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Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 14

Fluids - all with Video Answers

Educators

Chapter Questions

02:24

Problem 1

A fish maintains its depth in fresh water by adjusting the air content of porous bone or air sacs to make its average density the same as that of the water. Suppose that with its air sacs collapsed, a fish has a density of $1.08 \mathrm{~g} / \mathrm{cm}^{3} .$ To what fraction of its expanded body volume must the fish inflate the air sacs to reduce its density to that of water?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
01:06

Problem 2

A partially evacuated airtight container has a tight-fitting lid of surface area $77 \mathrm{~m}^{2}$ and negligible mass. If the force required to remove the lid is $480 \mathrm{~N}$ and the atmospheric pressure is $1.0 \times 10^{5}$ Pa, what is the internal air pressure?

Averell Hause
Averell Hause
Carnegie Mellon University
01:45

Problem 3

Find the pressure increase in the fluid in a syringe when a nurse applies a force of $42 \mathrm{~N}$ to the syringe's circular piston, which has a radius of $1.1 \mathrm{~cm}$.

James Erikson
James Erikson
Numerade Educator
02:39

Problem 4

Three liquids that will not mix are poured into a cylindrical container. The volumes and densities of the liquids are $0.50 \mathrm{~L}, 2.6 \mathrm{~g} / \mathrm{cm}^{3} ;$ $0.25 \mathrm{~L}, 1.0 \mathrm{~g} / \mathrm{cm}^{3}$; and $0.40 \mathrm{~L}, 0.80 \mathrm{~g} / \mathrm{cm}^{3} .$ What is the force on the
bottom of the container due to these liquids? One liter $=1 \mathrm{~L}=$ $1000 \mathrm{~cm}^{3}$. (Ignore the contribution due to the atmosphere.)

Averell Hause
Averell Hause
Carnegie Mellon University
02:12

Problem 5

An office window has dimensions $3.4 \mathrm{~m}$ by $2.1 \mathrm{~m}$. As a result of the passage of a storm, the outside air pressure drops to $0.96$ atm, but inside the pressure is held at $1.0$ atm. What net force pushes out on the window?

James Erikson
James Erikson
Numerade Educator
02:23

Problem 6

You inflate the front tires on your car to 28 psi. Later, you measure your blood pressure, obtaining a reading of $120 / 80$, the readings being in $\mathrm{mm} \mathrm{Hg}$. In metric countries (which is to say, most of the world), these pressures are customarily reported in kilopascals (kPa). In kilopascals, what are (a) your tire pressure and
(b) your blood pressure?

Matthew Baker
Matthew Baker
Numerade Educator
06:25

Problem 7

In 1654 Otto von Guericke, inventor of the air pump, gave a demonstration before the noblemen of the Holy Roman Empire in which two teams of eight horses could not pull apart two evacuated brass hemispheres. (a) Assuming the hemispheres have (strong) thin walls, so that $R$ in Fig. 14-29 may be considered both the inside and outside radius, show that the force $\vec{F}$ required to pull apart the hemispheres has magnitude $F=\pi R^{2} \Delta p$, where $\Delta p$ is the difference between the pressures outside and inside the sphere.
(b) Taking $R$ as $30 \mathrm{~cm}$, the inside pressure as $0.10 \mathrm{~atm}$, and the outside pressure as $1.00 \mathrm{~atm}$, find the force magnitude the teams of horses would have had to exert to pull apart the hemispheres.
(c) Explain why one team of horses could have proved the point just as well if the hemispheres were attached to a sturdy wall.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:18

Problem 8

The bends during flight. Anyone who scuba dives is advised not to fly within the next $24 \mathrm{~h}$ because the air mixture for diving can introduce nitrogen to the bloodstream. Without allowing the nitrogen to come out of solution slowly, any sudden air-pressure reduction (such as during airplane ascent) can result in the nitrogen forming bubbles in the blood, creating the bends, which can be painful and even fatal. Military special operation forces are especially at risk. What is the change in pressure on such a special-op soldier who must scuba dive at a depth of $20 \mathrm{~m}$ in seawater one day and parachute at an altitude of $7.6 \mathrm{~km}$ the next day? Assume that the average air density within the altitude range is $0.87 \mathrm{~kg} / \mathrm{m}^{3}$.

Averell Hause
Averell Hause
Carnegie Mellon University
06:03

Problem 9

Blood pressure in Argentinosaurus. (a) If this longnecked, gigantic sauropod had a head height of $21 \mathrm{~m}$ and a heart height of $9.0 \mathrm{~m}$, what (hydrostatic) gauge pressure in its blood was required at the heart such that the blood pressure at the brain was 80 torr (just enough to perfuse the brain with blood)? Assume the blood had a density of $1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$. (b) What was the blood pressure (in torr or $\mathrm{mm} \mathrm{Hg}$ ) at the feet?

Matthew Baker
Matthew Baker
Numerade Educator
00:50

Problem 10

The plastic tube in Fig. 14-30 has a cross-sectional area of $5.00 \mathrm{~cm}^{2}$. The tube is filled with water until the short arm (of length $d=0.800 \mathrm{~m}$ ) is full. Then the short arm is sealed and more water is gradually poured into the long arm. If the seal will pop off when the force on it exceeds $9.80 \mathrm{~N}$, what total height of water in the long arm will put the seal on the verge of popping?

