Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 14

Fluid Mechanics - all with Video Answers

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Chapter Questions

Problem 1

Calculate the mass of a solid gold rectangular bar that has dimensions of $4.50 \mathrm{cm} \times 11.0 \mathrm{cm} \times 26.0 \mathrm{cm} .$

Mayukh Banik

Mayukh Banik

Numerade Educator

Problem 2

The nucleus of an atom can be modeled as several protons and neutrons closely packed together. Each particle has a mass of $1.67 \times 10^{-27} \mathrm{kg}$ and radius on the order of $10^{-15} \mathrm{m}$ . (a) Use this model and the data provided to estimate the density of the nucleus of an atom. (b) Compare your result with the density of a material such as iron. What do your result and comparison suggest concerning the structure of matter?

Mayukh Banik

Numerade Educator

Problem 3

A 50.0-kg woman wearing high-heeled shoes is invited into a home in which the kitchen has vinyl floor covering. The heel on each shoe is circular and has a radius of 0.500 cm. (a) If the woman balances on one heel, what pressure does she exert on the floor? (b) Should the home- owner be concerned? Explain your answer.

Umar Sohail Qureshi

Umar Sohail Qureshi

Numerade Educator

Problem 4

Estimate the total mass of the Earth's atmosphere. (The radius of the Earth is $6.37 \times 10^{6} \mathrm{m},$ and atmospheric pressure at the surface is $1.013 \times 10^{5} \mathrm{Pa}$ .)

Mayukh Banik

Numerade Educator

Problem 5

A large man sits on a four-legged chair with his feet off the floor. The combined mass of the man and chair is 95.0 kg. If the chair legs are circular and have a radius of 0.500 cm at the bottom, what pressure does each leg exert on the floor?

Mayukh Banik

Numerade Educator

Problem 6

A swimming pool has dimensions $30.0 \mathrm{m} \times 10.0 \mathrm{m}$ and a flat bottom. When the pool is filled to a depth of 2.00 $\mathrm{m}$ with fresh water, what is the force exerted by the water on (a) the bottom? (b) On each end? (c) On each side?

Mayukh Banik

Numerade Educator

Problem 7

The spring of the pressure gauge shown in Figure P14.7 has a force constant of 1 250 N/m, and the piston has a diameter of 1.20 cm. As the gauge is lowered into water in a lake, what change in depth causes the piston to move in by 0.750 cm?

Eduard Sanchez

Numerade Educator

Problem 8

The small piston of a hydraulic lift (Fig. Pl4.8) has a cross- sectional area of $3.00 \mathrm{cm}^{2},$ and its large piston has a cross sectional area of $200 \mathrm{cm}^{2} .$ What downward force of magnitude $F_{1}$ must be applied to the small piston for the lift to raise a load whose weight is $F_{g}=15.0 \mathrm{kN} ?$

Mayukh Banik

Numerade Educator

Problem 9

What must be the contact area between a suction cup (completely evacuated) and a ceiling if the cup is to support the weight of an 80.0-kg student?

Mayukh Banik

Numerade Educator

Problem 10

(a) A very powerful vacuum cleaner has a hose 2.86 $\mathrm{cm}$ in diameter. With the end of the hose placed perpendicularly on the flat face of a brick, what is the weight of the heaviest brick that the cleaner can lift? (b) What If? An octopus uses one sucker of diameter 2.86 $\mathrm{cm}$ on each of the two
shells of a clam in an attempt to pull the shells apart. Find the greatest force the octopus can exert on a clamshell in salt water 32.3 m deep.

Mayukh Banik

Numerade Educator

Problem 11

(a) Calculate the absolute pressure at the bottom of a freshwater lake at a point whose depth is $27.5 \mathrm{m} .$ Assume the density of the water is $1.00 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$ and that the air above is at a pressure of 101.3 $\mathrm{kPa}$ . (b) What force is exerted by the water on the window of an underwater vehicle at this depth if the window is circular and has a diameter of 35.0 $\mathrm{cm}$ ?

Mayukh Banik

Numerade Educator

Problem 12

Why is the following situation impossible? Figure P14.12 shows Superman attempting to drink cold water through a straw of length $\ell=12.0 \mathrm{m} .$ The walls of the tubular straw are very strong and do not collapse. With his great strength, he achieves maximum possible suction and enjoys drinking the cold water.

Mayukh Banik

Numerade Educator

Problem 13

Review. The tank in Figure $\mathrm{P} 14.13$ is filled with water of depth $d=2.00 \mathrm{m} .$ At the bottom of one sidewall is a rectangular hatch of height $h=1.00 \mathrm{m}$ and width $w=2.00 \mathrm{m}$ that is hinged at the top of the hatch. (a) Determine the magnitude of the force the water exerts on the hatch. (b) Find the magnitude of the torque exerted by the water about the hinges.

Mayukh Banik

Numerade Educator

Problem 14

The tank in Figure $\mathrm{P} 14.13$ is filled with water of depth $d .$ At the bottom of one sidewall is a rectangular hatch of height $h$ and width $w$ that is hinged at the top of the hatch. (a) Determine the magnitude of the force the water exerts on the hatch. (b) Find the magnitude of the torque exerted by the water about the hinges.

Mayukh Banik

Numerade Educator

Problem 15

Review. A solid sphere of brass (bulk modulus of $14.0 \times$ $10^{10} \mathrm{N} / \mathrm{m}^{2} )$ with a diameter of 3.00 $\mathrm{m}$ is thrown into the ocean. By how much does the diameter of the sphere decrease as it sinks to a depth of 1.0 $\mathrm{km}$ ?

