Physics for Scientists and Engineers with Modern Physics

Paul Tipler, Gene Mosca

Chapter 14

Fluid Mechanics - all with Video Answers

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+ 10 more educators

Chapter Questions

Problem 1

Calculate the mass of a solid iron sphere that has a diameter of $3.00 \mathrm{cm} .$

Anand Jangid

Anand Jangid

Numerade Educator

Problem 2

Find the order of magnitude of the density of the nucleus of an atom. What does this result suggest concerning the structure of matter? Model a nucleus as protons and neutrons closely packed together. Each has mass $1.67 \times 10^{-27} \mathrm{kg}$ and radius on the order of $10^{-15} \mathrm{m}$ .

RK

Rajesh Kumar

Numerade Educator

Problem 3

A 50.0 -kg woman balances on one heel of a pair of high-heeled shoes. If the heel is circular and has a radius of $0.500 \mathrm{cm},$ what pressure does she exert on the floor?

Surjit Tewari

Numerade Educator

Problem 4

The four tires of an automobile are inflated to a gauge pressure of 200 $\mathrm{kPa}$ . Each tire has an area of 0.0240 $\mathrm{m}^{2}$ in contact with the ground. Determine the weight of the automobile.

Narayan Hari

Numerade Educator

Problem 5

What is 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 $1.013 \times 10^{5} \mathrm{N} / \mathrm{m}^{2} . )$

Surjit Tewari

Numerade Educator

Problem 6

(a) Calculate the absolute pressure at an ocean depth of 1000 $\mathrm{m}$ . Assume the density of seawater is 1024 $\mathrm{kg} / \mathrm{m}^{3}$ and that the air above exerts a pressure of 101.3 $\mathrm{kPa}$ . (b) At this depth, what force must the frame around a circular submarine porthole having a diameter of 30.0 $\mathrm{cm}$ exert to counterbalance the force exerted by the water?

Surjit Tewari

Numerade Educator

Problem 7

The spring of the pressure gauge shown in Figure 14.2 has a force constant of 1000 $\mathrm{N} / \mathrm{m}$ , and the piston has a diameter of $2.00 \mathrm{cm} .$ As the gauge is lowered into water, what change in depth causes the piston to move in by 0.500 $\mathrm{cm}$ ?

Surjit Tewari

Numerade Educator

Problem 8

The small piston of a hydraulic lift has a cross-sectional area of $3.00 \mathrm{cm}^{2},$ and its large piston has a cross-sectional area of 200 $\mathrm{cm}^{2}$ (Figure $14.4 ) .$ What force must be applied to the small piston for the lift to raise a load of 15.0 $\mathrm{kN}$ ? (In service stations, this force is usually exerted by compressed air.)

Bret Rosen

Numerade Educator

Problem 9

What must be the contact area between a suction cup (completely exhausted) and a ceiling if the cup is to support the weight of an $80.0-\mathrm{kg}$ student?

Surjit Tewari

Numerade Educator

Problem 10

(a) A very powerful vacuum cleaner has a hose 2.86 $\mathrm{cm}$ in diameter. With no nozzle on the hose, what is the weight of the heaviest brick that the cleaner can lift? (Fig. Pl4.10a) (b) What If? A very powerful 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 (Fig. Pl4.10b). Find the greatest force the octopus can exert in salt water 32.3 $\mathrm{m}$ deep. Caution: Experimental verification can be interesting, but do not drop a brick on your foot. Do not overheat the motor of a vacuum cleaner. Do not get an octopus mad at you.

Surjit Tewari

Numerade Educator

Problem 11

For the cellar of a new house, a hole is dug in the ground, with vertical sides going down $2.40 \mathrm{m} .$ A concrete foundation wall is built all the way across the $9.60-\mathrm{m}$ width of the excavation. This foundation wall is 0.183 $\mathrm{m}$ away from the front of the cellar hole. During a rainstorm, drainage from the street fills up the space in front of the concrete wall, but not the cellar behind the wall. The water does not soak into the clay soil. Find the force the water causes on the foundation wall. For comparison, the weight of the water is given $\quad$ by $2.40 \mathrm{m} \times 9.60 \mathrm{m} \times 0.183 \mathrm{m} \times 1000 \mathrm{kg} / \mathrm{m}^{3} \times$ $9.80 \mathrm{m} / \mathrm{s}^{2}=41.3 \mathrm{kN} .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 12

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 caused by the water on the bottom? On each end? On each side?