Averell Hause
Averell Hause
Carnegie Mellon University
03:24

Problem 11

Giraffe bending to drink. In a giraffe with its head $2.0 \mathrm{~m}$ above its heart, and its heart $2.0 \mathrm{~m}$ above its feet, the (hydrostatic) gauge pressure in the blood at its heart is 250 torr. Assume that the giraffe stands upright and the blood density is $1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3} .$ In torr (or $\mathrm{mm} \mathrm{Hg}$ ), find the (gauge) blood pressure (a) at the brain (the pressure is enough to perfuse the brain with blood, to keep the giraffe from fainting) and (b) at the feet (the pressure must be countered by tight-fitting skin acting like a pressure stocking). (c) If the giraffe were to lower its head to drink from a pond without splaying its legs and moving slowly, what would be the increase in the blood pressure in the brain? (Such action would probably be lethal.)

Averell Hause
Averell Hause
Carnegie Mellon University
02:38

Problem 12

The maximum depth $d_{\max }$ that a diver can snorkel is set by the density of the water and the fact that human lungs can func-tion against a maximum pressure difference (between inside and outside the chest cavity) of $0.050 \mathrm{~atm}$. What is the difference in $d_{\max }$ for fresh water and the water of the Dead Sea (the saltiest natural water in the world, with a density of $1.5 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ )?

Matthew Baker
Matthew Baker
Numerade Educator
02:06

Problem 13

At a depth of $10.9 \mathrm{~km}$, the Challenger Deep in the Marianas Trench of the Pacific Ocean is the deepest site in any ocean. Yet, in 1960 , Donald Walsh and Jacques Piccard reached the Challenger Deep in the bathyscaph Trieste. Assuming that seawater has a uniform density of $1024 \mathrm{~kg} / \mathrm{m}^{3}$, approximate the hydrostatic pressure (in atmospheres) that the Trieste had to withstand. (Even a slight defect in the Trieste structure would have been disastrous.)

Eric Mockensturm
Eric Mockensturm
Numerade Educator
00:41

Problem 14

Calculate the hydrostatic difference in blood pressure between the brain and the foot in a person of height $1.83 \mathrm{~m}$. The density of blood is $1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$.

Averell Hause
Averell Hause
Carnegie Mellon University
01:01

Problem 15

What gauge pressure must a machine produce in order to suck mud of density $1800 \mathrm{~kg} / \mathrm{m}^{3}$ up a tube by a height of $1.5 \mathrm{~m} ?$

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:28

Problem 16

Snorkeling by humans and elephants. When a person snorkels, the lungs are connected directly to the atmosphere through the snorkel tube and thus are at at-
mospheric pressure. In atmospheres, what is the difference $\Delta p$ between this internal air pressure and the water pressure against the body if the length of the snorkel
tube is (a) $20 \mathrm{~cm}$ (standard situation) and (b) $4.0 \mathrm{~m}$ (probably lethal situation)? In the latter, the pressure difference causes blood vessels on the walls of the lungs to rupture, releasing blood into the lungs. As depicted in Fig. 14-31, an elephant can safely snorkel through its trunk while swimming with its lungs $4.0 \mathrm{~m}$ below the water surface because the membrane around its lungs contains connective tissue that holds and protects the blood vessels, preventing rupturing.

Keshav Singh
Keshav Singh
Numerade Educator
02:37

Problem 17

Crew members attempt to escape from a damaged submarine $100 \mathrm{~m}$ below the surface. What force must be applied to a pop-out hatch, which is $1.2 \mathrm{~m}$ by $0.60 \mathrm{~m}$, to push it out at that depth? Assume that the density of the ocean water is 1024 $\mathrm{kg} / \mathrm{m}^{3}$ and the internal air pressure is at $1.00 \mathrm{~atm} .$

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:03

Problem 18

An open tube of length $L=1.8 \mathrm{~m}$ and cross-sectional area $A=$ $4.6 \mathrm{~cm}^{2}$ is fixed to the top of a cylindrical barrel of diameter $D=1.2 \mathrm{~m}$ and height $H=$ $1.8 \mathrm{~m}$. The barrel and tube are filled with water (to the top of the tube). Calculate the ratio of the hydrostatic force on the bottom of the barrel to the gravitational force on the water contained in the barrel. Why is that ratio not equal to $1.0 ?$ (You need not consider the atmospheric pressure.)

Averell Hause
Averell Hause
Carnegie Mellon University
03:50

Problem 19

A large aquarium of height $5.00$ $\mathrm{m}$ is filled with fresh water to a depth of $2.00 \mathrm{~m}$. One wall of the aquarium consists of thick plastic $8.00 \mathrm{~m}$ wide. By how much does the total force on that wall increase if the aquarium is next filled to a depth of $4.00 \mathrm{~m} ?$

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:36

Problem 20

The L-shaped fish tank shown in Fig. 14-33 is filled with water and is open at the top. If $d=5.0 \mathrm{~m}$, what is the (total) force exerted by the water (a) on face $A$ and (b) on face $B$ ?

