how high the blood will rise in the tube. Take the density of the blood to be $1050 \mathrm{~kg} / \mathrm{m}^3$.

Mayukh Banik

Numerade Educator

Problem 33

Consider a 1.8 -m-tall man standing vertically in water and completely submerged in a pool. Determine the difference between the pressures acting at the head and at the toes of this man, in kPa .

Ranjeet Singh

Numerade Educator

Problem 34

Consider a U-tube whose arms are open to the atmosphere. Now water is poured into the U-tube from one arm, and light oil ( $\left.\rho=790 \mathrm{~kg} / \mathrm{m}^3\right)$ from the other. One arm contains $70-\mathrm{cm}$-high water, while the other arm contains both fluids with an oil-to-water height ratio of 6 . Determine the height of each fluid in that arm.

Mayukh Banik

Numerade Educator

Problem 35

The hydraulic lift in a car repair shop has an output diameter of 30 cm and is to lift cars up to 2000 kg . Determine the fluid gage pressure that must be maintained in the reservoir.

Mayukh Banik

Numerade Educator

Problem 36

Freshwater and seawater flowing in parallel horizontal pipelines are connected to each other by a double U-tube manometer, as shown in Fig. P3-36. Determine the pressure difference between the two pipelines. Take the density of seawater at that location to be $\rho=1035 \mathrm{~kg} / \mathrm{m}^3$. Can the air column be ignored in the analysis?

Mayukh Banik

Numerade Educator

Problem 37

Repeat Prob. 3-36 by replacing the air with oil whose specific gravity is 0.72 .

Mayukh Banik

Numerade Educator

Problem 38

The pressure in a natural gas pipeline is measured by the manometer shown in Fig. P3-38E with one of the arms open to the atmosphere where the local atmospheric pressure is 14.2 psia . Determine the absolute pressure in the pipeline.

Mayukh Banik

Numerade Educator

Problem 39

Repeat Prob. 3-38E by replacing air by oil with a specific gravity of 0.69 .

Mayukh Banik

Numerade Educator

Problem 40

The gage pressure of the air in the tank shown in Fig. P3-40 is measured to be 65 kPa . Determine the differential height $h$ of the mercury column.

Ranjeet Singh

Numerade Educator

Problem 41

Repeat Prob. 3-40 for a gage pressure of 45 kPa .

Ranjeet Singh

Numerade Educator

Problem 42

The top part of a water tank is divided into two compartments, as shown in Fig. P3-42. Now a fluid with an unknown density is poured into one side, and the water level rises a certain amount on the other side to compensate for this effect. Based on the final fluid heights shown on the figure, determine the density of the fluid added. Assume the liquid does not mix with water.

Prabhat Tyagi

Numerade Educator

Problem 43

The $500-\mathrm{kg}$ load on the hydraulic lift shown in Fig. P3-43 is to be raised by pouring oil ( $\rho=780 \mathrm{~kg} / \mathrm{m}^3$ ) into a thin tube. Determine how high $h$ should be in order to begin to raise the weight.

Supratim Pal

Numerade Educator

Problem 44

Two oil tanks are connected to each other through a manometer. If the difference between the mercury levels in the two arms is 32 in , determine the pressure difference between the two tanks. The densities of oil and mercury are $45 \mathrm{lbm} / \mathrm{ft}^3$ and $848 \mathrm{lbm} / \mathrm{ft}^3$, respectively.

Sophie S

Numerade Educator

Problem 45

Pressure is often given in terms of a liquid column and is expressed as "pressure head." Express the standard atmospheric pressure in terms of (a) mercury ( $\mathrm{SG}=13.6$ ), (b) water ( $\mathrm{SG}=1.0$ ), and (c) glycerin ( $\mathrm{SG}=1.26$ ) columns. Explain why we usually use mercury in manometers.

David Collins

Numerade Educator

Problem 46

A simple experiment has long been used to demonstrate how negative pressure prevents water from being spilled out of an inverted glass. A glass that is fully filled by water and covered with a thin paper is inverted, as shown in Fig. P3-46. Determine the pressure at the bottom of the glass, and explain why water does not fall out.

Mayukh Banik

Numerade Educator

Problem 47

Two chambers with the same fluid at their base are separated by a piston whose weight is 25 N , as shown in Fig. P3-47. Calculate the gage pressures in chambers $A$ and $B$.

Sophie S

Numerade Educator

Problem 48

Consider a double-fluid manometer attached to an aur pipe shown in Fig. P3-48. If the specific gravity of one fluid is 13.55 , determine the specific gravity of the other fluid for the indicated absolute pressure of air. Take the atmospheric pressure to be 100 kPa .

Mayukh Banik

Numerade Educator

Problem 49

The pressure difference between an oil pipe and water pipe is measured by a double-fluid manometer, as shown in Fig. P3-49. For the given fluid heights and specific gravities, calculate the pressure difference $\Delta P=P_B-P_A$.

