Principles of Physics a Calculus Based Text

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

Chapter 15

Fluid Mechanics - all with Video Answers

Educators

Chapter Questions

Problem 1

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.

Surjit Tewari

Surjit Tewari

Numerade Educator

Problem 2

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 $0.500 \mathrm{cm} .$ (a) If the woman balances on one heel, what pressure does she exert on the floor? (b) Should the homeowner be concerned? Explain your answer.

Surjit Tewari

Numerade Educator

Problem 3

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

Averell Hause

Carnegie Mellon University

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 $\left.1.013 \times 10^{5} \text { Pa. }\right)$

Surjit Tewari

Numerade Educator

Problem 5

The spring of the pressure gauge shown in Figure $\mathrm{P} 15.5$ has a force constant of $1250 \mathrm{N} / \mathrm{m},$ and the piston has a diameter of $1.20 \mathrm{cm} .$ As the gauge is lowered into water in a lake, what change in depth causes the piston to move in by $0.750 \mathrm{cm}^{2}$

Surjit Tewari

Numerade Educator

Problem 6

The small piston of a hydraulic lift (Fig. P15.6) has a cross-sectional area of $3.00 \mathrm{cm}^{2},$ and its large piston has a crosssectional 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 $F_{g}=15.0 \mathrm{kN} ?$

Surjit Tewari

Numerade Educator

Problem 7

A container is filled to a depth of $20.0 \mathrm{cm}$ with water. On top of the water floats a 30.0 -cm-thick layer of oil with specific gravity $0.700 .$ What is the absolute pressure at the bottom of the container?

Salamat Ali

Numerade Educator

Problem 8

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 9

(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 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 $32.3 \mathrm{m}$ deep.

Mayukh Banik

Numerade Educator

Problem 11

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 $80.0-\mathrm{kg}$ student?

Mayukh Banik

Numerade Educator

Problem 12

Why is the following situation impossible? Figure $\mathrm{P} 15.12$ 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

The tank in Figure P15.13 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 $P 15.13$ 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

Piston O in Figure P15.1 5 has a diameter of 0.250 in. Piston (2) has a diameter of 1.50 in. Determine the magnitude $F$ of the force necessary to support the 500 lb load in the absence of friction.

Surjit Tewari

Numerade Educator

Problem 16

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 -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 by $2.40 \mathrm{m} \times$ $9.60 \mathrm{m} \times 0.183 \mathrm{m} \times 1000 \mathrm{kg} / \mathrm{m}^{3} \times 9.80 \mathrm{m} / \mathrm{s}^{2}=41.3 \mathrm{kN}$

Surjit Tewari

Numerade Educator

Problem 17

Mercury is poured into a U-tube as shown in Figure $\mathrm{Pl} 5.17$ 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 shown in Figure $P 15.17$ 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 18

Blaise Pascal duplicated Torricelli's barometer using a red Bordeaux wine, of density $984 \mathrm{kg} / \mathrm{m}^{3},$ as the working liquid (Fig. $\mathrm{P} 15.18$ ).
(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 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?

Mayukh Banik

Numerade Educator

Problem 21

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 22

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 23

A table-tennis ball has a diameter of $3.80 \mathrm{cm}$ and average density of $0.084 ~ 0 \mathrm{g} / \mathrm{cm}^{3} .$ What force is required to hold it completely submerged under water?

Surjit Tewari

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 submerged in water, the scale reads $3.50 \mathrm{N} \text { (Fig. } \mathrm{P} 15.24)$
Find the density of the object.

Surjit Tewari

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 P15.24b. The $12.0-\mathrm{cm}$ 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.

Surjit Tewari

Numerade Educator

Problem 26

A hydrometer is an instrument used to determine liquid density. A simple one is sketched in Figure $\mathrm{P} 15.26$ 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
$$\rho=\frac{\rho_{0} L}{L-h}$$

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?

Salamat Ali

Numerade Educator

Problem 28

Refer to Problem 26 and Figure P15.26. 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.

Dominador Tan

Numerade Educator

Problem 29

How many cubic meters of helium are required to lift a balloon with a 400 -kg payload to a height of $8000 \mathrm{m}$ ? 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.

Surjit Tewari

Numerade Educator

Problem 30

A long, cylindrical rod of radius $r$ is weighted on one end so that it floats upright in a fluid having a density $\rho .$ It is pushed down a distance $x$ from its equilibrium position and released. Show that the rod will execute simple harmonic motion if the resistive effects of the fluid are negligible and determine the period of the oscillations.