Mayukh Banik

Numerade Educator

Problem 16

Blaise Pascal duplicated Torricelli's barometer using a red Bordeaux wine, of density $984 \mathrm{kg} / \mathrm{m}^{3},$ as the working liquid (Fig. Pl4.16). (a) What was the height $h$ of the wine column for normal atmospheric pressure? (b) Would you expect the vacuum above the column to be as good as for
mercury?

Mayukh Banik

Numerade Educator

Problem 17

Normal atmospheric pressure is $1.013 \times 10^{5}$ Pa. The approach of a storm causes the height of a mercury barometer to drop by 20.0 $\mathrm{mm}$ from the normal height. What is the atmospheric pressure?

Mayukh Banik

Numerade Educator

Problem 18

The human brain and spinal cord are immersed in the cerebrospinal fluid. The fluid is normally continuous between the cranial and spinal cavities and exerts a pressure of 100 to 200 $\mathrm{mm}$ of $\mathrm{H}_{2} \mathrm{O}$ above the prevailing atmospheric pressure. In medical work, pressures are often measured in units of millimeters of $\mathrm{H}_{2} \mathrm{O}$ because body fluids, including the cerebrospinal fluid, typically have the same density as water. The pressure of the cerebrospinal fluid can be measured by means of a spinal tap as illustrated in Figure P14.18. A hollow tube is inserted into the spinal column, and the height to which the fluid rises is observed. If the fluid rises to a height of $160 \mathrm{mm},$ we write its gauge pressure as 160 $\mathrm{mm} \mathrm{H}_{2} \mathrm{O}$ . ( a ) Express this pressure in pascals, in atmospheres, and in millimeters of mercury. (b) Some conditions that block or inhibit the flow of cerebrospinal fluid can be investigated by means of Queckenstedt's test. In this procedure, the veins in the patient's neck are compressed to make the blood pressure rise in the brain, which in turn should be transmitted to the cerebrospinal fluid. Explain how the level of fluid in the spinal tap can be used as a diagnostic tool for the condition of the patient's spine.

Mayukh Banik

Numerade Educator

Problem 19

A backyard swimming pool with a circular base of diameter 6.00 $\mathrm{m}$ is filled to depth 1.50 $\mathrm{m}$ (a) Find the absolute - pressure at the bottom of the pool. (b) Two persons with combined mass 150 $\mathrm{kg}$ enter the pool and float quietly there. No water overflows. Find the pressure increase at the bottom of the pool after they enter the pool and float.

Mayukh Banik

Numerade Educator

Problem 20

A tank with a flat bottom of area $A$ and vertical sides is filled to a depth $h$ with water. The pressure is $P_{0}$ at the top surface. (a) What is the absolute pressure at the bottom of the tank? (b) Suppose an object of mass $M$ and density less than the density of water is placed into the tank and floats. No water overflows. What is the resulting increase in pressure at the bottom of the tank?

Penny Riley

Numerade Educator

Problem 21

Mercury is poured into a U-tube as shown in Figure Pl4. 2 la. The left arm of the tube has cross-sectional area $A_{1}$ of $10.0 \mathrm{cm}^{2},$ and the right arm has a cross-sectional area $A_{2}$ of 5.00 $\mathrm{cm}^{2}$ . One hundred grams of water are then poured into the right arm as shown in Figure Pl4.21b. (a) Determine the length of the water column in the right arm of the U-tube. (b) Given that the density of mercury is $13.6 \mathrm{g} / \mathrm{cm}^{3},$ what distance $h$ does the mercury rise in the left arm?

Mayukh Banik

Numerade Educator

Problem 22

A light balloon is filled with 400 $\mathrm{m}^{3}$ of helium at atmo- spheric pressure. (a) At $0^{\circ} \mathrm{C}$ , the balloon can lift a payload of what mass? (b) What If? In Table 14.1 , observe that the
density of hydrogen is nearly half the density of helium. What load can the balloon lift if filled with hydrogen?

Mayukh Banik

Numerade Educator

Problem 23

A table-tennis ball has a diameter of 3.80 $\mathrm{cm}$ and average density of $0.0840 \mathrm{g} / \mathrm{cm}^{3} .$ What force is required to hold it completely submerged under water?

Mayukh Banik

Numerade Educator

Problem 24

The gravitational force exerted on a solid object is 5.00 $\mathrm{N}$ . When the object is suspended from a spring scale and sub- merged in water, the scale reads 3.50 $\mathrm{N}$ (Fig. Pl4.24). Find the density of the object.

Mayukh Banik

Numerade Educator

Problem 25

A 10.0 -kg block of metal measuring 12.0 $\mathrm{cm}$ by 10.0 $\mathrm{cm}$ by 10.0 $\mathrm{cm}$ is suspended from a scale and immersed in water as shown in Figure $\mathrm{P} 14.24 \mathrm{b}$ . The 12.0 -cm dimension is vertical, and the top of the block is 5.00 $\mathrm{cm}$ below the surface of the water. (a) What are the magnitudes of the forces acting on the top and on the bottom of the block due to the surrounding water? (b) What is the reading of the spring scale? (c) Show that the buoyant force equals the difference between the forces at the top and bottom of the block.