Prabhat Tyagi

Numerade Educator

Problem 50

Consider the system shown in Fig. P3-50. If a change of 0.7 kPa in the pressure of air causes the brine-mercury interface in the right column to drop by 5 mm in the brine level in the right column while the pressure in the brine pipe remains constant, determine the ratio of $A_2 / A_1$.

Hariprasad Annamalai

Hariprasad Annamalai

Numerade Educator

Problem 51

Two water tanks are connected to each other through a mercury manometer with inclined tubes, as shown in Fig. P3-51. If the pressure difference between the two tanks is 20 kPa , calculate $a$ and $\theta$.

Mayukh Banik

Numerade Educator

Problem 52

A multifluid container is connected to a U-tube, as shown in Fig. P3-52. For the given specific gravities and fluid column heights, determine the gage pressure at $A$. Also determine the height of a mercury column that would create the same pressure at $A$.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 53

Define the resultant hydrostatic force acting on a submerged surface, and the center of pressure.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 54

Someone claims that she can determine the magnitude of the hydrostatic force acting on a plane surface submerged in water regardless of its shape and orientation if she knew the vertical distance of the centroid of the surface from the free surface and the area of the surface. Is this a valid claim? Explain.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 55

A submerged horizontal flat plate is suspended in water by a string attached at the centroid of its upper surface. Now the plate is rotated $45^{\circ}$ about an axis that passes through its centroid. Discuss the change on the hydrostatic force acting on the top surface of this plate as a result of this rotation. Assume the plate remains submerged at all times.

Yuou Sun

Numerade Educator

Problem 56

You may have noticed that dams are much thicker at the bottom. Explain why dams are built that way.

Surjit Tewari

Numerade Educator

Problem 57

Consider a submerged curved surface. Explain how you would determine the horizontal component of the hydrostatic force acting on this surface.

Narayan Hari

Numerade Educator

Problem 58

Consider a submerged curved surface. Explain how you would determine the vertical component of the hydrostatic force acting on this surface.

Narayan Hari

Numerade Educator

Problem 59

Consider a circular surface subjected to hydrostatic forces by a constant density liquid. If the magnitudes of the horizontal and vertical components of the resultant hydrostatic force are determined, explain how you would find the line of action of this force.

Narayan Hari

Numerade Educator

Problem 60

Consider a heavy car submerged in water in a lake with a flat bottom. The driver's side door of the car is 1.1 m high and 0.9 m wide, and the top edge of the door is 8 m below the water surface. Determine the net force acting on the door (normal to its surface) and the location of the pres-sure center if (a) the car is well-sealed and it contains air at atmospheric pressure and (b) the car is filled with water.

Rashmi Sinha

Numerade Educator

Problem 61

A long, solid cylinder of radius 2 ft hinged at point $A$ is used as an automatic gate, as shown in Fig. P3-61E. When the water level reaches 15 ft , the cylindrical gate opens by turning about the hinge at point $A$. Determine (a) the hydrostatic force acting on the cylinder and its line of action when the gate opens and (b) the weight of the cylinder per ft length of the cylinder.

Hunza Gilgit

Numerade Educator

Problem 62

Consider a 4-m-long, 4-m-wide, and 1.5-m-high aboveground swimming pool that is filled with water to the rim. (a) Determine the hydrostatic force on each wall and the distance of the line of action of this force from the ground. (b) If the height of the walls of the pool is doubled and the pool is filled, will the hydrostatic force on each wall double or quadruple? Why?

Chris Trentman

Numerade Educator

Problem 63

Consider a 200 -ft-high, 1200 -ft-wide dam filled to capacity. Determine (a) the hydrostatic force on the dam and (b) the force per unit area of the dam near the top and near the bottom.

CH

Chloe Hacker

Numerade Educator

Problem 64

A room in the lower level of a cruise ship has a $30-\mathrm{cm}$-diameter circular window. If the midpoint of the window is 5 m below the water surface, determine the hydrostatic force acting on the window, and the pressure center. Take the specific gravity of seawater to be 1.025 .

Narayan Hari

Numerade Educator

Problem 65

The water side of the wall of a $100-\mathrm{m}$-long dam is a quarter circle with a radius of 10 m . Determine the hydrostatic force on the dam and its line of action when the dam is filled to the rim.

Janielle Madlansacay

Janielle Madlansacay

Numerade Educator

Problem 66

A 4-m-high, 5-m-wide rectangular plate blocks the end of a 4-m-deep freshwater channel, as shown in Fig. P3-66. The plate is hinged about a horizontal axis along its upper edge through a point $A$ and is restrained from opening by a fixed ridge at point $B$. Determine the force exerted on the plate by the ridge.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 67

Reconsider Prob. 3-66. Using EES (or other)
software, investigate the effect of water depth on the force exerted on the plate by the ridge. Let the water depth vary from 0 m to 5 m in increments of 0.5 m . Tabulate and plot your results.