Sheh Lit Chang

University of Washington

Problem 31

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.

Surjit Tewari

Numerade Educator

Problem 32

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) Find the buoyant force on the block. (b) 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 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 $60.0 \mathrm{N}$. 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?

Victor Salazar

Numerade Educator

Problem 33

Decades ago, it was thought that huge herbivorous dinosaurs such as Apatosaurus and Brachiosaurus habitually walked on the bottom of lakes, extending their long necks up to the surface to breathe. Brachiosaurus had its nostrils on the top of its head. In $1977,$ Knut Schmidt-Nielsen pointed out that breathing would be too much work for such a creature. For a simple model, consider a sample consisting of $10.0 \mathrm{L}$ of air at absolute pressure 2.00 atm, with density $2.40 \mathrm{kg} / \mathrm{m}^{3}$ located at the surface of a freshwater lake. Find the work required to transport it to a depth of $10.3 \mathrm{m},$ with its temperature, volume, and pressure remaining constant. This energy investment is greater than the energy that can be obtained by metabolism of food with the oxygen in that quantity of air.

Surjit Tewari

Numerade Educator

Problem 34

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 35

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 36

A balloon is filled with $400 \mathrm{m}^{3}$ of helium at atmospheric pressure. (a) At $0^{\circ} \mathrm{C}$, the balloon can lift a payload of what mass? (b) What If? In Table 15.1 , 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 37

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 $8.00 \times 10^{4}$ 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 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 $8.00 \times 10^{4}$ 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?
$$P=R g h$$
where $g$ is the free-fall acceleration. (b) Each hydroelectric unitat the Grand Coulee Dam takes in waterata 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?

Dominador Tan

Numerade Educator

Problem 39

A large storage tank with an open top is filled to a height $h_{0} .$ The tank is punctured at a height $h$ above the bottom of the tank (Fig. $P 15.39$ ). Find an expression for how far from the tank the exiting stream lands.

Surjit Tewari

Numerade Educator

Problem 40

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. P15.40).
(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 Ber-
noulli's equation to determine the speed of the water as it leaves ground level.
(c) How does the answer from part (a) comparewith the answerfrom part (b)?
(d) What's 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.

Dominador Tan

Numerade Educator

Problem 41

(a) A water hose $2.00 \mathrm{cm}$ in diameter is used to fill a $20.0-\mathrm{L}$ bucket. If it takes $1.00 \mathrm{min}$ to fill the bucket, what is the speed $v$ at which water moves through the hose? (Note: $1 \mathrm{L}=$ $1000 \mathrm{cm}^{3} .$ (b) The hose has a nozzle $1.00 \mathrm{cm}$ in diameter. Find the speed of the water at the nozzle.

Surjit Tewari

Numerade Educator

Problem 42

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 43

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 44

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 1 ft? Assume the hole remained constant in size.

Surjit Tewari

Numerade Educator

Problem 45

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 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 46

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 47

Figure $P 15.47$ 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.

Mayukh Banik

Numerade Educator

Problem 48

An airplane is cruising at altitude $10 \mathrm{km} .$ The pressure outside the craft is 0.287 atm; within the passenger compartment, the pressure is 1.00 atm and the temperature is $20^{\circ} \mathrm{C}$. A small leak occurs in one of the window seals in

Surjit Tewari

Numerade Educator

Problem 49

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 50

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 $7.00 \times 10^{4}$ 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 51

A siphon is used to drain water from a tank as illustrated in Figure $P 15.51$ 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 52

A common parameter that can be used to predict turbulence in fluid flow is called the Reynolds number. The Reynolds number for fluid flow in a pipe is a dimensionless quantity defined as
$$\operatorname{Re}=\frac{\rho v d}{\mu}$$
where $\rho$ is the density of the fluid, $v$ is its speed, $d$ is the inner diameter of the pipe, and $\mu$ is the viscosity of the fluid. Viscosity is a measure of the internal resistance of a liquid to flow and has units of $\mathrm{Pa} \cdot \mathrm{s}$. The criteria for the type of flow are as follows:
$\cdot$ If $\operatorname{Re} < 2300,$ the flow is laminar.
$\cdot$ If $2300 < \mathrm{Re} < 4000$, the flow is in a transition region between laminar and turbulent.
$\cdot$ If $\mathrm{Re} > 4000,$ the flow is turbulent.
(a) Let's model blood of density $1.06 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$ and viscosity $3.00 \times 10^{-3} \mathrm{Pa} \cdot \mathrm{s}$ as a pure liquid, that is, ignore the fact that it contains red blood cells. Suppose it is flowing in a large artery of radius $1.50 \mathrm{cm}$ with a speed of $0.0670 \mathrm{m} / \mathrm{s}$. Show that the flow is laminar. (b) Imagine that the artery ends in a single capillary so that the radius of the artery reduces to a much smaller value. What is the radius of the capillary that would cause the flow to become turbulent?
(c) Actual capillaries have radii of about $5-10$ micrometers, much smaller than the value in part (b). Why doesn't the flow in actual capillaries become turbulent?