Surjit Tewari

Numerade Educator

Problem 13

A sealed spherical shell of diameter $d$ is rigidly attached to a cart, which is moving horizontally with an acceleration $a$ as in Figure $\mathrm{P} 14.13$ . The sphere is nearly filled with a fluid having density $\rho$ and also contains one small bubble of air at atmospheric pressure. Determine the pressure $P$ at the center of the sphere.

Jarrad Pond

Numerade Educator

Problem 14

The tank in Figure $\mathrm{P} 14.14$ is filled with water 2.00 $\mathrm{m}$ deep. At the bottom of one side wall is a rectangular hatch 1.00 $\mathrm{m}$ high and 2.00 $\mathrm{m}$ wide, which is hinged at the top of the hatch. (a) Determine the force the water exerts on the hatch. (b) Find the torque exerted by the water about the hinges.

Ajay Singhal

Numerade Educator

Problem 15

Review problem. The Abbott of Aberbrothock paid to have a bell moored to the Inchcape Rock to warn seamen of the hazard. Assume the bell was 3.00 $\mathrm{m}$ in diameter, cast from brass with a bulk modulus of $14.0 \times 10^{10} \mathrm{N} / \mathrm{m}^{2}$ . The pirate Ralph the Rover cut loose the warning bell and threw it into the ocean. By how much did the diameter of the bell decrease as it sank to a depth of 10.0 $\mathrm{km}$ ? Years later, Ralph drowned when his ship collided with the rock.
Note: The brass is compressed uniformly, so you may model the bell as a sphere of diameter $3.00 \mathrm{m} .$

Surjit Tewari

Numerade Educator

Problem 16

Figure $\mathrm{P} 14.16$ shows Superman attempting to drink water through a very long straw. With his great strength he achieves maximum possible suction. The walls of the tubular straw do not collapse. (a) Find the maximum height through which he can lift the water. (b) What If? Still thirsty, the Man of Steel repeats his attempt on the Moon, which has no atmosphere. Find the difference between the water levels inside and outside the straw.

Surjit Tewari

Numerade Educator

Problem 17

Blaise Pascal duplicated Torricelli's barometer using a red Bordeaux wine, of density $984 \mathrm{kg} / \mathrm{m}^{3},$ as the working liquid (Fig. Pl4.17). What was the height $h$ of the wine column for normal atmospheric pressure? Would you expect the vacuum above the column to be as good as for
mercury?

Surjit Tewari

Numerade Educator

Problem 18

Mercury is poured into a U-tube as in Figure $\mathrm{P} 14.18 \mathrm{a}$ . 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 in Figure $\mathrm{P} 14.18 \mathrm{b}$ . ( 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?

Dominador Tan

Numerade Educator

Problem 19

Normal atmospheric pressure is $1.013 \times 10^{5} \mathrm{Pa}$ . 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? (The density of mercury is $13.59 \mathrm{g} / \mathrm{cm}^{3} . )$

Kon Aoki

Numerade Educator

Problem 20

A U-tube of uniform cross-sectional area, open to the atmosphere, is partially filled with mercury. Water is then poured into both arms. If the equilibrium configuration of the tube is as shown in Figure $\mathrm{P} 14.20$ , with $h_{2}=1.00 \mathrm{cm},$ determine the value of $h_{1} .$

Surjit Tewari

Numerade Educator

Problem 21

The human brain and spinal cord are immersed in the cerebrospinal fluid. The fluid is normally continuous between the cranial and spinal cavities. It normally 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 P14.21. 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) Sometimes it is necessary to determine if an accident victim has suffered a crushed vertebra that is blocking flow of the cerebrospinal fluid in the spinal column. In
other cases a physician may suspect a tumor or other growth is blocking the spinal column and inhibiting flow of cerebrospinal fluid. Such conditions can be investigated by means of the Queckensted test. In this procedure, the veins in the patient's neck are compressed, to make the blood pressure rise in the brain. The increase in pressure in the blood vessels is transmitted to the cerebrospinal fluid. What should be the normal effect on the height of the fluid in the spinal tap? (c) Suppose that compressing the veins had no effect on the fluid level. What might account for this?