Carson Merrill

Numerade Educator

Problem 68

The flow of water from a reservoir is controlled by a 5 -ft-wide L-shaped gate hinged at point $A$, as shown in Fig. P3-68E. If it is desired that the gate open when the water height is 12 ft , determine the mass of the required weight $W$.

Surendra Kumar

Numerade Educator

Problem 69

Repeat Prob. 3-68E for a water height of 8 ft .

Check back soon!

Problem 70

A water trough of semicircular cross section of radius 0.5 m consists of two symmetric parts hinged to each other at the bottom, as shown in Fig. P3-70. The two parts are held together by a cable and turnbuekle placed every 3 m along the length of the trough. Calculate the tension in each cable when the trough is filled to the rim.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 71

The two sides of a V-shaped water trough are hinged to each other at the bottom where they meet, as shown in Fig. P3-71, making an angle of $45^{\circ}$ with the ground from both sides. Each side is 0.75 m wide, and the two parts are held together by a cable and turnbuckle placed every 6 m along the length of the trough. Calculate the tension in each cable when the trough is filled to the rim.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 72

Repeat Prob. 3-71 for the case of a partially filled trough with a water height of 0.4 m directly above the hinge.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 73

A retaining wall against a mud slide is to be constructed by placing $0.8-\mathrm{m}$-high and $0.2-\mathrm{m}$-wide rectangular concrete blocks ( $\left.\rho=2700 \mathrm{~kg} / \mathrm{m}^3\right)$ side by side, as shown in Fig. P3-73. The friction coefficient between the ground and the concrete blocks is $f=0.3$, and the density of the mud is about $1800 \mathrm{~kg} / \mathrm{m}^3$. There is concern that the concrete blocks may slide or tip over the lower left edge as the mud level rises. Determine the mud height at which (a) the blocks will overcome friction and start sliding and (b) the blocks will tip over.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 74

Repeat Prob. 3-73 for 0.4-m-wide concrete blocks.

Check back soon!

Problem 75

(8) A 4-m-long quarter-circular gate of radius 3 m and of negligible weight is hinged about its upper edge A, as shown in Fig. P3-75. The gate controls the flow of water over the ledge at $B$, where the gate is pressed by a spring. Determine the minimum spring force required to keep the gate closed when the water level rises to $A$ at the upper edge of the gate.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 76

Repeat Prob. 3-75 for a radius of 4 m for the gate.

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 77

What is buoyant force? What causes it? What is the magnitude of the buoyant force acting on a submerged body whose volume is V ? What are the direction and the line of action of the buoyant force?

Shital Rijal

Numerade Educator

Problem 78

Consider two identical spherical balls submerged in water at different depths. Will the buoyant forces acting on these two balls be the same or different? Explain.

Averell Hause

Carnegie Mellon University

Problem 79

Consider two 5 -cm-diameter spherical balls-one made of aluminum, the other of iron-submerged in water. Will the buoyant forces acting on these two balls be the same or different? Explain.

Mayukh Banik

Numerade Educator

Problem 80

Consider a 3-kg copper cube and a 3-kg copper ball submerged in a liquid. Will the buoyant forces acting on these two bodies be the same or different? Explain.

Mayukh Banik

Numerade Educator

Problem 81

Discuss the stability of (a) a submerged and (b) a floating body whose center of gravity is above the center of buoyancy.

Averell Hause

Carnegie Mellon University

Problem 82

The density of a liquid is to be determined by an old 1 -cm-diameter cylindrical hydrometer whose division marks are completely wiped out. The hydrometer is first dropped in water, and the water level is marked. The hydrometer is then dropped into the other liquid, and it is observed that the mark for water has risen 0.5 cm above the liquid-air interface. If the height of the water mark is 10 cm , determine the density of the liquid.

Mayukh Banik

Numerade Educator

Problem 83

A crane is used to lower weights into a lake for an underwater construction project. Determine the tension in the rope of the crane due to a 3-ft-diameter spherical steel block (density $=494 \mathrm{lbm} / \mathrm{ft}^3$ ) when it is (a) suspended in the air and (b) completely immersed in water.

Vipender Yadav

Numerade Educator

Problem 84

The volume and the average density of an irregularly shaped body are to be determined by using a spring scale. The body weighs 7200 N in air and 4790 N in water. Determine the volume and the density of the body. State your assumptions.

Prabhu Ramji

Numerade Educator

Problem 85

Consider a large cubic ice block floating in seawater. The specific gravities of ice and seawater are 0.92 and 1.025 , respectively. If a $10-\mathrm{cm}$-high portion of the ice block extends above the surface of the water, determine the height of the ice block below the surface.

Prabhu Ramji

Numerade Educator

Problem 86

A $170-\mathrm{kg}$ granite rock ( $\rho=2700 \mathrm{~kg} / \mathrm{m}^3$ ) is dropped into a lake. A man dives in and tries to lift the rock. Determine how much force the man needs to apply to lift it from the bottom of the lake. Do you think he can do it?