Mayukh Banik

Numerade Educator

Problem 26

The United States possesses the ten largest warships in the world, aircraft carriers of the Nimitz class. Suppose one of the ships bobs up to float 11.0 $\mathrm{cm}$ higher in the ocean water when 50 fighters take off from it in a time interval of 25 $\mathrm{min}$ , at a location where the free-fall acceleration is 9.78 $\mathrm{m} / \mathrm{s}^{2}$ . The planes have an average laden mass of 29000 $\mathrm{kg}$ . Find the horizontal area enclosed by the waterline of the ship.

Mayukh Banik

Numerade Educator

Problem 27

A cube of wood having an edge dimension of 20.0 $\mathrm{cm}$ and a density of 650 $\mathrm{kg} / \mathrm{m}^{3}$ floats on water. (a) What is the distance from the horizontal top surface of the cube to the water level? (b) What mass of lead should be placed on the cube so that the top of the cube will be just level with the water surface?

Mayukh Banik

Numerade Educator

Problem 28

A spherical vessel used for deep-sea exploration has a radius of 1.50 $\mathrm{m}$ and a mass of $1.20 \times 10^{4} \mathrm{kg}$ . To dive, the vessel takes on mass in the form of seawater. Determine the
mass the vessel must take on if it is to descend at a constant speed of 1.20 $\mathrm{m} / \mathrm{s}$ , when the resistive force on it is 1100 $\mathrm{N}$ in the upward direction. The density of seawater is equal to $1.03 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$ .

Mayukh Banik

Numerade Educator

Problem 29

A plastic sphere floats in water with 50.0% of its volume submerged. This same sphere floats in glycerin with 40.0% of its volume submerged. Determine the densities of (a) the glycerin and (b) the sphere.

Mayukh Banik

Numerade Educator

Problem 30

The weight of a rectangular block of low-density material is 15.0 N. With a thin string, the center of the horizontal bottom face of the block is tied to the bottom of a beaker partly filled with water. When 25.0% of the block’s volume is submerged, the tension in the string is 10.0 N. (a) Find the buoyant force on the block. (b) Oil of density 800 kg/m3 is now steadily added to the beaker, forming a layer above the water and surrounding the block. The oil exerts forces on each of the four sidewalls of the block that the oil touches. What are the directions of these forces? (c) What happens to the string tension as the oil is added? Explain how the oil has this effect on the string tension. (d) The string breaks when its tension reaches 60.0 N. At this moment, 25.0% of the block’s volume is still below the water line. What additional fraction of the block’s volume is below the top surface of the oil?

Mayukh Banik

Numerade Educator

Problem 31

A wooden block of volume $5.24 \times 10^{-4} \mathrm{m}^{3}$ floats in water, and a small steel object of mass $m$ is placed on top of the block. When $m=0.310 \mathrm{kg}$ , the system is in equilibrium
and the top of the wooden block is at the level of the water. (a) What is the density of the wood? (b) What happens to the block when the steel object is replaced by an object whose mass is less than 0.310 $\mathrm{kg}$ ? (c) What happens to the block when the steel object is replaced by an object whose mass is greater than 0.310 $\mathrm{kg}$ ?

Mayukh Banik

Numerade Educator

Problem 32

On October 21, 2001, Ian Ashpole of the United Kingdom achieved a record altitude of 3.35 km (11 000 ft)
powered by 600 toy balloons filled with helium. Each filled balloon had a radius of about 0.50 m and an estimated mass of 0.30 kg. (a) Estimate the total buoyant force on the 600 balloons. (b) Estimate the net upward force on all 600 balloons. (c) Ashpole parachuted to the Earth after the balloons began to burst at the high altitude and the buoyant force decreased. Why did the balloons burst?

Mayukh Banik

Numerade Educator

Problem 33

A large weather balloon whose mass is 226 kg is filled with helium gas until its volume is 325 $\mathrm{m}^{3}$ . Assume the density of air is 1.20 $\mathrm{kg} / \mathrm{m}^{3}$ and the density of helium is $0.179 \mathrm{kg} / \mathrm{m}^{3} .$ (a) Calculate the buoyant force acting on the balloon. (b) Find the net force on the balloon and determine whether the balloon will rise or fall after it is released. (c) What additional mass can the balloon support in equilibrium?

Surendra Kumar

Numerade Educator

Problem 34

A hydrometer is an instrument used to determine liquid density. A simple one is sketched in Figure $\mathrm{P} 14.34$ . The bulb of a syringe is squeezed and released to let the atmosphere lift a sample of the liquid of interest into a tube containing a calibrated rod of known density. The rod, of length $L$ and average density $\rho_{0}$ , floats partially immersed in the liquid of density $\rho .$ A length $h$ of the rod protrudes above the surface of the liquid. Show that the density of the liquid is given by
$$
\rho=\frac{\rho_{0} L}{L-h}
$$

Penny Riley

Numerade Educator

Problem 35

Refer to Problem 34 and Figure P14.34. A hydrometer is to be constructed with a cylindrical floating rod.
Nine fiduciary marks are to be placed along the rod to indicate densities of $0.98 \mathrm{g} / \mathrm{cm}^{3}, 1.00 \mathrm{g} / \mathrm{cm}^{3}, 1.02 \mathrm{g} / \mathrm{cm}^{3},$
$1.04 \mathrm{g} / \mathrm{cm}^{3}, \ldots, 1.14 \mathrm{g} / \mathrm{cm}^{3} .$ The row of marks is to start 0.200 $\mathrm{cm}$ from the top end of the rod and end 1.80 $\mathrm{cm}$ from the top end. (a) What is the required length of the rod? (b) What must be its average density? (c) Should the marks be equally spaced? Explain your answer.