Mayukh Banik

Numerade Educator

Problem 22

(a) A light balloon is filled with 400 $\mathrm{m}^{3}$ of helium. 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?

Surjit Tewari

Numerade Educator

Problem 23

A Ping-Pong 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?

Surjit Tewari

Numerade Educator

Problem 24

A Styrofoam slab has thickness $h$ and density $\rho_{s} .$ When a swimmer of mass $m$ is resting on it, the slab floats in fresh water with its top at the same level as the water surface. Find the area of the slab.

Surjit Tewari

Numerade Educator

Problem 25

A piece of aluminum with mass 1.00 $\mathrm{kg}$ and density 2700 $\mathrm{kg} / \mathrm{m}^{3}$ is suspended from a string and then completely immersed in a container of water (Figure Pl4.25). Calculate the tension in the string (a) before and (b) after the metal is immersed.

Surjit Tewari

Numerade Educator

Problem 26

The weight of a rectangular block of low-density material is 15.0 $\mathrm{N}$ . 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 $\mathrm{N}$ . (a) Sketch a free-body diagram for the block, showing all forces acting on it. (b) Find the buoyant force on the block. (c) Oil of density 800 $\mathrm{kg} / \mathrm{m}^{3}$ 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 side walls of the block that the oil touches. What are the directions of these forces? (d) What happens to the string tension as the oil is added? Explain how the oil has
this effect on the string tension. (e) The string breaks when its tension reaches 60.0 $\mathrm{N}$ . 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? ( $(\mathrm{f})$ After the string breaks, the block comes to a new equilibrium position in the beaker. It is now in contact only with the oil. What fraction of the block's volume is submerged?

Victor Salazar

Numerade Educator

Problem 27

A 10.0 -kg block of metal measuring $12.0 \mathrm{cm} \times 10.0 \mathrm{cm} \times$ 10.0 $\mathrm{cm}$ is suspended from a scale and immersed in water as in Figure $\mathrm{P} 14.25 \mathrm{b}$ . The 12.0 -m dimension is vertical, and the top of the block is 5.00 $\mathrm{cm}$ below the surface of the water. (a) What are the forces acting on the top and on the bottom of the block? (Take $P_{0}=1.0130 \times 10^{5} \mathrm{N} / \mathrm{m}^{2} . )$ (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.

Prabhu Ramji

Numerade Educator

Problem 28

To an order of magnitude, how many helium-filled toy balloons would be required to lift you? Because helium is an irreplaceable resource, develop a theoretical answer rather than an experimental answer. In your solution state what physical quantities you take as data and the values you measure or estimate for them.

Surjit Tewari

Numerade Educator

Problem 29

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) How much lead weight has to be placed on top of the cube so that its top is just level with the water?

Kon Aoki

Numerade Educator

Problem 30

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

Pawan Yadav

Numerade Educator

Problem 31

Determination of the density of a fluid has many important applications. A car battery contains sulfuric acid, for which density is a measure of concentration. For the battery to function properly, the density must be inside a range specified by the manufacturer. Similarly, the effectiveness of antifreeze in your car's engine coolant depends on the density of the mixture (usually ethylene glycol and water). When you donate blood to a blood bank, its screening includes determination of the density of the blood, since higher density correlates with higher hemoglobin content. A hydrometer is an instrument used to determine liquid density. A simple one is sketched in Figure P14.31. 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
$$\rho=\frac{\rho_{0} L}{L-h}$$

Donald Albin

Numerade Educator

Problem 32

Refer to Problem 31 and Figure $\mathrm{P} 14.31 .$ 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.

Surjit Tewari

Numerade Educator

Problem 33

How many cubic meters of helium are required to lift a balloon with a $400-\mathrm{kg}$ payload to a height of 8000 $\mathrm{m}$ ? (Take $\rho_{\mathrm{He}}=0.180 \mathrm{kg} / \mathrm{m}^{3} . )$ Assume that the balloon maintains a constant volume and that the density of air decreases with
the altitude $z$ according to the expression $\rho_{\text { air }}=\rho_{0} e^{-z / 8} 000$ where $z$ is in meters and $\rho_{0}=1.25 \mathrm{kg} / \mathrm{m}^{3}$ is the density of air at sea level.