Prabhu Ramji

Numerade Educator

Problem 87

It is said that Archimedes discovered his principle during a bath while thinking about how he could determine if King Hiero's crown was actually made of pure gold. While in the bathtub, he conceived the idea that he could determine the average density of an irregularly shaped object by weighing it in air and also in water. If the crown weighed 3.20 kgf $(=31.4 \mathrm{~N})$ in air and $2.95 \mathrm{kgf}(=28.9 \mathrm{~N})$ in water, determine if the crown is made of pure gold. The density of gold is $19,300 \mathrm{~kg} / \mathrm{m}^3$. Discuss how you can solve this problem without weighing the crown in water but by using an ordinary bucket with no calibration for volume. You may weigh anything in air.

Shazia Naz

Numerade Educator

Problem 88

One of the common procedures in fitness programs is to determine the fat-to-muscle ratio of the body. This is based on the principle that the muscle tissue is denser than the fat tissue, and, thus, the higher the average density of the body, the higher is the fraction of muscle tissue. The average density of the body can be determined by weighing the person in air and also while submerged in water in a tank. Treating all tissues and bones (other than fat) as muscle with an equivalent density of $\rho_{\text {muscle }}$, obtain a relation for the volume fraction of body fat $x_{\text {firt }}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 89

The hull of a boat has a volume of $150 \mathrm{~m}^3$, and the total mass of the boat when empty is 8560 kg . Determine how much load this boat can carry without sinking (a) in a lake and (b) in seawater with a specific gravity of 1.03 .

Nathan Silvano

Numerade Educator

Problem 90

Under what conditions can a moving body of fluid be treated as a rigid body?

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Problem 91

Consider a glass of water. Compare the water pressures at the bottom surface for the following cases: the glass is (a) stationary, (b) moving up at constant velocity, (c) moving down at constant velocity, and (d) moving horizontally at constant velocity.

Saman Zulfiqar

Numerade Educator

Problem 92

Consider two identical glasses of water, one stationary and the other moving on a horizontal plane with constant acceleration. Assuming no splashing or spilling occurs, which glass will have a higher pressure at the (a) front, (b) midpoint, and (c) back of the bottom surface?

Alexander Allen

Alexander Allen

Numerade Educator

Problem 93

Consider a vertical cylindrical container partially filled with water. Now the cylinder is rotated about its axis at a specified angular velocity, and rigid-body motion is established. Discuss how the pressure will be affected at the midpoint and at the edges of the bottom surface due to rotation.

Hast Aggarwal

Numerade Educator

Problem 94

A water tank is being towed by a truck on a level road, and the angle the free surface makes with the horizontal is measured to be $15^{\circ}$. Determine the acceleration of the truck.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 95

Consider two water tanks filled with water. The first tank is 8 m high and is stationary, while the second tank is 2 m high and is moving upward with an acceleration of $5 \mathrm{~m} / \mathrm{s}^2$. Which tank will have a higher pressure at the bottom?

Sophie S

Numerade Educator

Problem 96

A water tank is being towed on an uphill road that makes $20^{\circ}$ with the horizontal with a constant acceleration of $5 \mathrm{~m} / \mathrm{s}^2$ in the direction of motion. Determine the angle the free surface of water makes with the horizontal. What would your answer be if the direction of motion were downward on the same road with the same acceleration?

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 97

A 2-ft-diameter vertical cylindrical tank open to the atmosphere contains 1 -ft-high water. The tank is now rotated about the centerline, and the water level drops at the center while it rises at the edges. Determine the angular velocity at which the bottom of the tank will first be exposed. Also determine the maximum water height at this moment.

Surendra Kumar

Numerade Educator

Problem 98

A 60-cm-high, 40-cm-diameter cylindrical water tank is being transported on a level road. The highest acceleration anticipated is $4 \mathrm{~m} / \mathrm{s}^2$. Determine the allowable initial water height in the tank if no water is to spill out during acceleration.

Penny Riley

Numerade Educator

Problem 99

A 40-cm-diameter, 90-cm-high vertical cylindrical container is partially filled with $60-\mathrm{cm}$-high water. Now the cylinder is rotated at a constant angular speed of 120 rpm . Determine how much the liquid level at the center of the cylinder will drop as a result of this rotational motion.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 100

A fish tank that contains 40 -cm-high water is moved in the cabin of an elevator. Determine the pressure at the bot-tom of the tank when the elevator is (a) stationary, (b) moving up with an upward acceleration of $3 \mathrm{~m} / \mathrm{s}^2$, and (c) moving down with a downward acceleration of $3 \mathrm{~m} / \mathrm{s}^2$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 101

A 3-m-diameter vertical cylindrical milk tank rotates at a constant rate of 12 rpm . If the pressure at the center of the bottom surface is 130 kPa , determine the pressure at the edge of the bottom surface of the tank. Take the density of the milk to be $1030 \mathrm{~kg} / \mathrm{m}$.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 102

Milk with a density of $1020 \mathrm{~kg} / \mathrm{m}^3$ is transported on a level road in a 7 -m-long, 3 -m-diameter cylindrical tanker. The tanker is completely filled with milk (no air space), and it accelerates at $2.5 \mathrm{~m} / \mathrm{s}^2$. If the minimum pressure in the tanker is 100 kPa , determine the maximum pressure and its location.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 103

Repeat Prob. 3-102 for a deceleration of $2.5 \mathrm{~m} / \mathrm{s}^2$.