Averell Hause
Averell Hause
Carnegie Mellon University
05:55

Problem 21

Two identical cylindrical vessels with their bases at the same level each contain a liquid of density $1.30 \times 10^{3}$ $\mathrm{kg} / \mathrm{m}^{3}$. The area of each base is $4.00 \mathrm{~cm}^{2}$, but in one vessel the liquid height is $0.854$ $\mathrm{m}$ and in the other it is $1.560 \mathrm{~m}$. Find the work done by the gravitational force in equalizing the levels when the two vessels are connected.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
02:52

Problem 22

In dogfights. When a pilot takes a tight turn at high speed in a modern fighter airplane, the blood pressure at the brain level decreases, blood no longer perfuses the brain, and the blood in the brain drains. If the heart maintains the (hydrostatic) gauge pressure in the aorta at 120 torr (or $\mathrm{mm} \mathrm{Hg}$ ) when the pilot undergoes a horizontal centripetal acceleration of $4 g$, what is the blood pressure (in torr) at the brain, $30 \mathrm{~cm}$ radially inward from the heart? The perfusion in the brain is small enough that the vision switches to black and white and narrows to "tunnel vision" and the pilot can undergo $g$ - $\operatorname{LOC}(" g$ induced loss of consciousness"). Blood density is $1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
04:02

Problem 23

In analyzing certain geological features, it is often appropriate to assume that the pressure at some horizontal level of compensation, deep inside Earth, is the same over a large region and is equal to the pressure due to the gravitational force on the overlying material. Thus, the pressure on the level of compensation is given by the fluid pressure formula. This model requires, for one thing, that mountains have roots of continental rock extending into the denser mantle Consider a mountain of height $H=6.0 \mathrm{~km}$ on a continent of thickness $T=32 \mathrm{~km} .$ The continental rock has a density of $2.9 \mathrm{~g} / \mathrm{cm}^{3}$, and beneath this rock the mantle has a density of $3.3 \mathrm{~g} / \mathrm{cm}^{3} .$ Calculate the depth $D$ of the root. (Hint: Set the pressure at points $a$ and $b$ equal; the depth $y$ of the level of compensation will cancel out.)

Eric Mockensturm
Eric Mockensturm
Numerade Educator
04:08

Problem 24

water stands at depth $D=35.0 \mathrm{~m}$ behind the vertical upstream face of a dam of width $W=314 \mathrm{~m}$. Find (a) the net horizontal force on the dam from the gauge pressure of the water and
(b) the net torque due to that force about a horizontal line through $O$
parallel to the (long) width of the dam. This torque tends to rotate the dam around that line, which would cause the dam to fail. (c) Find the moment arm of the torque.

Averell Hause
Averell Hause
Carnegie Mellon University
02:59

Problem 25

In one observation, the column in a mercury barometer (as is shown in Fig. $14-5 a$ ) has a measured height $h$ of $740.35 \mathrm{~mm}$. The temperature is $-5.0^{\circ} \mathrm{C}$, at which temperature the density of mercury $\rho$ is $1.3608 \times 10^{4} \mathrm{~kg} / \mathrm{m}^{3} .$ The free-fall acceleration $g$ at the site of the barom-eter is $9.7835 \mathrm{~m} / \mathrm{s}^{2}$. What is the atmospheric pressure at that site in pascals and in torr (which is the common unit for barometer readings)?

Matthew Baker
Matthew Baker
Numerade Educator
02:05

Problem 26

To suck lemonade of density $1000 \mathrm{~kg} / \mathrm{m}^{3}$ up a straw to a maximum height of $4.0 \mathrm{~cm}$, what minimum gauge pressure (in atmospheres) must you produce in your lungs?

Matthew Baker
Matthew Baker
Numerade Educator
04:09

Problem 27

What would be the height of the atmosphere if the air density (a) were uniform and (b) decreased linearly to zero with height? Assume that at sea level the air pressure is $1.0 \mathrm{~atm}$ and the air density is $1.3 \mathrm{~kg} / \mathrm{m}^{3}$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:15

Problem 28

A piston of cross-sectional area $a$ is used in a hydraulic press to exert a small force of magnitude $f$ on the enclosed liquid. A connecting pipe leads to a larger piston of crosssectional area $A$ (Fig. 14-36). (a) What force magnitude $F$ will the larger piston sustain without moving? (b) If the piston diameters are $3.80 \mathrm{~cm}$ and $53.0 \mathrm{~cm}$, what force magnitude on the small piston will balance a $20.0$ $\mathrm{kN}$ force on the large piston?

Averell Hause
Averell Hause
Carnegie Mellon University
06:00

Problem 29

A spring of spring constant $3.00 \times 10^{4} \mathrm{~N} / \mathrm{m}$ is between a rigid beam and the output piston of a hydraulic lever. An empty container with negligible mass sits on the input piston. The input piston has area $A_{i}$, and the output piston has area $18.0 A_{i}$. Initially the spring is at its rest length. How many kilograms of sand must be (slowly) poured into the container to compress the spring by $5.00 \mathrm{~cm}$ ?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:01

Problem 30

A $5.00 \mathrm{~kg}$ object is released from rest while fully submerged in a liquid. The liquid displaced by the submerged object has a mass of $3.00 \mathrm{~kg}$. How far and in what direction does the object move in $0.200 \mathrm{~s}$, assuming that it moves freely and that the drag force on it from the liquid is negligible?

Averell Hause
Averell Hause
Carnegie Mellon University
04:42

Problem 31

A block of wood floats in fresh water with two-thirds of its volume $V$ submerged and in oil with $0.90 \mathrm{~V}$ submerged. Find the density of (a) the wood and (b) the oil.

Matthew Baker
Matthew Baker
Numerade Educator
05:30

Problem 32

a cube of edge length $L=0.600 \mathrm{~m}$ and mass $450 \mathrm{~kg}$ is suspended by a rope in an open tank of liquid of density $1030 \mathrm{~kg} / \mathrm{m}^{3}$. Find (a) the magnitude of the total downward force on the top of the cube from the liquid and the atmosphere, assuming atmospheric pressure is $1.00 \mathrm{~atm}$, (b) the magnitude of the total upward force on the bottom of the cube, and (c) the tension
in the rope. (d) Calculate the magnitude of the buoyant force on the cube using Archimedes' principle. What relation exists among all these quantities?