Surjit Tewari

Numerade Educator

Problem 34

A frog in a hemispherical pod (Fig. Pl4.34) just floats without sinking into a sea of blue-green ooze with density $1.35 \mathrm{g} / \mathrm{cm}^{3} .$ If the pod has radius 6.00 $\mathrm{cm}$ and negligible
mass, what is the mass of the frog?

Surjit Tewari

Numerade Educator

Problem 35

A plastic sphere floats in water with 50.0 percent of its volume submerged. This same sphere floats in glycerin with 40.0 percent of its volume submerged. Determine the densities of the glycerin and the sphere.

Joe Lesueur

Numerade Educator

Problem 36

A bathysphere 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, this submarine takes on mass in the form of seawater. Determine the amount of mass the submarine 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 $1.03 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$ .

Surjit Tewari

Numerade Educator

Problem 37

The United States possesses the eight largest warships in the world-aircraft carriers of the Nimitz class-and is building two more. Suppose one of the ships bobs up to float 11.0 $\mathrm{cm}$ higher in the water when 50 fighters take off from it in 25 $\mathrm{min}$ , at a location where the free-fall acceleration is 9.78 $\mathrm{m} / \mathrm{s}^{2}$ . Bristling with bombs and missiles, the planes have average mass 29000 $\mathrm{kg}$ . Find the horizontal area enclosed by the waterline of the $\$ 4$ -billion ship. By comparison, its flight deck has area $18000 \mathrm{m}^{2} .$ Below decks are passageways hundreds of meters long, so narrow that two large men cannot pass each other.

Surjit Tewari

Numerade Educator

Problem 38

A horizontal pipe 10.0 $\mathrm{cm}$ in diameter has a smooth reduction to a pipe 5.00 $\mathrm{cm}$ in diameter. If the pressure of the water in the larger pipe is $8.00 \times 10^{4} \mathrm{Pa}$ and the pressure in the smaller pipe is $6.00 \times 10^{4} \mathrm{Pa}$ , at what rate does water flow through the pipes?

Surjit Tewari

Numerade Educator

Problem 39

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. If the rate of flow from the leak is equal to $2.50 \times 10^{-3} \mathrm{m}^{3} / \mathrm{min}$ , determine (a) the speed at which the water leaves the hole and (b) the diameter of the hole.

Mayukh Banik

Numerade Educator

Problem 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 by the nozzle on the stopper. (b) The stopper is removed. What mass of water flows from the notzle 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

Water flows through a fire hose of diameter 6.35 $\mathrm{cm}$ at a rate of 0.0120 $\mathrm{m}^{3} / \mathrm{s}$ . The fire hose ends in a nozzle of inner diameter $2.20 \mathrm{cm} .$ What is the speed with which the water exits the nozzle?

Surjit Tewari

Numerade Educator

Problem 42

Water falls over a dam of height $h$ with a mass flow rate of $R,$ in units of $\mathrm{kg} / \mathrm{s}$ . ( a) Show that the power available from the water is
$$\mathscr{P}=R g h$$
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 is produced by each hydroelectric unit?

Mayukh Banik

Numerade Educator

Problem 43

Figure $\mathrm{P} 14.43$ 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 $\mathrm{s}$ . Find the diameter of the stream 13.0 $\mathrm{cm}$ below the opening of the faucet.

Sheh Lit Chang

University of Washington

Problem 44

A legendary Dutch boy saved Holland by plugging a hole in a dike with his finger, which is 1.20 $\mathrm{cm}$ in diameter. 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, how long would it take the released water to fill 1 acre of land to a depth of 1 $\mathrm{ft}$ , assuming the hole remained constant in size? (A typical U.S. family of four uses 1 acre-foot of water, $1234 \mathrm{m}^{3},$ in 1 year.)

Surjit Tewari

Numerade Educator

Problem 45

Through a pipe 15.0 $\mathrm{cm}$ in diameter, water is pumped from the Colorado River up to Grand Canyon Village, located on the rim of the canyon. The river is an elevation of $564 \mathrm{m},$ and the village is at an elevation of 2096 $\mathrm{m}$ . (a) What is the minimum pressure at which the water must be pumped if it is to arrive at the village? (b) If 4500 $\mathrm{m}^{3}$ are pumped per day, what is the speed of the water in the pipe? (c) What additional pressure is necessary to deliver this flow? Note: Assume that the free-fall acceleration and the density of air are constant over this range of elevations.