Averell Hause

Carnegie Mellon University

Problem 104

The distance between the centers of the two arms of a U-tube open to the atmosphere is 25 cm , and the U-tube contains $20-\mathrm{cm}$-high alcohol in both arms. Now the U-tube is rotated about the left arm at $4.2 \mathrm{rad} / \mathrm{s}$. Determine the elevation difference between the fluid surfaces in the two arms.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 105

A 1.2-m-diametrex, 3-m-high sealed vertical cylinder is completely filled with gasoline whose density is $740 \mathrm{~kg} / \mathrm{m}^3$. The tank is now rotated about its vertical axis at a rate of 70 rpm . Determine (a) the difference between the pressures at the centers of the bottom and top surfaces and (b) the differenee between the pressures at the center and the edge of the bottom surface.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 106

Reconsider Prob. 3-105. Using EES (or other) software, investigate the effect of rotational speed on the pressure difference between the center and the edge of the bottom surface of the cylinder. Let the rotational speed vary from 0 rpm to 500 rpm in increments of 50 rpm . Tabulate and plot your results.

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Problem 107

A 20 -ft-long, 8-ft-high rectangular tank open to the atmosphere is towed by a truck on a level road. The tank is filled with water to a depth of 6 ft . Determine the maximum acceleration or deceleration allowed if no water is to spill during towing.

Penny Riley

Numerade Educator

Problem 108

An 8-ft-long tank open to the atmosphere initially contains 3-ft-high water. It is being towed by a truck on a level road. The truck driver applies the brakes and the water level at the front rises 0.5 ft above the initial level. Determine the deceleration of the truck.

Penny Riley

Numerade Educator

Problem 109

A 3-m-diameter, 7-m-long cylindrical tank is completely filled with water. The tank is pulled by a truck on a level road with the $7-\mathrm{m}$-long axis being horizontal. Determine the pressure difference between the front and back ends of the tank along a horizontal line when the truck (a) accelerates at $3 \mathrm{~m} / \mathrm{s}^2$ and (b) decelerates at $4 \mathrm{~m} / \mathrm{s}^2$.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 110

An air-conditioning system requires a 20 -m-long section of $15-\mathrm{cm}$-diameter ductwork to be laid underwater. Determine the upward force the water will exert on the duct. Take the densities of air and water to be $1.3 \mathrm{~kg} / \mathrm{m}^3$ and 1000 $\mathrm{kg} / \mathrm{m}^3$, respectively.

Mayukh Banik

Numerade Educator

Problem 111

Balloons are often filled with helium gas because it weighs only about one-seventh of what air weighs under identical conditions. The buoyancy force, which can be expressed as $F_b=\rho_{\text {air }} g V_{\text {talloon }}$, will push the balloon upward. If the balloon has a diameter of 10 m and carries two people, 70 kg each, determine the acceleration of the balloon when it is first released. Assume the density of air is $\rho=1.16 \mathrm{~kg} / \mathrm{m}^3$, and neglect the weight of the ropes and the cage.

Mayukh Banik

Numerade Educator

Problem 112

Reconsider Prob. 3-111. Using EES (or other)
software, investigate the effect of the number of people carried in the balloon on acceleration. Plot the acceleration against the number of people, and discuss the results.

Hariprasad Annamalai

Hariprasad Annamalai

Numerade Educator

Problem 113

Determine the maximum amount of load, in kg, the balloon described in Prob. 3-111 can carry.

Mayukh Banik

Numerade Educator

Problem 114

The pressure in a steam boiler is given to be 75 $\mathrm{kgf} / \mathrm{cm}^2$. Express this pressure in $\mathrm{psi}, \mathrm{kPa}, \mathrm{atm}$, and bars.

Saman Zulfiqar

Numerade Educator

Problem 115

The basic barometer can be used as an altitudemeasuring device in airplanes. The ground control reports a barometric reading of 753 mmHg while the pilot's reading is 690 mmHg . Estimate the altitude of the plane from ground level if the average air density is $1.20 \mathrm{~kg} / \mathrm{m}^3$.

Mayukh Banik

Numerade Educator

Problem 116

The lower half of a $10-\mathrm{m}$-high cylindrical container is filled with water ( $\rho=1000 \mathrm{~kg} / \mathrm{m}^3$ ) and the upper half with oil that has a specific gravity of 0.85 . Determine the pressure difference between the top and bottom of the cylinder.