Mayukh Banik

Numerade Educator

Problem 36

How many cubic meters of helium are required to lift a balloon with a 400-kg payload to a height of 8 000 m? Take $\rho_{\mathrm{He}}=0.179 \mathrm{kg} / \mathrm{m}^{3} .$ Assume the balloon maintains a constant volume and the density of air decreases with the altitude $z$ according to the expression $\rho_{\text { air }}=\rho_{0} e^{-z / 8000}$ , where $z$ is in meters and $\rho_{0}=1.20 \mathrm{kg} / \mathrm{m}^{3}$ is the density of air at sea level.

Mayukh Banik

Numerade Educator

Problem 37

A large storage tank, open at the top and filled with water, develops a small hole in its side at a point 16.0 $\mathrm{m}$ below the water level. The rate of flow from the leak is found to be $2.50 \times 10^{-3} \mathrm{m}^{3} / \mathrm{min}$ . Determine (a) the speed at which the water leaves the hole and (b) the diameter of the hole.

Mayukh Banik

Numerade Educator

Problem 38

Water flowing through a garden hose of diameter 2.74 $\mathrm{cm}$ fills a $25-\mathrm{L}$ bucket in 1.50 $\mathrm{min}$ (a) What is the speed of the water leaving the end of the hose? (b) A nozzle is now
attached to the end of the hose. If the nozzle diameter is one-third the diameter of the hose, what is the speed of the water leaving the nozzle?

Mayukh Banik

Numerade Educator

Problem 39

Water moves through a constricted pipe in steady, ideal flow. At the lower point shown in Figure $\mathrm{P} 14.39$ , the pressure is $P_{1}=1.75 \times 10^{4} \mathrm{Pa}$ and the pipe diameter is $6.00 \mathrm{cm} .$ At another point $y=0.250 \mathrm{m}$ higher, the pressure is $P_{2}=$ $1.20 \times 10^{4} \mathrm{Pa}$ and the pipe diameter is $3.00 \mathrm{cm} .$ Find the speed of flow (a) in the lower section and $(\mathrm{b})$ in the upper section. (c) Find the volume flow rate through the pipe.

Mayukh Banik

Numerade Educator

Problem 40

A village maintains a large tank with an open top, containing water for emergencies. The water can drain from the tank through a hose of diameter $6.60 \mathrm{cm} .$ The hose ends with a nozzle of diameter $2.20 \mathrm{cm} .$ A rubber stopper is inserted into the nozzle. The water level in the tank is kept 7.50 $\mathrm{m}$ above the nozzle. (a) Calculate the friction force exerted on the stopper by the nozzle. (b) The stopper is removed. What mass of water flows from the nozzle in 2.00 $\mathrm{h}$ ? (c) Calculate the gauge pressure of the flowing water in the hose just behind the nozzle.

Mayukh Banik

Numerade Educator

Problem 41

Figure $\mathrm{P} 14.41$ shows a stream of water in steady flow from a kitchen faucet. At the faucet, the diameter of the stream is $0.960 \mathrm{cm} .$ The stream fills a $125-\mathrm{cm}^{3}$ container in 16.3 s. Find the diameter of the stream 13.0 $\mathrm{cm}$ below the opening of the faucet.

Mayukh Banik

Numerade Educator

Problem 42

Water falls over a dam of height $h$ with a mass flow rate of $R,$ in units of kilograms per second. (a) Show that the power available from the water is
$$
P=R g h
$$
where $g$ is the free-fall acceleration. (b) Each hydroelectric unit at the Grand Coulee Dam takes in water at a rate of $8.50 \times 10^{5} \mathrm{kg} / \mathrm{s}$ from a height of 87.0 $\mathrm{m}$ . The power developed by the falling water is converted to electric power with an efficiency of $85.0 \% .$ How much electric power does each hydroelectric unit produce?

Mayukh Banik

Numerade Educator

Problem 43

A legendary Dutch boy saved Holland by plugging a hole of diameter 1.20 $\mathrm{cm}$ in a dike with his finger. If the hole was 2.00 $\mathrm{m}$ below the surface of the North Sea (density 1030 $\mathrm{kg} / \mathrm{m}^{3}$ , (a) what was the force on his finger? (b) If he pulled his finger out of the hole, during what time interval would the released water fill 1 acre of land to a depth of $\mathrm{ft}$ ? Assume the hole remained constant in size.

Mayukh Banik

Numerade Educator

Problem 44

In ideal flow, a liquid of density 850 $\mathrm{kg} / \mathrm{m}^{3}$ moves from a horizontal tube of radius 1.00 $\mathrm{cm}$ into a second horizontal tube of radius 0.500 $\mathrm{cm}$ at the same elevation as the first tube. The pressure differs by $\Delta P$ between the liquid in one tube and the liquid in the second tube. (a) Find the volume flow rate as a function of $\Delta P$ . Evaluate the volume flow rate for $(\mathrm{b}) \Delta P=6.00 \mathrm{kPa}$ and $(\mathrm{c}) \Delta P=12.0 \mathrm{kPa}$

Mayukh Banik

Numerade Educator

Problem 45

Water is pumped up from the Colorado River to supply Grand Canyon Village, located on the rim of the canyon. The river is at an elevation of 564 $\mathrm{m}$ , and the village is at an elevation of 2096 $\mathrm{m}$ . Imagine that the water is pumped through a single long pipe 15.0 $\mathrm{cm}$ in diameter, driven by a single pump at the bottom end. (a) What is the minimum pressure at which the water must be pumped if it is to arrive at the village? (b) If 4500 $\mathrm{m}^{3}$ of water are pumped per day, what is the speed of the water in the pipe? Note: Assume the free-fall acceleration and the density of air are constant over this range of elevations. The pressures you calculate are too high for an ordinary pipe. The water is actually lifted in stages by several pumps through shorter pipes.