Surjit Tewari

Numerade Educator

Problem 46

Old Faithful Geyser in Yellowstone Park (Fig. Pl4.46) erupts at approximately 1 -h intervals, and the height of the water column reaches 40.0 $\mathrm{m}$ (a) Model the rising stream as a series of separate drops. Analyze the free-fall motion of one of the drops 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) What is the pressure (above atmospheric) in the heated underground chamber if its depth is 175 $\mathrm{m}$ ? You may assume that the chamber is large compared with the gevser's vent.

Surjit Tewari

Numerade Educator

Problem 47

A Venturi tube may be used as a fluid flow meter (see Fig. 14.20 ). If the difference in pressure is $P_{1}-P_{2}=21.0$ kPa, find the fluid flow rate in cubic meters per second, given that the radius of the outlet tube is $1.00 \mathrm{cm},$ the radius of the inlet tube is $2.00 \mathrm{cm},$ and the fluid is gasoline $\left(\rho=700 \mathrm{kg} / \mathrm{m}^{3}\right) .$

Khoobchandra Agrawal

Khoobchandra Agrawal

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 $7.00 \times 10^{4} \mathrm{Pa}$ . Determine the pressure on the upper wing surface.

Surjit Tewari

Numerade Educator

Problem 49

A Pitot tube can be used to determine the velocity of air flow by measuring the difference between the total pressure and the static pressure (Fig. Pl4.49). If the fluid in the tube is mercury, density $\rho_{\mathrm{Hg}}=13600 \mathrm{kg} / \mathrm{m}^{3},$ and $\Delta h=5.00 \mathrm{cm},$ find the speed of air flow. (Assume that the air is stagnant at point $A,$ and take $\rho_{\text { air }}=1.25 \mathrm{kg} / \mathrm{m}^{3} . )$

Surjit Tewari

Numerade Educator

Problem 50

An airplane is cruising at an altitude of 10 $\mathrm{km}$ . The pressure outside the craft is 0.287 atm; within the passenger compartment 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 find the speed of the stream of air flowing through the leak.

Kon Aoki

Numerade Educator

Problem 51

A siphon is used to drain water from a tank, as illustrated in Figure $\mathrm{P} 14.51 .$ The siphon has a uniform diameter. Assume steady flow without friction.
(a) If the distance $h=1.00 \mathrm{~m}$, 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 water surface? (For the flow of the liquid to be continuous, the pressure must not drop below the vapor pressure of the liquid.)

Dading Chen

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 window panes, which fell many stories to the sidewalk below. (a) Suppose that a horizontal wind blows in streamline flow 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 uniform at 1.30 $\mathrm{kg} / \mathrm{m}^{3}$ . The air inside the building is at atmospheric pressure. What is the total force exerted by air on the window pane? (b) What If: If a second skyscraper is built nearby, the air speed can be especially high where wind passes through the narrow separation between the buildings. Solve part (a) again if the wind speed is 22.4 $\mathrm{m} / \mathrm{s}$ , twice as high.

Surjit Tewari

Numerade Educator

Problem 53

A hypodermic syringe contains a medicine with the density of water (Figure P14.53). 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 atm. A force $\mathbf{F}$ 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.

Anand Jangid

Numerade Educator

Problem 54

Figure Pl4.54 shows a water tank with a valve at the bottom. If this valve is opened, what is the maximum height attained by the water stream coming out of the right side of the tank? Assume that $h=10.0 \mathrm{m}, L=2.00 \mathrm{m},$ and $\theta=$ $30.0^{\circ},$ and that the cross-sectional area at $A$ is very large compared with that at $B$ .

Salamat Ali

Numerade Educator

Problem 55

A helium-filled balloon is tied to a $2.00-\mathrm{m}-$ long $, 0.0500$ -kg uniform string. The balloon is spherical with a radius of $0.400 \mathrm{m} .$ When released, it lifts a length $h$ of string and then remains in equilibrium, as in Figure $\mathrm{P} 14.55 .$ Determine the value of $h .$ The envelope of the balloon has mass $0.250 \mathrm{kg} .$

Surjit Tewari

Numerade Educator

Problem 56

Water is forced out of a fire extinguisher by air pressure, as shown in Figure $\mathrm{P} 14.56$ . How much gauge air pressure in the tank (above atmospheric) is required for the water jet to have a speed of 30.0 $\mathrm{m} / \mathrm{s}$ when the water level in the tank is 0.500 $\mathrm{m}$ below the nozzle?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 57

