Physics for Scientists and Engineers with Modern Physics

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

Chapter 17

Superposition and Standing Waves - all with Video Answers

Educators

Chapter Questions

Problem 1

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

Hubert Agamasu

Hubert Agamasu

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

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 4

Why is the following situation impossible? Figure $\mathrm{P} 14.4$ 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 5

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 6

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 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 7

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 $1.00 \mathrm{~km} ?$

Rashmi Sinha

Numerade Educator

Problem 8

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 $\mathrm{P} 14.8 .$ 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 9

Blaise Pascal duplicated Torricelli's barometer using a red Bordeaux wine, of density $984 \mathrm{~kg} / \mathrm{m}^{3},$ as the working liquid (Fig. $\mathrm{P} 14.9$ ). (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 10

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 11

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 $3.50 \mathrm{~N}$ (Fig. $\mathrm{P} 14.11$ ). Find the density of the object.

Surjit Tewari

Numerade Educator

Problem 12

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 P14.11b. 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.

Surjit Tewari

Numerade Educator

Problem 13

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 14

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?

Eduard Sanchez

Numerade Educator

Problem 15

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 16

A hydrometer is an instrument used to determine liquid density. A simple one is sketched in Figure $\mathrm{P} 14.16 .$ 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 17

Refer to Problem 16 and Figure $\mathrm{P} 14.16 .$ 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.

Eduard Sanchez

Numerade Educator

Problem 18

On October $21,2001,$ Ian Ashpole of the United Kingdom achieved a record altitude of $3.35 \mathrm{~km}(11000 \mathrm{ft})$ powered by 600 toy balloons filled with helium. Each filled balloon had a radius of about $0.50 \mathrm{~m}$ and an estimated mass of $0.30 \mathrm{~kg}$.
(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?

Eduard Sanchez

Numerade Educator

Problem 19

You have a job in a company that produces party supplies. You are designing helium-filled balloons to be sold as gifts. To save money on production costs, part of this design is to choose from an available selection the least massive token
necessary to be tied to the lower end of the string hanging from the balloon in order to keep the balloon from rising off a table, hospital food tray, bedroom dresser, etc. In this way, the token remains stationary on the flat surface and the balloon is buoyed above the token at a fixed height, with the string straight. You are working on a balloon whose envelope is very thin and has a mass of $0.150 \mathrm{~kg}$. The envelope is filled to a volume of $0.2300 \mathrm{~m}^{3}$ with helium at atmospheric pressure. The string has a mass of $0.0700 \mathrm{~kg}$. Among the selection of tokens are those with masses $10.0 \mathrm{~g}, 20.0 \mathrm{~g}, 30.0 \mathrm{~g}$ $40.0 \mathrm{~g},$ and $50.0 \mathrm{~g} .$ Chose the appropriate token for the balloon you are working on.

Dominador Tan

Numerade Educator

Problem 20

Water flowing through a garden hose of diameter $2.74 \mathrm{~cm}$ fills a 25-L bucket in 1.50 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 21

Water falls over a dam of height $h$ with a mass flow rate of $I_{V}$ in units of kilograms per second. (a) Show that the power available from the water is $P=I_{V} g h$ 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?

Eduard Sanchez

Numerade Educator

Problem 22

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 $\left.1030 \mathrm{~kg} / \mathrm{m}^{3}\right),$ (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 $1 \mathrm{ft}$ ? Assume the hole remained constant in size.

Surjit Tewari

Numerade Educator

Problem 23

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 is pumped per day, what is the speed of the water in the pipe? Note: Assume the free-fall acceleration and thedensity 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.

Eduard Sanchez

Numerade Educator

Problem 24

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
(b) $\Delta P=6.00 \mathrm{kPa}$ and
(c) $\Delta P=12.0 \mathrm{kPa}$.

Mayukh Banik

Numerade Educator

Problem 25

Old Faithful Geyser in Yellowstone National
Park erupts at approximately one-hour intervals, and the height of the water $\begin{array}{llll}\text { column } & \text { reaches } & 40.0 & \mathrm{~m}\end{array}$
(Fig. $\mathrm{P} 14.25)$
(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.

Eduard Sanchez

Numerade Educator

Problem 26

You are working as an expert witness for the owner of a skyscraper complex in a downtown area. The owner is being sued by pedestrians on the streets below his buildings who were injured by falling glass when windows popped outward from the sides of the building. The Bernoulli effect can have important consequences for windows in such buildings. For example, wind can blow around a skyscraper at remarkably high speed, creating low pressure on the outside surface of the windows. The higher atmospheric pressure in the still air inside the buildings can cause windows to pop out. (a) In your research into the case, you find some overhead views of your client's project, as shown below. The project includes two tall skyscrapers and some park area on a square plot. Plan
(i) (Fig. $\mathrm{P} 14.26$ (i), page 382 ) was submitted by the original architects and planners. At the last minute, the owner decided he didn't want the park grounds to be divided into two areas and submitted Plan
(ii) (Fig. $\mathrm{P} 14.26$ (ii), which is the way the project was built. Explain to your client why Plan (ii) is a much more dangerous situation in terms of windows popping out than Plan (i).
(b) Your client is not convinced by your conceptual argument in part (a), so you provide a numerical argument. 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. Calculate the total force exerted by air on the windowpane for your client.
(c) What If? To further convince your client of the problems with the building design, calculate the total force exerted by air on the windowpane if the wind speed between the buildings is $22.4 \mathrm{~m} / \mathrm{s}$, twice as high as in part (b).