Mayukh Banik

Numerade Educator

Problem 46

Review. Old Faithful Geyser in Yellowstone National Park erupts at approximately one-hour intervals, and the height of the water column reaches $40.0 \mathrm{m} .$ (Fig. Pl4.46, page 428 ). (a) Model the rising stream as a series of separate droplets. Analyze the free-fall motion of one of the droplets to determine the speed at which the water leaves the ground. (b) What If? Model the rising stream as an ideal fluid in streamline flow. Use Bernoulli's equation to determine the speed of the water as it leaves ground level. (c) How does the answer from part (a) compare with the answer from part (b)? (d) What is the pressure (above atmospheric) in the heated underground chamber if its depth is 175 $\mathrm{m}$ ? Assume the chamber is large compared with the geyser's vent.

Mayukh Banik

Numerade Educator

Problem 47

The Venturi tube discussed in Example 14.8 and shown in Figure $\mathrm{P} 14.47$ may be used as a fluid flowmeter. Suppose the device is used at a service station to measure the flow rate of gasoline $\left(\rho=7.00 \times 10^{2} \mathrm{kg} / \mathrm{m}^{3}\right)$ through a hose having an outlet radius of $1.20 \mathrm{cm} .$ If the difference in pressure is measured to be $P_{1}-P_{2}=1.20 \mathrm{kPa}$ and the radius of the inlet tube to the meter is $2.40 \mathrm{cm},$ find ( a ) the speed of
the gasoline as it leaves the hose and (b) the fluid flow rate in cubic meters per second.

Surendra Kumar

Numerade Educator

Problem 48

An airplane has a mass of $1.60 \times 10^{4} \mathrm{kg},$ and each wing has an area of $40.0 \mathrm{m}^{2} .$ During level flight, the pressure on the lower wing surface is $7.00 \times 10^{4} \mathrm{Pa}$ . (a) Suppose the lift on the airplane were due to a pressure difference alone. Determine the pressure on the upper wing surface. (b) More realistically, a significant part of the lift is due to deflection of air downward by the wing. Does the inclusion of this force mean that the pressure in part (a) is higher or lower? Explain.

Mayukh Banik

Numerade Educator

Problem 49

An airplane is cruising at altitude 10 $\mathrm{km}$ . The pressure outside the craft is 0.287 atm; within the passenger com- partment, the pressure is 1.00 $\mathrm{atm}$ and the temperature is $20^{\circ} \mathrm{C}$ . A small leak occurs in one of the window seals in the passenger compartment. Model the air as an ideal fluid to estimate the speed of the airstream flowing through the leak.

Mayukh Banik

Numerade Educator

Problem 50

A siphon is used to drain water from a tank as illustrated in Figure $\mathrm{P} 14.50 .$ Assume steady flow without friction. (a) If $h=1.00 \mathrm{m}$ , find the speed of outflow at the end of the siphon. (b) What If? What is the limitation on the height of the top of the siphon above the end of the siphon? Note: For the flow of the liquid to be continuous, its pressure must not drop below its vapor pressure. Assume the water is at $20.0^{\circ} \mathrm{C},$ at which the vapor pressure is 2.3 $\mathrm{kPa}$ .

Mayukh Banik

Numerade Educator

Problem 51

A hypodermic syringe contains a medicine with the density of water (Fig. Pl4.51). The barrel of the syringe
has a cross-sectional area $A=2.50 \times 10^{-5} \mathrm{m}^{2},$ and the needle has a cross-sectional area $a=1.00 \times 10^{-8} \mathrm{m}^{2} .$ In the absence of a force on the plunger, the pressure everywhere is 1.00 atm. A force $\overrightarrow{\mathbf{F}}$ of magnitude 2.00 $\mathrm{N}$ acts on the plunger, making medicine squirt horizontally from the needle. Determine the speed of the medicine as it leaves the needle's tip.

Mayukh Banik

Numerade Educator

Problem 52

The Bernoulli effect can have important consequences for the design of buildings. For example, wind can blow around a skyscraper at remarkably high speed, creating low pressure. The higher atmospheric pressure in the still air inside the buildings can cause windows to pop out. As originally constructed, the John Hancock Building in Boston popped windowpanes that fell many stories to the sidewalk below. (a) Suppose a horizontal wind blows with a speed of 11.2 $\mathrm{m} / \mathrm{s}$ outside a large pane of plate glass with dimensions $4.00 \mathrm{m} \times 1.50 \mathrm{m}$ . Assume the density of the air to be constant at 1.20 $\mathrm{kg} / \mathrm{m}^{3}$ . The air inside the building is at atmospheric pressure. What is the total force exerted by air on the windowpane? (b) What If? If a second skyscraper is built nearby, the airspeed can be especially high
where wind passes through the narrow separation between the buildings. Solve part (a) again with a wind speed of 22.4 $\mathrm{m} / \mathrm{s}$ , twice as high.