The true weight of an object can be measured in a vacuum, where buoyant forces are absent. An object of volume $V$ is weighed in air on a balance with the use of weights of density $\rho$ . If the density of air is $\rho$ air and the balance reads $F_{g}^{\prime},$ show that the true weight $F_{g}$ is
$$F_{g}=F_{g}^{\prime}+\left(V-\frac{F_{g}^{\prime}}{\rho g}\right) \rho_{\operatorname{air}} g$$

Surjit Tewari

Numerade Educator

Problem 58

A wooden dowel has a diameter of $1.20 \mathrm{cm} .$ It floats in water with 0.400 $\mathrm{cm}$ of its diameter above water (Fig. Pl4.58). Determine the density of the dowel.

Oliver Mcneely

Numerade Educator

Problem 59

A light spring of constant $k=90.0 \mathrm{N} / \mathrm{m}$ is attached vertically to a table (Fig. Pl4.59a). A $2.00-\mathrm{g}$ balloon is filled with helium (density $=0.180 \mathrm{kg} / \mathrm{m}^{3} )$ to a volume of 5.00 $\mathrm{m}^{3}$ and is then connected to the spring, causing it to stretch as in Figure $\mathrm{P} 14.59 \mathrm{b}$ . Determine the extension distance $L$ when the balloon is in equilibrium.

Rashmi Sinha

Numerade Educator

Problem 60

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 that the density is constant at 1.29 $\mathrm{kg} / \mathrm{m}^{3}$ up to some altitude $h,$ and zero above that altitude, then $h$ would represent the depth of the ocean of air. Use this model to determine the value of $h$ that gives a pressure of 1.00 $\mathrm{atm}$ at the surface of the Earth. Would the peak of Mount Everest rise above the surface of such an atmosphere?

Surjit Tewari

Numerade Educator

Problem 61

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

Surjit Tewari

Numerade Educator

Problem 62

In about 1657 Otto von Guericke, inventor of the air pump, evacuated a sphere made of two brass hemispheres. 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 (Fig. Pl4.62). (a) Show that the force $F$ required to pull the evacuated hemispheres apart is $\pi R^{2}\left(P_{0}-P\right),$ where $R$ is the radius of the hemispheres and $P$ is the pressure inside the hemispheres, which is much less than $P_{0} .$ (b) Determine the force if $P=0.100 P_{0}$ and $R=0.300 \mathrm{m} .$

Mayukh Banik

Numerade Educator

Problem 63

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 $2.00-\mathrm{kg}$ block of iron is suspended from a spring scale and completely submerged in the oil as in Figure $\mathrm{P} 14.63$ . Determine the equilibrium readings of both scales.

Prashant Bana

Numerade Educator

Problem 64

A beaker of mass $m_{\text { beaker }}$ containing oil of mass $m_{\text { oil }}$ (density $=\rho_{\text { oil }} )$ rests on a scale. A block of iron of mass $m_{\text { iron }}$ is suspended from a spring scale and completely submerged in the oil as in Figure $\mathrm{P} 14.63 .$ Determine the equilibrium readings of both scales.

Surjit Tewari

Numerade Educator

Problem 65

In 1983 , the United States began coining the cent piece out of copper-clad zinc rather than pure copper. The mass of the old copper penny is 3.083 g, while that of the new cent is 2.517 g. Calculate the percentage of zinc (by volume) in the new cent. The density of copper is 8.960 $\mathrm{g} / \mathrm{cm}^{3}$ and that of zinc is $7.133 \mathrm{g} / \mathrm{cm}^{3} .$ The new and old coins have the same volume.

Surjit Tewari

Numerade Educator

Problem 66

A thin spherical shell of mass 4.00 $\mathrm{kg}$ and diameter 0.200 $\mathrm{m}$ is filled with helium (density $=0.180 \mathrm{kg} / \mathrm{m}^{3} ) .$ It is then released from rest on the bottom of a pool of water that is 4.00 $\mathrm{m}$ deep. (a) Neglecting frictional effects, show that the shell rises with constant acceleration and determine the value of that acceleration. (b) How long will it take for the top of the shell to reach the water surface?