Mayukh Banik

Numerade Educator

Problem 117

A vertical, frictionless piston-cylinder device contains a gas at 500 kPa . The atmospheric pressure outside is 100 kPa , and the piston area is $30 \mathrm{~cm}^2$. Determine the mass of the piston.

Mayukh Banik

Numerade Educator

Problem 118

A pressure cooker cooks a lot faster than an ordinary pan by maintaining a higher pressure and temperature inside. The lid of a pressure cooker is well sealed, and steam can escape only through an opening in the middle of the lid. A separate metal piece, the petcock, sits on top of this opening and prevents steam from escaping until the pressure force overcomes the weight of the petcock. The periodic escape of the steam in this manner prevents any potentially dangerous pressure buildup and keeps the pressure inside at a constant value. Determine the mass of the petcock of a pressure cooker whose operation pressure is 100 kPa gage and has an opening cross-sectional area of $4 \mathrm{~mm}^2$. Assume an atmospheric pressure of 101 kPa , and draw the free-body diagram of the petcock.

Mayukh Banik

Numerade Educator

Problem 119

A glass tube is attached to a water pipe, as shown in Fig. P3-119. If the water pressure at the bottom of the tube is 115 kPa and the local atmospheric pressure is 92 kPa , determine how high the water will rise in the tube, in m . Assume $g=9.8 \mathrm{~m} / \mathrm{s}^2$ at that location and take the density of water to be $1000 \mathrm{~kg} / \mathrm{m}^3$.

Mayukh Banik

Numerade Educator

Problem 120

The average atmospheric pressure on earth is approximated as a function of altitude by the relation $P_{\mathrm{um}}$ $=101.325(1-0.02256 z)^{5.256}$, where $P_{\text {mam }}$ is the atmospheric pressure in kPa and $z$ is the altitude in km with $z=0$ at sea level. Determine the approximate atmospheric pressures at Atlanta $(z=306 \mathrm{~m})$, Denver $(z=1610 \mathrm{~m})$, Mexico City $(z$ $=2309 \mathrm{~m}$ ), and the top of Mount Everest $(z=8848 \mathrm{~m})$.

Mayukh Banik

Numerade Educator

Problem 121

When measuring small pressure differences with a manometer, often one arm of the manometer is inclined to improve the accuracy of reading. (The pressure difference is still proportional to the vertical distance and not the actual length of the fluid along the tube.) The air pressure in a cir-cular duct is to be measured using a manometer whose open arm is inclined $35^{\circ}$ from the horizontal, as shown in Fig. P3-121. The density of the liquid in the manometer is 0.81 $\mathrm{kg} / \mathrm{L}$, and the vertical distance between the fluid levels in the two arms of the manometer is 8 cm . Determine the gage pressure of air in the duct and the length of the fluid column in the inclined arm above the fluid level in the vertical arm.

Ranjeet Singh

Numerade Educator

Problem 122

Consider a U-tube whose arms are open to the atmosphere. Now equal volumes of water and light oil ( $\rho$ $=49.3 \mathrm{lbm} / \mathrm{ft}^3$ ) are poured from different arms. A person blows from the oil side of the U-tube until the contact surface of the two fluids moves to the bottom of the U-tube, and thus the liquid levels in the two arms are the same. If the fluid height in each arm is 30 in , determine the gage pressure the person exerts on the oil by blowing.

Mayukh Banik

Numerade Educator

Problem 123

Intravenous infusions are usually driven by gravity by hanging the fluid bottle at sufficient height to counteract the blood pressure in the vein and to force the fluid into the body. The higher the bottle is raised, the higher the flow rate of the fluid will be. (a) If it is observed that the fluid and the blood pressures balance each other when the bottle is 1.2 m above the arm level, determine the gage pressure of the blood. (b) If the gage pressure of the fluid at the arm level needs to be 20 kPa for sufficient flow rate, determine how high the bottle must be placed. Take the density of the fluid to be $1020 \mathrm{~kg} / \mathrm{m}^3$.

Averell Hause

Carnegie Mellon University

Problem 124

A gasoline line is connected to a pressure gage through a double-U manometer, as shown in Fig. P3-124. If the reading of the pressure gage is 370 kPa , determine the gage pressure of the gasoline line.

Ranjeet Singh

Numerade Educator

Problem 125

Repeat Prob. 3-124 for a pressure gage reading of 240 kPa .

Ranjeet Singh

Numerade Educator

Problem 126

A water pipe is connected to a double-U manometer as shown in Fig. P3-1026E at a location where the local atmospheric pressure is 14.2 psia . Determine the absolute pressure at the center of the pipe.

Mukesh Devi

Numerade Educator

Problem 127

The pressure of water flowing through a pipe is measured by the arrangement shown in Fig. P3-127. For the values given, calculate the pressure in the pipe.
3-128 Consider a U-tube filled with mercury except the $18-\mathrm{cm}$-high portion at the top, as shown in Fig.