Keshav Singh
Keshav Singh
Numerade Educator
02:17

Problem 33

An iron anchor of density $7870 \mathrm{~kg} / \mathrm{m}^{3}$ appears $200 \mathrm{~N}$ lighter in water than in air. (a) What is the volume of the anchor?
(b) How much does it weigh in air?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:57

Problem 34

A boat floating in fresh water displaces water weighing
$35.6 \mathrm{kN}$. (a) What is the weight of the water this boat displaces when floating in salt water of density $1.10 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3} ?(\mathrm{~b})$ What is the difference between the volume of fresh water displaced and the volume of salt water displaced?

Averell Hause
Averell Hause
Carnegie Mellon University
02:33

Problem 35

Three children, each of weight $356 \mathrm{~N}$, make a log raft by lashing together logs of diameter $0.30 \mathrm{~m}$ and length $1.80 \mathrm{~m}$. How many logs will be needed to keep them afloat in fresh water? Take the density of the logs to be $800 \mathrm{~kg} / \mathrm{m}^{3}$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
05:35

Problem 36

A rectangular block is gradually pushed face-down into a liquid. The block has height $d$; on the bottom and top the face area is $A=5.67 \mathrm{~cm}^{2}$. Figure 14-39b gives the apparent weight $W_{\text {app }}$ of the block as a function of the depth $h$ of its lower face. The scale on the vertical axis is set by $W_{s}=0.20 \mathrm{~N}$. What is the density of the liquid?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:51

Problem 37

A hollow spherical iron shell floats almost completely submerged in water. The outer diameter is $60.0 \mathrm{~cm}$, and the density of iron is $7.87 \mathrm{~g} / \mathrm{cm}^{3}$. Find the inner diameter.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
06:43

Problem 38

A small solid ball is released from rest while fully submerged in a liquid and then its kinetic energy is measured when it has moved $4.0 \mathrm{~cm}$ in the liquid. Figure $14-40$ gives the results after many liquids are used:
The kinetic energy $K$ is plotted versus the liquid density $\rho_{\text {liq }}$, and $K_{s}=1.60 \mathrm{~J}$ sets the scale on the vertical axis. What are (a) the density and (b) the volume of the ball?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
04:21

Problem 39

A hollow sphere of inner radius $8.0 \mathrm{~cm}$ and outer radius $9.0 \mathrm{~cm}$ floats half-submerged in a liquid of density $800 \mathrm{~kg} / \mathrm{m}^{3}$.
(a) What is the mass of the sphere?
(b) Calculate the density of the material of which the sphere is made.

Matthew Baker
Matthew Baker
Numerade Educator
02:14

Problem 40

Lurking alligators. An alligator waits for prey by floating with only the top of its head exposed, so that the prey cannot easily see it. One way it can adjust the extent of sinking is by controlling the size of its
lungs. Another way may be by swallowing stones (gastrolithes) that then reside in the stomach. Figure $14-41$ shows a highly simplified model (a "rhombohedron gater") of mass $130 \mathrm{~kg}$ that roams with its head partially exposed. The top head surface has area $0.20 \mathrm{~m}^{2}$. If the alligator were to swallow stones with a total mass of $1.0 \%$ of its body mass (a typical amount), how far would it sink?

Averell Hause
Averell Hause
Carnegie Mellon University
02:03

Problem 41

What fraction of the volume of an iceberg (density $917 \mathrm{~kg} / \mathrm{m}^{3}$ ) would be visible if the iceberg floats (a) in the ocean (salt water, density $\left.1024 \mathrm{~kg} / \mathrm{m}^{3}\right)$ and $(\mathrm{b})$ in a river (fresh water, density $1000 \mathrm{~kg} / \mathrm{m}^{3}$ )? (When salt water freezes to form ice, the salt is excluded. So, an iceberg could provide fresh water to a community.)

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:06

Problem 42

A flotation device is in the shape of a right cylinder, with a height of $0.500 \mathrm{~m}$ and a face area of $4.00 \mathrm{~m}^{2}$ on top and bottom, and its density is $0.400$ times that of fresh water. It is initially held fully submerged in fresh water, with its top face at the water surface. Then it is allowed to ascend gradually until it begins to float. How much work does the buoyant force do on the device during the ascent?

Averell Hause
Averell Hause
Carnegie Mellon University
03:16

Problem 43

When researchers find a reasonably complete fossil of a dinosaur, they can determine the mass and weight of the living dinosaur with a scale model sculpted from plastic and based on the dimensions of the fossil bones. The scale of the model is $1 / 20 ;$ that is, lengths are $1 / 20$ actual length, areas are $(1 / 20)^{2}$ actual areas, and volumes are $(1 / 20)^{3}$ actual
volumes. First, the model is suspended from one arm of a balance and weights are added to the other arm until equilibrium is reached. Then the model is fully submerged in water and enough weights are removed from the second arm to reestablish equilibrium (Fig. 14-42). For a model of a particular T. rex fossil, $637.76 \mathrm{~g}$ had to be removed to reestablish equilibrium. What was the volume of (a) the model and (b) the actual T. rex? (c) If the density of
T. rex was approximately the density of water, what was its mass?

Keshav Singh
Keshav Singh
Numerade Educator
04:53

Problem 44

A wood block (mass $3.67 \mathrm{~kg}$, density $600 \mathrm{~kg} / \mathrm{m}^{3}$ ) is fitted with lead (density $1.14 \times 10^{4} \mathrm{~kg} / \mathrm{m}^{3}$ ) so that it floats in water with $0.900$ of its volume submerged. Find the lead mass if the lead is fitted to the block's (a) top and (b) bottom.