Dominador Tan

Numerade Educator

Problem 53

Water is forced out of a fire extinguisher by air pressure as shown in Figure P15.53. 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 54

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
$$F_{\varepsilon}=F_{g}^{\prime}+\left(V-\frac{F_{g}^{\prime}}{\rho g}\right) \rho_{\text {air }} g$$

Surjit Tewari

Numerade Educator

Problem 55

A light spring of constant $k=90.0 \mathrm{N} / \mathrm{m}$ is attached vertically to a table (Fig. P15.55a). 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 shown in Figure $P 15.55 b .$ Determine the extension distance $L$ when the balloon is in equilibrium.

Surjit Tewari

Numerade Educator

Problem 56

The hull of an experimental boat is to be lifted above the water by a hydrofoil mounted below its keel as shown in Figure $P 15.56 .$ 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
$$F \approx \frac{1}{2}\left(n^{2}-1\right) \rho v_{b}^{2} A$$
(b) The boat has mass $M$. Show that the liftoff speed is given by
$$v \approx \sqrt{\frac{2 M g}{\left(n^{2}-1\right) A \rho}}$$

Eduard Sanchez

Numerade Educator

Problem 57

Evangelista Torricelli was the first person to realize that we live at the bottom of an ocean of air. He correctly surmisedthat 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?

Surjit Tewari

Numerade Educator

Problem 58

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 $m=0.050$ 0 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 $\mathrm{P} 15.58 .$ 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 $\rho_{11 c^{*}}$ (d) Find the numerical value of the mass $m_{s^{*}}$ (e) Find the length $h$ numerically.

Dominador Tan

Numerade Educator

Problem 59

A copper cylinder hangs at the bottom of a steel wire of negligible mass. The top end of the wire is fixed. When the wire is struck, it emits sound with a fundamental frequency of 300 Hz. The copper cylinder is then submerged in water so that half its volume is below the waterline. Determine the new fundamental frequency.

Surjit Tewari

Numerade Educator

Problem 60

With reference to the dam studied in Example 15.2 and shown in Figure $15.6,$ (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 61

An incompressible, nonviscous fluid is initially at rest in the vertical portion of the pipe shown in Figure $\mathrm{P} 15.61 \mathrm{a}$, where $L=$ $2.00 \mathrm{m} .$ When the valve is opened, the fluid flows into the horizontal section of the pipe. What is the fluid's speed when all the fluid is in the horizontal section as shown in Figure P15.61b? Assume the cross-sectional area of the entire pipe is constant.

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 (Fig. $P 15.62) .$ 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 $P_{0^{*}}$

Surjit Tewari

Numerade Educator

Problem 63

A 1.00 -kg beaker containing 2.00 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 $P 15.63 .$ Determine the equilibrium readings of both scales.

Prashant Bana

Numerade Educator

Problem 64

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 $\mathrm{P} 15.63 .$ Determine the equilibrium readings of both scales.

Vipender Yadav

Numerade Educator

Problem 65

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 66

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 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 15.4 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 $16,$ is equivalent to assuming the temperature of the air is the same at all altitudes.

Dominador Tan

Numerade Educator

Problem 67

A U-tube open at both ends is partially filled with water (Fig. P15.67a). 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. $P 15.67$ b). (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. $P 15.67 c$ ). Determine the speed of the air being blown across the left arm. Take the density of air as constant at $1.20 \mathrm{kg} / \mathrm{m}^{3}$

Dominador Tan

Numerade Educator

Problem 68

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 rof 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.

Dominador Tan

Numerade Educator

Problem 69

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 s. (a) What is the speed at which the water leaves the faucet? (b) What is the gauge pressure in the $6-\mathrm{cm}$ main pipe? Assume the faucet is the only "leak" in the building.

Mayukh Banik

Numerade Educator

Problem 70

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. P15.70). 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 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) Assuming that 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 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