Dominador Tan

Numerade Educator

Problem 27

A thin 1.50 -mm coating of glycerin has been placed between two microscope slides of width $1.00 \mathrm{~cm}$ and length $4.00 \mathrm{~cm} .$ Find the force required to pull one of the microscope slides at a constant speed of $0.300 \mathrm{~m} / \mathrm{s}$ relative to the other slide.

Narayan Hari

Numerade Educator

Problem 28

A hypodermic needle is $3.00 \mathrm{~cm}$ in length and $0.300 \mathrm{~mm}$ in diameter. What pressure difference between the input and output of the needle is required so that the flow rate of water through it will be $1.00 \mathrm{~g} / \mathrm{s} ?$ (Use $1.00 \times 10^{-3} \mathrm{~Pa} \cdot \mathrm{s}$ as the viscosity of water.)

Narayan Hari

Numerade Educator

Problem 29

What radius needle should be used to inject a volume of $500 \mathrm{~cm}^{3}$ of a solution into a patient in $30.0 \mathrm{~min}$ ? Assume the length of the needle is $2.50 \mathrm{~cm}$ and the solution is elevated $1.00 \mathrm{~m}$ above the point of injection. Further, assume the viscosity and density of the solution are those of pure water, and that the pressure inside the vein is atmospheric.

Narayan Hari

Numerade Educator

Problem 30

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 31

A siphon is used to drain water from a tank as illustrated in Figure P14.31. 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 $2.3 \mathrm{kPa}$.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 32

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 $2.00 \mathrm{~atm},$ 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 33

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 $\mathrm{P} 14.33 .$ 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 $\rho_{\mathrm{He}^{-}}$ (d) Find the numerical value of the mass $m_{s}$. (e) Find the length $h$ numerically.

Eduard Sanchez

Numerade Educator

Problem 34

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

Mayukh Banik

Numerade Educator

Problem 35

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 36

Assume a certain liquid, with density $1230 \mathrm{~kg} / \mathrm{m}^{3}$ 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.

Eduard Sanchez

Numerade Educator

Problem 37

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 38

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 $\operatorname{Re}=\frac{\rho v d}{\eta}$ where $\rho$ is the density of the fluid, $v$ is its speed, $d$ is the inner diameter of the pipe, and $\eta$ is the viscosity of the fluid. The criteria for the type of flow are as follows:
$\bullet$If $\operatorname{Re}<2300,$ the flow is laminar.
$\bullet$If $2300<$ Re $<4000$, the flow is in a transition region between laminar and turbulent.
$\bullet$If $\operatorname{Re}>4000,$ 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?

Eduard Sanchez

Numerade Educator

Problem 39

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 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 40

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 41

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. $\mathrm{P} 14.41$ ). 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 (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?

Eduard Sanchez

Numerade Educator

Problem 42

A woman is draining her fish tank by siphoning the water into an outdoor drain as shown in Figure $\mathrm{P} 14.42$. 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 crosssectional 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 $\Delta t=\frac{A h}{A^{\prime} \sqrt{2 g d}}$

Eduard Sanchez

Numerade Educator

Problem 43

You and your father are designing a waterfall for your backyard swimming pool. At the top of the waterfall is a tank containing water that is kept to a depth $d=0.280 \mathrm{~m}$ by a pump. As shown in Figure $\mathrm{P} 14.43,$ there is a small hatch of height $h=0.100 \mathrm{~m}$ and width $w=0.150 \mathrm{~m},$ hinged at the top, that can be used to turn on and turn off the supply of water to the waterfall. You want to attach a simple latch at the center of the bottom of the hatch and are trying to decide what type of latch to buy at your local hardware store. To make that decision, you need to determine the force that the latch must be able to withstand to keep the hatch closed.

Victor Salazar

Numerade Educator

Problem 44

You and your father are designing a waterfall for your backyard swimming pool. At the top of the waterfall is a tank containing water that is kept to a depth $d$ by a pump. As shown in Figure $\mathrm{P} 14.43$, there is a small hatch of height $h$ and width $w$, hinged at the top, that can be used to turn on and turn off the supply of water to the waterfall. You want to attach a simple latch at the center of the bottom of the hatch and are trying to decide what type of latch to buy at your local hardware store. To make that decision, you need to determine the force that the latch must be able to withstand to keep the hatch closed.

Victor Salazar

Numerade Educator

Problem 45

A uniform disk of mass $10.0 \mathrm{~kg}$ and radius $0.250 \mathrm{~m}$ spins at 300 rev $/$ min on a low-friction axle. It must be brought to a stop in 1.00 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 46

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 $\mathrm{P} 14.46 .$ The radius of the large tube is $1.00 \mathrm{~cm}$ and that of the small tube is $1.00 \mathrm{~mm} .$ The smaller tube is $3.00 \mathrm{~cm}$ above
the larger tube. (a) If the pistol is fired horizontally at a height of $1.50 \mathrm{~m},$ determine the time interval required for the water to travel from the nozzle to the ground. Neglect air resistance and assume atmospheric pressure is $1.00 \mathrm{~atm} .$ (b) If the desired range of the stream is $8.00 \mathrm{~m},$ with what speed $v_{2}$ must the stream leave the nozzle? (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 47

An incompressible, nonviscous fluid is initially at rest in the vertical portion of the pipe shown in Figure $\mathrm{P} 14.47 \mathrm{a},$ 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 P14.47b? Assume the cross-sectional area of the entire pipe is constant.

Surjit Tewari

Numerade Educator

Problem 48

The hull of an experimental boat is to be lifted above the water by a hydrofoil mounted below its keel as shown in Figure P14.48. 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 49

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 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 18 , is equivalent to assuming the temperature of the air is the same at all altitudes.

Eduard Sanchez

Numerade Educator

Problem 50

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