Keshav Singh
Keshav Singh
Numerade Educator
04:55

Problem 45

An iron casting containing a number of cavities weighs $6000 \mathrm{~N}$ in air and $4000 \mathrm{~N}$ in water. What is the total cavity volume in the casting? The density of solid iron is $7.87 \mathrm{~g} / \mathrm{cm}^{3}$.

Matthew Baker
Matthew Baker
Numerade Educator
02:54

Problem 46

Suppose that you release a small ball from rest at a depth of $0.600 \mathrm{~m}$ below the surface in a pool of water. If the density of the ball is $0.300$ that of water and if the drag force on the ball from the water is negligible, how high above the water surface will the ball shoot as it emerges from the water? (Neglect any transfer of energy to the splashing and waves produced by the emerging ball.)

Averell Hause
Averell Hause
Carnegie Mellon University
05:27

Problem 47

The volume of air space in the passenger compartment of an $1800 \mathrm{~kg}$ car is $5.00 \mathrm{~m}^{3}$. The volume of the motor and front wheels is $0.750 \mathrm{~m}^{3}$, and the volume of the rear wheels, gas tank, and trunk is $0.800$ $\mathrm{m}^{3} ;$ water cannot enter these two regions. The car rolls into a lake. (a) At first, no water enters the passenger compartment. How much of the car, in cubic meters, is below the water surface with the car floating (Fig. 14-43)? (b) As water slowly enters, the car sinks. How many cubic meters of water are in the car as it disappears below the water surface? (The car, with a heavy load in the trunk, remains horizontal.)

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:06

Problem 48

Shows an iron ball suspended by thread of negligible mass from an upright cylinder that floats partially submerged in water. The cylinder has a height of $6.00 \mathrm{~cm}$, a face area of $12.0$ $\mathrm{cm}^{2}$ on the top and bottom, and a density of $0.30 \mathrm{~g} / \mathrm{cm}^{3}$, and $2.00 \mathrm{~cm}$ of its height is above the water surface. What is the radius of the iron ball?

Keshav Singh
Keshav Singh
Numerade Educator
03:18

Problem 49

Canal effect. shows an anchored barge that extends across a canal by distance $d=30 \mathrm{~m}$ and into the water by distance $b=12 \mathrm{~m} .$ The canal has a width $D=55 \mathrm{~m}$, a water depth $H=14 \mathrm{~m}$, and a uniform water-flow speed $v_{i}=1.5 \mathrm{~m} / \mathrm{s}$. Assume that the flow around the barge is uniform. As the water passes the bow, the water level undergoes a dramatic dip known as the canal effect. If the dip has depth $h=0.80 \mathrm{~m}$, what is the water speed alongside the boat through the vertical cross sections at (a) point $a$ and (b) point $b$ ? The erosion due to the speed increase is a common concern to hydraulic engineers.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
04:31

Problem 50

Shows two sections of an old pipe system that runs through a hill, with distances $d_{A}=d_{B}=30 \mathrm{~m}$ and
$D=110 \mathrm{~m} .$ On each side of the hill, the pipe radius is $2.00 \mathrm{~cm}$. However, the radius of the pipe inside the hill is no longer known. To determine it, hydraulic engineers first establish that water flows through the left and right sections at $2.50 \mathrm{~m} / \mathrm{s}$. Then they release a dye in the water at point $A$ and find that it takes $88.8 \mathrm{~s}$ to reach point $B$. What is the average radius of the pipe within the hill?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:51

Problem 51

A garden hose with an internal diameter of $1.9 \mathrm{~cm}$ is connected to a (stationary) lawn sprinkler that consists merely of a container with 24 holes, each $0.13 \mathrm{~cm}$ in diameter. If the water in the hose has a speed of $0.91 \mathrm{~m} / \mathrm{s}$, at what speed does it leave the sprinkler holes?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:05

Problem 52

Two streams merge to form a river. One stream has a width of $8.2 \mathrm{~m}$, depth of $3.4 \mathrm{~m}$, and current speed of $2.3 \mathrm{~m} / \mathrm{s}$. The other stream is $6.8 \mathrm{~m}$ wide and $3.2 \mathrm{~m}$ deep, and flows at $2.6 \mathrm{~m} / \mathrm{s}$. If the river has width $10.5 \mathrm{~m}$ and speed $2.9 \mathrm{~m} / \mathrm{s}$, what is its depth?

Averell Hause
Averell Hause
Carnegie Mellon University
01:49

Problem 53

Water is pumped steadily out of a flooded basement at $5.0 \mathrm{~m} / \mathrm{s}$ through a hose of radius $1.0 \mathrm{~cm}$, passing through a window 3.0 m above the waterline. What is the pump's power?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:26

Problem 54

The water flowing through a $1.9 \mathrm{~cm}$ (inside diameter) pipe flows out through three $1.3 \mathrm{~cm}$ pipes. (a) If the flow rates in the three smaller pipes are 26,19 , and $11 \mathrm{~L} / \mathrm{min}$, what is the flow rate in the $1.9 \mathrm{~cm}$ pipe? (b) What is the ratio of the speed in the $1.9 \mathrm{~cm}$ pipe to that in the pipe carrying $26 \mathrm{~L} / \mathrm{min}$ ?