Surjit Tewari

Numerade Educator

Problem 67

Review problem. A uniform disk of mass 10.0 $\mathrm{kg}$ and radius 0.250 $\mathrm{m}$ spins at 300 $\mathrm{rev} / \min$ 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 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.

Surjit Tewari

Numerade Educator

Problem 68

Show that the variation of atmospheric pressure with altitude is given by $P=P_{0} e^{-\alpha y},$ where $\alpha=\rho_{0} g / P_{0}, P_{0}$ is atmospheric pressure at some reference level $y=0,$ and $\rho_{0}$ is the atmospheric density at this level. Assume that the decrease in atmospheric pressure over an infinitesimal change in altitude (so that the density is approximately uniform) is given by $d P=-\rho g d y,$ and that the density of air is proportional to the pressure.

Dominador Tan

Numerade Educator

Problem 69

An incompressible, nonviscous fluid is initially at rest in the vertical portion of the pipe shown in Figure $\mathrm{P} 14.69 \mathrm{a}$ where $L=2.00 \mathrm{m}$ . When the valve is opened, the fluid flows into the horizontal section of the pipe. What is the speed of the fluid when all of it is in the horizontal section, as in Figure $\mathrm{P} 14.69 \mathrm{b}$ ? Assume the cross-sectional area of the entire pipe is constant.

Surjit Tewari

Numerade Educator

Problem 70

A cube of ice whose edges measure 20.0 $\mathrm{mm}$ is floating in a glass of ice-cold water with one of its faces 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 will be 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 71

A U-tube open at both ends is partially filled with water (Fig. Pl4.71a). Oil having a density of 750 $\mathrm{kg} / \mathrm{m}^{3}$ is then poured into the right arm and forms a column $L=5.00 \mathrm{cm}$ high (Fig. Pl4.71b). (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. Pl4.71c). Determine the speed of the air being blown across the left arm. Take the density of air as 1.29 $\mathrm{kg} / \mathrm{m}^{3}$ .

Surjit Tewari

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 $25.0-\mathrm{L}$ 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

The spirit-in-glass thermometer, invented in Florence, Italy, around $1654,$ consists of a tube of liquid (the spirit) containing a number of submerged glass spheres with slightly different masses (Fig. Pl4.73). At sufficiently low temperatures all the spheres float, but as the temperature rises, the spheres sink one after another. The device is a crude but interesting tool for measuring temperature. Suppose that the tube is filled with ethyl alcohol, whose density is 0.78945 $\mathrm{g} / \mathrm{cm}^{3}$ at $20.0^{\circ} \mathrm{C}$ and decreases to 0.78097 $\mathrm{g} / \mathrm{cm}^{3}$ at $30.0^{\circ} \mathrm{C} .$ (a) If one of the spheres has a radius of 1.000 $\mathrm{cm}$ and is in equilibrium halfway up the tube at $20.0^{\circ} \mathrm{C}$ , determine its mass. (b) When the temperature increases to $30.0^{\circ} \mathrm{C},$ what mass must a second sphere of the same radius have in order to be in equilibrium at the halfway point? (c) At $30.0^{\circ} \mathrm{C}$ the first sphere has fallen to the bottom of the tube. What upward force does the
bottom of the tube exert on this sphere?

Prashant Bana

Numerade Educator

Problem 74

A woman is draining her fish tank by siphoning the water into an outdoor drain, as shown in Figure P14.74. 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 crossectional area of the siphon tube is $A^{\prime} .$ Model the water as flowing without friction. (a) Show that the time interval required to empty the tank is
given by
$$\Delta t=\frac{A h}{A^{\prime} \sqrt{2 g d}}$$
(b) Evaluate the time interval required to empty the tank if it is a cube 0.500 $\mathrm{m}$ on each edge, if $A^{\prime}=2.00 \mathrm{cm}^{2},$ and $d=10.0 \mathrm{m} .$

Surjit Tewari

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 p1475 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 neglecting the buoyant force show that the upward lift force exerted by the water on the hydrofoil has a magnitude given by $f$ approx frac12leftn2 1right rho v_b2 a b the boat has mass m show that the liftoff speed is given by $v $approx sqrtfrac2 m gleftn2 1right a rho c assume that an 800 mathrmkg boat is to lift off at 950 mathrmm mathrms evaluate the area a required for the hydrofoil if its design yields n105.

Surjit Tewari

Numerade Educator