Mayukh Banik

Numerade Educator

Problem 53

(a) Calculate the absolute pressure at an ocean depth of 1000 $\mathrm{m}$ . Assume the density of seawater is 1030 $\mathrm{kg} / \mathrm{m}^{3}$ and the air above exerts a pressure of 101.3 $\mathrm{kPa}$ . (b) At this depth, what is the buoyant force on a spherical submarine having a diameter of 5.00 $\mathrm{m}$ ?

Mayukh Banik

Numerade Educator

Problem 54

In about 1657 , Otto von Guericke, inventor of the air pump, evacuated a sphere made of two brass hemispheres (Fig. Pl4.54). Two teams of eight horses each could pull the hemispheres apart only on some trials and then "with greatest difficulty," with the resulting sound likened to a cannon firing. Find the force $F$ required to pull the thin- walled evacuated hemispheres apart in terms of $R,$ the radius of the hemispheres; $P$ , the pressure inside the hemispheres; and atmospheric pressure $P_{0}$ .

Mayukh Banik

Numerade Educator

Problem 55

A spherical aluminum ball of mass 1.26 $\mathrm{kg}$ contains an empty spherical cavity that is concentric with the ball. The ball barely floats in water. Calculate (a) the outer radius of the ball and (b) the radius of the cavity.

Mayukh Banik

Numerade Educator

Problem 56

A helium-filled balloon (whose envelope has a mass of $m_{b}=0.250 \mathrm{kg}$ ) is tied to a uniform string of length $\ell=$ 2.00 $\mathrm{m}$ and mass $m=0.0500 \mathrm{kg}$ . The balloon is spherical with a radius of $r=0.400 \mathrm{m} .$ When released in air of temperature $20^{\circ} \mathrm{C}$ and density $\rho_{\text { air }}=1.20 \mathrm{kg} / \mathrm{m}^{3},$ it lifts a length $h$ of string and then remains stationary as shown in Figure $\mathrm{P} 14.56 .$ We wish to find the length of string lifted by the balloon. (a) When the balloon remains stationary, what is the appropriate analysis model to describe it? (b) Write a force equation for the balloon from this model in terms of the buoyant force $B$ , the weight $F_{b}$ of the balloon, the weight $F_{\mathrm{He}}$ of the helium, and the weight $F_{s}$ of the segment of string of length $h$ . (c) Make an appropriate substitution for each of these forces and solve symbolically for the mass $m_{s}$ of the segment of string of length $h$ in terms of $m_{b}, r, \rho_{\text { air }},$ and the density of helium $\rho_{\mathrm{Hc}} .$ (d) Find the numerical value of the mass $m_{s \cdot}$ (e) Find the length $h$ numerically.

Mayukh Banik

Numerade Educator

Problem 57

Review. Figure $\mathrm{P} 14.57$ shows a valve separating a reservoir from a water tank. If this valve is opened, what is the maximum height above point $B$ attained by the water stream coming out of the right side of the tank? Assume $h=$ $10.0 \mathrm{m}, L=2.00 \mathrm{m},$ and $\theta=30.0^{\circ}$ , and assume the cross- sectional area at $A$ is very large compared with that at $B$ .

Mayukh Banik

Numerade Educator

Problem 58

The true weight of an object can be measured in a vacuum, where buoyant forces are absent. A measurement in air, however, is disturbed by buoyant forces. An object of volume V is weighed in air on an equal-arm balance with the use of counterweights of density $\rho .$ Representing the density of air as $\rho_{\text { air }}$ and the balance reading as $F_{g}^{\prime},$ show that the true weight $F_{g}$ is
$$
F_{g}=F_{g}^{\prime}+\left(V-\frac{F_{g}^{\prime}}{\rho g}\right) \rho_{\text { air }} g
$$

Mayukh Banik

Numerade Educator

Problem 59

Water is forced out of a fire extinguisher by air pressure as shown in Figure $\mathrm{P} 14.59 .$ How much gauge air pressure in the tank is required for the water jet to have a speed of 30.0 $\mathrm{m} / \mathrm{s}$ when the water level is 0.500 $\mathrm{m}$ below the nozzle?

Mayukh Banik

Numerade Educator

Problem 60

Review. Assume a certain liquid, with density 1 230 $\mathrm{kg} / \mathrm{m}^{3}$ , exerts no friction force on spherical objects. A ball of mass 2.10 $\mathrm{kg}$ and radius 9.00 $\mathrm{cm}$ is dropped
from rest into a deep tank of this liquid from a height of 3.30 $\mathrm{m}$ above the surface. (a) Find the speed at which the ball enters the liquid. (b) Evaluate the magnitudes of the two forces that are exerted on the ball as it moves through the liquid. (c) Explain why the ball moves down only a limited distance into the liquid and calculate this distance. (d) With what speed will the ball pop up out of the liquid? (e) How does the time interval $\Delta t_{\text { down }},$ during which the ball moves from the surface down to its lowest point, compare with the time interval $\Delta t_{\text { up }}$ for the return trip between the same two points? (f) What If? Now modify the model to suppose the liquid exerts a small friction force on the ball, opposite in direction to its motion. In this
case, how do the time intervals $\Delta t_{\text { down }}$ and $\Delta t_{\text { up }}$ compare? Explain your answer with a conceptual argument rather than a numerical calculation.

Mayukh Banik

Numerade Educator

Problem 61

Review. A light spring of constant $k=90.0 \mathrm{N} / \mathrm{m}$ is attached vertically to a table (Fig. Pl4.6 la). A 2.00 -g balloon is filled with helium (density $=0.179 \mathrm{kg} / \mathrm{m}^{3} )$ to a volume of 5.00 $\mathrm{m}^{3}$ and is then connected with a light cord to the spring, causing the spring to stretch as shown in Figure Pl4.61b. Determine the extension distance $L$ when the balloon is in
equilibrium.