Matthew Baker
Matthew Baker
Numerade Educator
01:28

Problem 55

How much work is done by pressure in forcing $1.4 \mathrm{~m}^{3}$ of water through a pipe having an internal diameter of $13 \mathrm{~mm}$ if the difference in pressure at the two ends of the pipe is $1.0 \mathrm{~atm}$ ?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:44

Problem 56

Suppose that two tanks, 1 and 2 , each with a large opening at the top, contain different liquids. A small hole is made in the side of each tank at the same depth $h$ below the liquid surface, but the hole in tank 1 has half the cross-sectional area of the hole in tank $2 .$
(a) What is the ratio $\rho_{1} / \rho_{2}$ of the densities of the liquids if the mass flow rate is the same for the two holes? (b) What is the ratio $R_{V_{1}} / R_{V 2}$ of the volume flow rates from the two tanks? (c) At one instant, the liquid in tank 1 is $12.0 \mathrm{~cm}$ above the hole. If the tanks are to have equal volume flow rates, what height above the hole must the liquid in tank 2 be just then?

Averell Hause
Averell Hause
Carnegie Mellon University
03:46

Problem 57

A cylindrical tank with a large diameter is filled with water to a depth $D=0.30 \mathrm{~m} .$ A hole of cross-sectional area $A=6.5 \mathrm{~cm}^{2}$ in the bottom of the tank allows water to drain out. (a) What is the drainage rate in cubic meters per second? (b) At what distance below the bottom of the tank is the cross-sectional area of the stream equal to one-half the area of the hole?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:18

Problem 58

The intake has cross-sectional area of $0.74 \mathrm{~m}^{2}$ and water flow at $0.40 \mathrm{~m} / \mathrm{s}$. At the outlet, distance $D=180 \mathrm{~m}$ below the intake, the cross-sectional area is smaller than at the intake and the water flows out at $9.5 \mathrm{~m} / \mathrm{s}$ into
equipment. What is the pressure difference between inlet and outlet?

Matthew Baker
Matthew Baker
Numerade Educator
03:22

Problem 59

Water is moving with a speed of $5.0 \mathrm{~m} / \mathrm{s}$ through a pipe with a cross-sectional area of $4.0 \mathrm{~cm}^{2}$. The water gradually descends $10 \mathrm{~m}$ as the pipe cross-sectional area increases to $8.0 \mathrm{~cm}^{2}$. (a) What is the speed at the lower level? (b) If the pressure at the upper level is $1.5 \times 10^{5} \mathrm{~Pa}$, what is the pressure at the lower level?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:39

Problem 60

Models of torpedoes are sometimes tested in a horizontal pipe of flowing water, much as a wind tunnel is used to test model airplanes. Consider a circular pipe of internal diameter $25.0 \mathrm{~cm}$ and a torpedo model aligned along the long axis of the pipe. The model has a $5.00 \mathrm{~cm}$ diameter and is to be tested with water flowing past it at $2.50 \mathrm{~m} / \mathrm{s}$. (a) With what speed must the water flow in the part of the pipe that is unconstricted by the model? (b) What will the pressure difference be between the constricted and unconstricted parts of the pipe?

Averell Hause
Averell Hause
Carnegie Mellon University
03:41

Problem 61

A water pipe having a $2.5 \mathrm{~cm}$ inside diameter carries water into the basement of a house at a speed of $0.90 \mathrm{~m} / \mathrm{s}$ and a pressure of $170 \mathrm{kPa}$. If the pipe tapers to $1.2 \mathrm{~cm}$ and rises to the second floor $7.6 \mathrm{~m}$ above the input point, what are the
(a) speed and
(b) water pressure at the second floor?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:23

Problem 62

A pitot tube (Fig. 14-48) is used to determine the airspeed of an airplane. It consists of an outer tube with a number of small holes $B$ (four are shown) that allow air into the tube; that tube is connected to one arm of a U-tube. The other arm of the U-tube is connected to hole $A$ at the front end of the device, which points in the direction the plane is headed. At $A$ the air becomes stagnant so that $v_{A}=0$. At $B$, however, the speed of the air presumably equals the airspeed $v$ of the plane. (a) Use Bernoulli's equation to show that
$$
v=\sqrt{\frac{2 \rho g h}{\rho_{\text {air }}}} \text { , }
$$
where $\rho$ is the density of the liquid in the $\mathrm{U}$ -tube and $h$ is the difference in the liquid levels in that tube. (b) Suppose that the tube contains alcohol and the level difference $h$ is $26.0 \mathrm{~cm}$. What is the plane's speed relative to the air? The density of the air is $1.03 \mathrm{~kg} / \mathrm{m}^{3}$ and that of alcohol is $810 \mathrm{~kg} / \mathrm{m}^{3}$.

Keshav Singh
Keshav Singh
Numerade Educator
01:29

Problem 63

A pitot tube (see Problem 62) on a high-altitude aircraft measures a differential pressure of $180 \mathrm{~Pa}$. What is the aircraft's airspeed if the density of the air is $0.031 \mathrm{~kg} / \mathrm{m}^{3}$ ?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
08:03

Problem 64

Water flows through a horizontal pipe and then out into the atmosphere at a speed $v_{1}=15$ $\mathrm{m} / \mathrm{s}$. The diameters of the left and right sections of the pipe are $5.0 \mathrm{~cm}$ and $3.0$ $\mathrm{cm}$. (a) What volume of water flows into the atmosphere during a 10 min period? In the left section of the pipe, what are (b) the speed $v_{2}$ and (c) the gauge pressure?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:25