Eduard Sanchez

Numerade Educator

Problem 128

The diameter of the right arm of the U-tube is $D=2 \mathrm{~cm}$, and the diameter of the left arm is twice that. Oil with a specific gravity of 2.72 is poured into the left arm, forcing some mercury from the left arm into the right one. Determine the maximum amount of oil that can be added into the left arm.

Tanner Manwaring

Tanner Manwaring

Numerade Educator

Problem 129

A teapot with a brewer at the top is used to brew tea, as shown in Fig. P3-129. The brewer may partially block the vapor from escaping, causing the pressure in the teapot to rise and an overflow from the service tube to occur. Disregarding thermal expansion and the variation in the amount of water in the service tube to be negligible relative to the amount of water in the teapot, determine the maximum coldwater height that would not cause an overflow at gage pressures of up to 0.32 kPa for the vapor.

Hast Aggarwal

Numerade Educator

Problem 130

Repeat Prob. 3-129 by taking the thermal expansion of water into consideration as it is heated from $20^{\circ} \mathrm{C}$ to the boiling temperature of $100^{\circ} \mathrm{C}$.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 131

It is well known that the temperature of the atmosphere varies with altitude. In the troposphere, which extends to an altitude of 11 km , for example, the variation of temperature can be approximated by $T=T_0-\beta z$, where $T_0$ is the temperature at sea level, which can be taken to be 288.15 K , and $\beta=0.0065 \mathrm{~K} / \mathrm{m}$. The gravitational acceleration also changes with altitude as $g(z)=g_0 /(1+z / 6,370,320)^2$ where $g_0=9.807 \mathrm{~m} / \mathrm{s}^2$ and $z$ is the elevation from sea level in m . Obtain a relation for the variation of pressure in the troposphere (a) by ignoring and (b) by considering the variation of $g$ with altitude.

Hariprasad Annamalai

Hariprasad Annamalai

Numerade Educator

Problem 132

The variation of pressure with density in a thick gas layer is given by $P=C \rho^n$, where $C$ and $n$ are constants. Not-ing that the pressure change across a differential fluid layer of thickness $d z$ in the vertical $z$-direction is given as $d P$ $=-\rho g d z$, obtain a relation for pressure as a function of elevation $z$. Take the pressure and density at $z=0$ to be $P_0$ and $\rho_0$, respectively.

Hariprasad Annamalai

Hariprasad Annamalai

Numerade Educator

Problem 133

Pressure transducers are commonly used to measure pressure by generating analog signals usually in the range of 4 mA to 20 mA or $0 \mathrm{~V}-\mathrm{dc}$ to $10 \mathrm{~V}-\mathrm{dc}$ in response to applied pressure. The system whose schematic is shown in Fig. P3-133 can be used to calibrate pressure transducers. A rigid container is filled with pressurized air, and pressure is measured by the manometer attached. A valve is used to regulate the pressure in the container. Both the pressure and the electric signal are measured simultaneously for various settings, and the results are tabulated. For the given set of measurements, obtain the calibration curve in the form of $P=a I$ $+b$, where $a$ and $b$ are constants, and calculate the pressure that corresponds to a signal of 10 mA .

Ranjeet Singh

Numerade Educator

Problem 134

A system is equipped with two pressure gages and a manometer, as shown in Fig. P3-134. For $\Delta h=8 \mathrm{~cm}$, determine the pressure difference $\Delta P=P_2-P_1$.

Victor Salazar

Numerade Educator

Problem 135

An oil pipeline and a $1.3-\mathrm{m}^3$ rigid air tank are connected to each other by a manometer, as shown in Fig. P3-135. If the tank contains 15 kg of air at $80^{\circ} \mathrm{C}$, determine (a) the absolute pressure in the pipeline and (b) the change in $\Delta h$ when the temperature in the tank drops to $20^{\circ} \mathrm{C}$. Assume the pressure in the oil pipeline to remain constant, and the air volume in the manometer to be negligible relative to the volume of the tank.

Surendra Kumar

Numerade Educator

Problem 136

The density of a floating body can be determined by tying weights to the body until both the body and the weights are completely submerged, and then weighing them sepa-rately in air Consider a wood log that weighs 1540 N in air If it takes 34 kg of lead $\left(\rho=11,300 \mathrm{~kg} / \mathrm{m}^3\right)$ to completely sink the $\log$ and the lead in water, determine the average density of the log.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 137

The $200-\mathrm{kg}$, 5-m-wide rectangular gate shown in Fig. P3-137 is hinged at $B$ and leans against the floor at $A$ making an angle of $45^{\circ}$ with the horizontal. The gate is to be opened from its lower edge by applying a normal force at its center. Determine the minimum force $F$ required to open the water gate.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 138

Repeat Prob. 3-137 for a water height of 1.2 m above the hinge at $B$.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 139

A 3-m-high, 6-m-wide rectangular gate is hinged at the top edge at $A$ and is restrained by a fixed ridge at $B$. Determine the hydrostatic force exerted on the gate by the $5-\mathrm{m}$-high water and the location of the pressure center.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 140

Repeat Prob. 3-139 for a total water height of 2 m .