Mayukh Banik

Numerade Educator

Problem 62

A 42.0 -kg boy uses a solid block of Styrofoam as a raft while fishing on a pond. The Styrofoam has an area of 1.00 $\mathrm{m}^{2}$ and is 0.0500 $\mathrm{m}$ thick. While sitting on the surface of the raft, the boy finds that the raft just supports him so that the top of the raft is at the level of the pond. Determine the density of the Styrofoam.

Mayukh Banik

Numerade Educator

Problem 63

Evangelista Torricelli was the first person to realize that we live at the bottom of an ocean of air. He correctly surmised that the pressure of our atmosphere is attributable to the weight of the air. The density of air at $0^{\circ} \mathrm{C}$ at the Earth's surface is 1.29 $\mathrm{kg} / \mathrm{m}^{3}$ . The density decreases with increasing altitude (as the atmosphere thins). On the other hand, if we assume the density is constant at 1.29 $\mathrm{kg} / \mathrm{m}^{3}$ up to some altitude $h$ and is zero above that altitude, then $h$ would represent the depth of the ocean of air. (a) Use this model to determine the value of $h$ that gives a pressure of 1.00 atm at the surface of the Earth. (b) Would the peak of Mount Everest rise above the surface of such an atmosphere?

Mayukh Banik

Numerade Educator

Problem 64

Review. With reference to the dam studied in Example 14.4 and shown in Figure 14.5, (a) show that the total torque exerted by the water behind the dam about a horizontal axis through $O$ is $\frac{1}{6} \rho g w H^{3} .$ (b) Show that the effective line of action of the total force exerted by the water is at a distance $\frac{1}{3} H$ above $O$ .

Mayukh Banik

Numerade Educator

Problem 65

A 1.00 -kg beaker containing 2.00 $\mathrm{kg}$ of oil (density = 916.0 $\mathrm{kg} / \mathrm{m}^{3}$ ) rests on a scale. A. 2.00 -kg block of iron suspended from a spring scale is completely submerged in the oil as shown in Figure $\mathrm{P} 14.65 .$ Determine the equilibrium readings of both scales.

Mayukh Banik

Numerade Educator

Problem 66

A beaker of mass $m_{b}$ containing oil of mass $m_{o}$ and density $\rho_{o}$ rests on a scale. A block of iron of mass $m_{\mathrm{Fe}}$ suspended from a spring scale is completely submerged in the
oil as shown in Figure $\mathrm{P} 14.65 .$ Determine the equilibrium readings of both scales.

Mayukh Banik

Numerade Educator

Problem 67

In 1983 , the United States began coining the one-cent piece out of copper-clad zinc rather than pure copper. The mass of the old copper penny is 3.083 $\mathrm{g}$ and that of the new cent is 2.517 $\mathrm{g}$ . The density of copper is 8.920 $\mathrm{g} / \mathrm{cm}^{3}$ and that of $\mathrm{zinc}$ is $7.133 \mathrm{g} / \mathrm{cm}^{3} .$ The new and old coins have the same volume. Calculate the percent of zinc (by volume) in the new cent.

Mayukh Banik

Numerade Educator

Problem 68

Review. Figure $\mathrm{P} 14.68$ shows the essential parts of a hydraulic brake system. The area of the piston in the master cylinder is 1.8 $\mathrm{cm}^{2}$ and that of the piston in the brake cylinder is $6.4 \mathrm{cm}^{2} .$ The coefficient of friction between shoe and wheel drum is $0.50 .$ If the wheel has a radius of $34 \mathrm{cm},$ determine the frictional torque about the axle when a force of 44 $\mathrm{N}$ is exerted on the brake pedal.

Mayukh Banik

Numerade Educator

Problem 69

Review. A uniform disk of mass 10.0 $\mathrm{kg}$ and radius 0.250 $\mathrm{m}$ spins at 300 $\mathrm{rev} / \mathrm{min}$ on a low-friction axle. It must be brought to a stop in 1.00 $\mathrm{min}$ by a brake pad that makes contact with the disk at an average distance 0.220 $\mathrm{m}$ from the axis. The coefficient of friction between pad and disk is $0.500 .$ A piston in a cylinder of diameter 5.00 $\mathrm{cm}$ presses the brake pad against the disk. Find the pressure required for the brake fluid in the cylinder.

Mayukh Banik

Numerade Educator

Problem 70

Review. In a water pistol, a piston drives water through a large tube of area $A_{1}$ into a smaller tube of area $A_{2}$ as shown in Figure P14.70. The radius of the large tube is 1.00 cm and that of the small tube is 1.00 mm. The smaller tube is 3.00 cm above the larger tube. (a) If the pistol is fired horizontally at a height of 1.50 m, determine the time interval required for the water to travel from the nozzle to the ground. Neglect air resistance and assume atmospheric pressure is 1.00 atm. (b) If the desired range of the stream is $8.00 \mathrm{m},$ with what speed $v_{2}$ must the stream leave the nozzle? ( $\mathrm{c} )$ At what speed $v_{1}$ must the plunger be moved to achieve the desired range? (d) What is the pressure at the nozzle? (e) Find the pressure needed in the larger tube. (f) Calculate the force that must be exerted on the trigger to achieve the desired range. (The force that must be exerted is due to pressure over and above atmospheric pressure.)