Problem 65

A venturi meter is used to measure the flow speed of a fluid in a pipe. The meter is connected between two sections of the pipe (Fig. $14-50$ ); the cross-sectional area $A$ of the entrance and exit of the meter matches the pipe's cross-sectional area. Between the entrance and exit, the fluid flows from the pipe with speed $V$ and then through a narrow "throat" of crosssectional area $a$ with speed $v .$ A manometer connects the wider portion of the meter to the narrower portion. The change in the fluid's speed is accompanied by a change $\Delta p$ in the fluid's pressure, which causes a height difference $h$ of the liquid in the two arms of the manometer. (Here $\Delta p$ means pressure in the throat minus pressure in the pipe.) (a) By applying Bernoulli's equation and the equation of continuity to points 1 and 2 in Fig. $14-50$, show that
$$
V=\sqrt{\frac{2 a^{2} \Delta p}{\rho\left(a^{2}-A^{2}\right)}}
$$
where $\rho$ is the density of the fluid. (b) Suppose that the fluid is fresh water, that the cross-sectional areas are $64 \mathrm{~cm}^{2}$ in the pipe and $32 \mathrm{~cm}^{2}$ in the throat, and that the pressure is $55 \mathrm{kPa}$ in the pipe and $41 \mathrm{kPa}$ in the throat. What is the rate of water flow in cubic meters per second?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:00

Problem 66

Consider the venturi tube of Problem 65 and without the manometer. Let $A$ equal $5 a$. Suppose the pressure $p_{1}$ at $A$ is $2.0 \mathrm{~atm} .$ Compute the values of (a) the speed $V$ at $A$ and $(\mathrm{b})$ the speed $v$ at $a$ that make the pressure $p_{2}$ at $a$ equal to zero. (c) Compute the corresponding volume flow rate if the diameter at $A$ is $5.0 \mathrm{~cm}$. The phenomenon that occurs at $a$ when $p_{2}$ falls to nearly zero is known as cavitation. The water vaporizes into small bubbles.

Averell Hause
Averell Hause
Carnegie Mellon University
04:22

Problem 67

The fresh water behind a reservoir dam has depth $D=15 \mathrm{~m}$. A horizontal pipe $4.0 \mathrm{~cm}$ in diameter passes through the dam at depth $d=6.0 \mathrm{~m} .$ A plug secures the pipe
opening. (a) Find the magnitude of the frictional force between plug and pipe wall. (b) The plug is removed. What water volume exits the pipe in $3.0 \mathrm{~h}$ ?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
02:33

Problem 68

Fresh water flows horizontally from pipe section 1 of cross-sectional area $A_{1}$ into pipe section 2 of cross-sec tional area $A_{2}$. Figure $14-52$ gives a plot of the pressure difference $p_{2}-p_{1}$ versus the inverse area squared $A_{1}^{-2}$ that would be expected for a volume flow rate of a certain value if the water flow were laminar under all circumstances. The scale on the vertical axis is set by $\Delta p_{s}=300 \mathrm{kN} / \mathrm{m}^{2} .$ For the conditions
of the figure, what are the values of
(a) $A_{2}$ and (b) the volume flow rate?
${ }^{\infty} 69$ A liquid of density $900 \mathrm{~kg} / \mathrm{m}^{3}$
flows through a horizontal pipe that has a cross-sectional area of $1.90 \times 10^{-2} \mathrm{~m}^{2}$ in region $A$ and a cross-sectional area of $9.50 \times 10^{-2} \mathrm{~m}^{2}$ in region $B$. The pressure difference between the two regions is $7.20 \times 10^{3} \mathrm{~Pa}$. What are (a) the volume flow rate and (b) the mass flow rate?

Keshav Singh
Keshav Singh
Numerade Educator
04:12

Problem 69

A liquid of density $900 \mathrm{~kg} / \mathrm{m}^{3}$ flows through a horizontal pipe that has a cross-sectional area of $1.90 \times 10^{-2} \mathrm{~m}^{2}$ in region $A$ and a cross-sectional area of $9.50 \times 10^{-2} \mathrm{~m}^{2}$ in region $B$. The pressure difference between the two regions is $7.20 \times 10^{3} \mathrm{~Pa}$. What are (a) the volume flow rate and (b) the mass flow rate?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
06:07

Problem 70

water flows steadily from the left pipe section (radius $r_{1}=2.00 R$ ), through the middle section (radius $R$ ), and into the right section (radius $r_{3}=3.00 R$ ). The speed of the water in the middle sec-tion is $0.500 \mathrm{~m} / \mathrm{s}$. What is the net work done on $0.400 \mathrm{~m}^{3}$ of the water as it moves from the left section to the right section?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
05:01

Problem 71

Shows a stream of water flowing through a hole at depth $h=10 \mathrm{~cm}$ in a tank holding water to height $H=40 \mathrm{~cm} .$ (a) At what distance $x$ does the stream strike the floor? (b) At what depth should a second hole be made to give the same value of $x ?(\mathrm{c})$ At what depth should a hole be made to maximize $x$ ?

Keshav Singh
Keshav Singh
Numerade Educator
02:26

Problem 72

A very simplified schematic of the rain drainage system for a home is shown in Fig. $14-55$. Rain falling on the slanted roof runs off into gutters around the roof edge; it then drains through downspouts (only one is shown) into a main drainage pipe $M$ below the basement, which carries the water to an even larger pipe below the street. In Fig. $14-55$, a floor drain in the basement is also connected to drainage pipe $M$. Suppose the following apply:
(1) the downspouts have height $h_{1}=11 \mathrm{~m},(2)$ the
floor drain has height $h_{2}=$ $1.2 \mathrm{~m},(3)$ pipe $M$ has radius $3.0 \mathrm{~cm},(4)$ the house has side width $w=30 \mathrm{~m}$ and front length $L=60 \mathrm{~m},(5)$ all
the water striking the roof goes through pipe $M,(6)$ the initial speed of the water in a downspout is negligible, and $(7)$ the wind speed is negligible (the rain falls vertically).
At what rainfall rate, in centimeters per hour, will water from pipe $M$ reach the height of the floor drain and threaten to flood the basement?