Mayukh Banik

Numerade Educator

Problem 141

A semicircular 30 - ft -diameter tunnel is to be built under a 150 -ft-deep, 800 -ft-long lake, as shown in Fig. P3-141E. Determine the total hydrostatic force acting on the roof of the tunnel.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 142

A 50-ton, 6-m-diameter hemispherical dome on a level surface is filled with water, as shown in Fig. P3-142. Someone claims that he can lift this dome by making use of Pascal's law by attaching a long tube to the top and filling it with water. Determine the required height of water in the tube to lift the dome. Disregard the weight of the tube and the water in it.

Surendra Kumar

Numerade Educator

Problem 143

The water in a 25 -m-deep reservoir is kept inside by a $150-\mathrm{m}$-wide wall whose cross section is an equilateral triangle, as shown in Fig. P3-143. Determine (a) the total force (hydrostatic + atmospheric) acting on the inner surface of the wall and its line of action and (b) the magnitude of the horizontal component of this force. Take $P_{\text {atm }}=100 \mathrm{kPa}$.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 144

A U-tube contains water in the right arm, and another liquid in the left arm. It is observed that when the Utube rotates at 30 rpm about an axis that is 15 cm from the right arm and 5 cm from the left arm, the liquid levels in both arms become the same. Determine the density of the fluid in the left arm.

Tanner Manwaring

Tanner Manwaring

Numerade Educator

Problem 145

A 1-m-diameter, 2-m-high vertical cylinder is completely filled with gasoline whose density is $740 \mathrm{~kg} / \mathrm{m}^3$. The tank is now rotated about its vertical axis at a rate of 90 rpm , while being accelerated upward at $5 \mathrm{~m} / \mathrm{s}^2$. Determine (a) the difference between the pressures at the centers of the bottom and top surfaces and (b) the difference between the pressures at the center and the edge of the bottom surface.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 146

A 5-m-long, 4-m-high tank contains 2.5 -m-deep water when not in motion and is open to the atmosphere through a vent in the middle. The tank is now accelerated to the right on a level surface at $2 \mathrm{~m} / \mathrm{s}^2$. Determine the maximum pressure in the tank relative to the atmospheric pressure.

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Problem 147

$\Subset$ Reconsider Prob. 3-146. Using EES (or other) software, investigate the effect of acceleration on the slope of the free surface of water in the tank. Let the acceleration vary from $0 \mathrm{~m} / \mathrm{s}^2$ to $5 \mathrm{~m} / \mathrm{s}^2$ in increments of 0.5 $\mathrm{m} / \mathrm{s}^2$. Tabulate and plot your results.

Hariprasad Annamalai

Hariprasad Annamalai

Numerade Educator

Problem 148

An elastic air balloon having a diameter of 30 cm is attached to the base of a container partially filled with water at $+4^{\circ} \mathrm{C}$, as shown in Fig. P3-148. If the pressure of air above water is gradually increased from 100 kPa to 1.6 MPa , will the force on the cable change? If so, what is the percent change in the force? Assume the pressure on the free surface and the diameter of the balloon are related by $P=C D^n$, where $C$ is a constant and $n=-2$. The weight of the balloon and the air in it is negligible.

Vipender Yadav

Numerade Educator

Problem 149

Reconsider Prob. 3-148. Using EES (or other) software, investigate the effect of air pressure above water on the cable force. Let this pressure vary from 0.1 MPa to 10 MPa . Plot the cable force versus the air pressure.

Vidhi Bhatt

Numerade Educator

Problem 150

The average density of icebergs is about $917 \mathrm{~kg} / \mathrm{m}^3$. (a) Determine the percentage of the total volume of an iceberg submerged in seawater of density $1042 \mathrm{~kg} / \mathrm{m}^3$. (b) Although icebergs are mostly submerged, they are observed to turn over. Explain how this can happen.

Jill Tolbert

Numerade Educator

Problem 151

A cylindrical container whose weight is 79 N is inverted and pressed into the water, as shown in Fig. P3-151. Determine the differential height $h$ of the manometer and the force $F$ needed to hold the container at the position shown.

Prashant Bana

Numerade Educator

Problem 152

Shoes are to be designed to enable people of up to 80 kg to walk on freshwater or seawater. The shoes are to be made of blown plastic in the shape of a sphere, a (American) football, or a loaf of French bread. Determine the equivalent diameter of each shoe and comment on the proposed shapes from the stability point of view. What is your assessment of the marketability of these shoes?

Shahab Ullah

Numerade Educator

Problem 153

The volume of a rock is to be determined without using any volume measurement devices. Explain how you would do this with a waterproof spring scale.

Averell Hause

Carnegie Mellon University