Victor Salazar

Numerade Educator

Problem 71

An incompressible, nonviscous fluid is initially at rest in the vertical portion of the pipe shown in Figure Pl4.71a, where $L=2.00 \mathrm{m} .$ When the valve is opened, the fluid flows into the horizontal section of the pipe. What is the fluid's speed when all the fluid is in the horizontal section as shown in Figure $\mathrm{P} 14.71 \mathrm{b}$ ? Assume the cross-sectional area of the entire pipe is constant.

Mayukh Banik

Numerade Educator

Problem 72

The water supply of a building is fed through a main pipe 6.00 $\mathrm{cm}$ in diameter. A 2.00 -cm-diameter faucet tap, located 2.00 $\mathrm{m}$ above the main pipe, is observed to fill a 25.0 -L container in 30.0 $\mathrm{s}$ (a) What is the speed at which the water leaves the faucet? (b) What is the gauge pressure in the 6 -cm main pipe? Assume the faucet is the only "leak" in the building.

Mayukh Banik

Numerade Educator

Problem 73

A U-tube open at both ends is partially filled with water (Fig. Pl4.73a). Oil having a density 750 $\mathrm{kg} / \mathrm{m}^{3}$ is then poured into the right arm and forms a column $L=5.00 \mathrm{cm}$ high (Fig. Pl4.73b). (a) Determine the difference $h$ in the heights of the two liquid surfaces. (b) The right arm is then shielded from any air motion while air is blown across the top of the left arm until the surfaces of the two liquids are at the same height (Fig. Pl4.73c). Determine the speed of the air being blown across the left arm. Take the density of air as constant at 1.20 $\mathrm{kg} / \mathrm{m}^{3}$ .

Mayukh Banik

Numerade Educator

Problem 74

A woman is draining her fish tank by siphoning the water into an outdoor drain as shown in Figure $\mathrm{P} 14.74$ . The rectangular tank has footprint area $A$ and depth $h$ . The drain is located a distance $d$ below the surface of the water in the tank, where $d>>h$ . The cross-sectional area of the siphon tube is $A^{\prime} .$ Model the water as flowing without friction. Show that the time interval required to empty the tank is given by
$$\Delta t=\frac{A h}{A^{\prime} \sqrt{2 g d}}$$

Mayukh Banik

Numerade Educator

Problem 75

The hull of an experimental boat is to be lifted above the water by a hydrofoil mounted below its keel as shown in Figure P14.75 on page 432. The hydrofoil has a shape like that of an airplane wing. Its area projected onto a horizontal surface is A. When the boat is towed at sufficiently high speed, water of density $\rho$ moves in streamline flow so that its average speed at the top of the hydrofoil is $n$ times
larger than its speed $v_{b}$ below the hydrofoil. (a) Ignoring the buoyant force, show that the upward lift force exerted by the water on the hydrofoil has a magnitude
$$
F \approx \frac{1}{3}\left(n^{2}-1\right) \rho v_{b}^{2} A
$$
(b) The boat has mass $M$ . Show that the liftoff speed is given by
$$
v \approx \sqrt{\frac{2 M g}{\left(n^{2}-1\right) A \rho}}
$$

Mayukh Banik

Numerade Educator

Problem 76

Show that the variation of atmospheric pressure with altitude is given by $P=P_{0} e^{-\alpha y},$ where $\alpha=\rho_{0} g / \rho P_{0}, P_{0}$ is atmospheric pressure at some reference level $y=0,$ and $\rho_{0}$ is the atmospheric density at this level. Assume the decrease in atmospheric pressure over an infinitesimal change in altitude (so that the density is approximately uniform over the infinitesimal change) can be expressed from Equation 14.4 as $d P=-\rho g d y .$ Also assume the density of air is proportional to the pressure, which, as we will see in Chapter $20,$ is equivalent to assuming the temperature of the air is the same at all altitudes.

Mayukh Banik

Numerade Educator

Problem 77

An ice cube whose edges measure 20.0 $\mathrm{mm}$ is floating in a glass of ice-cold water, and one of the ice cube's faces is parallel to the water's surface. (a) How far below the water surface is the bottom face of the block? (b) Ice-cold ethyl alcohol is gently poured onto the water surface to form a layer 5.00 $\mathrm{mm}$ thick above the water. The alcohol does not mix with the water. When the ice cube again attains hydrostatic equilibrium, what is the distance from the top of the water to the bottom face of the block? (c) Additional cold ethyl alcohol is poured onto the water's surface until the top surface of the alcohol coincides with the top surface of the ice cube (in hydrostatic equilibrium). How thick is the required layer of ethyl alcohol?

Oliver Mcneely

Numerade Educator

Problem 78

Why is the following situation impossible? A barge is carrying a load of small pieces of iron along a river. The iron pile is in the shape of a cone for which the radius $r$ of the base of the cone is equal to the central height $h$ of the cone. The barge is square in shape, with vertical sides of length 2$r$ so that the pile of iron comes just up to the edges of the barge. The barge approaches a low bridge, and the captain realizes that the top of the pile of iron is not going to make it under the bridge. The captain orders the crew to shovel iron pieces from the pile into the water to reduce the height of the pile. As iron is shoveled from the pile, the pile always has the shape of a cone whose diameter is equal to the side length of the barge. After a certain volume of iron is removed from the barge, it makes it under the bridge without the top of the pile striking the bridge.

Mayukh Banik

Numerade Educator