Keshav Singh
Keshav Singh
Numerade Educator
01:51

Problem 73

About one-third of the body of a person floating in the Dead Sea will be above the waterline. Assuming that the human body density is $0.98 \mathrm{~g} / \mathrm{cm}^{3}$, find the density of the water in the Dead Sea. (Why is it so much greater than $1.0 \mathrm{~g} / \mathrm{cm}^{3} ?$ )

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:09

Problem 74

A simple open U-tube contains mercury. When $11.2 \mathrm{~cm}$ of water is poured into the right arm of the tube, how high above its initial level does the mercury rise in the left arm?

Averell Hause
Averell Hause
Carnegie Mellon University
03:03

Problem 75

If a bubble in sparkling water accelerates upward at the rate of $0.225 \mathrm{~m} / \mathrm{s}^{2}$ and has a radius of $0.500 \mathrm{~mm}$, what is its mass? Assume that the drag force on the bubble is negligible.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:40

Problem 76

Suppose that your body has a uniform density of $0.95$ times that of water. (a) If you float in a swimming pool, what fraction of your body's volume is above the water surface?
Quicksand is a fluid produced when water is forced up into sand, moving the sand grains away from one another so they are no longer locked together by friction. Pools of quicksand can form when water drains underground from hills into valleys where there are sand pockets. (b) If you float in a deep pool of quicksand that has a density $1.6$ times that of water, what fraction of your body's volume is above the quicksand surface? (c) Are you unable to breathe?

Averell Hause
Averell Hause
Carnegie Mellon University
12:00

Problem 77

A glass ball of radius $2.00 \mathrm{~cm}$ sits at the bottom of a container of milk that has a density of $1.03 \mathrm{~g} / \mathrm{cm}^{3} .$ The normal force on the ball from the container's lower surface has magnitude $9.48 \times 10^{-2} \mathrm{~N}$. What is the mass of the ball?

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
01:35

Problem 78

Caught in an avalanche, a skier is fully submerged in flowing snow of density $96 \mathrm{~kg} / \mathrm{m}^{3}$. Assume that the average density of the skier, clothing, and skiing equipment is $1020 \mathrm{~kg} / \mathrm{m}^{3} .$ What percentage of the gravitational force on the skier is offset by the buoyant force from the snow?

Matthew Baker
Matthew Baker
Numerade Educator
02:39

Problem 79

An object hangs from a spring balance. The balance registers $30 \mathrm{~N}$ in air, $20 \mathrm{~N}$ when this object is immersed in water, and $24 \mathrm{~N}$ when the object is immersed in another liquid of unknown density. What is the density of that other liquid?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:12

Problem 80

In an experiment, a rectangular block with height $h$ is allowed to float in four separate liquids. In the first liquid, which is water, it floats fully submerged. In liquids $A, B$, and $C$, it floats with heights $h / 2,2 h / 3$, and $h / 4$ above the liquid surface, respectively. What are the relative densities (the densities relative to that of water) of
(a) $A,(\mathrm{~b}) B$, and $(\mathrm{c}) C ?$

Averell Hause
Averell Hause
Carnegie Mellon University
02:07

Problem 81

Shows a modified U-tube: the right arm is shorter than the left arm. The open end of the right arm is height $d=10.0 \mathrm{~cm}$ above the laboratory bench. The radius throughout the tube is $1.50 \mathrm{~cm}$. Water is gradually poured into the open end of the left arm until the water begins to flow out the open end of the right arm. Then a liquid of density $0.80 \mathrm{~g} / \mathrm{cm}^{3}$ is gradually added to the left arm until its height in that arm is $8.0 \mathrm{~cm}$ (it does not mix with the water). How much water flows out of the right arm?

Keshav Singh
Keshav Singh
Numerade Educator
01:04

Problem 82

What is the acceleration of a rising hot-air balloon if the ratio of the air density outside the balloon to that inside is $1.39 ?$ Neglect the mass of the balloon fabric and the basket.

Averell Hause
Averell Hause
Carnegie Mellon University
05:59

Problem 83

Shows a siphon, which is a device for removing liquid from a container. Tube $A B C$ must initially be filled, but once this has been done, liquid will flow through the tube until the liquid surface in the container is level with the tube opening at $A$. The liquid has density $1000 \mathrm{~kg} / \mathrm{m}^{3}$ and negligible viscosity. The distances shown are $h_{1}=25 \mathrm{~cm}, d=$ $12 \mathrm{~cm}$, and $h_{2}=40 \mathrm{~cm} .$ (a) With what speed does the liquid emerge from the tube at $C ?(b)$ If the atmospheric pressure is $1.0 \times 10^{5} \mathrm{~Pa}$, what is the pressure in the liquid at the topmost point $B ?$ (c) Theoretically, what is the greatest possible height $h_{1}$ that a siphon can lift water?

Keshav Singh
Keshav Singh
Numerade Educator
02:10

Problem 84

When you cough, you expel air at high speed through the trachea and upper bronchi so that the air will remove excess mucus lining the pathway. You produce the high speed by this procedure: You breathe in a large amount of air, trap it by closing the glottis (the narrow opening in the larynx), increase the air pressure by contracting the lungs, partially collapse the trachea and upper bronchi to narrow the pathway, and then expel the air through the pathway by suddenly reopening the glottis. Assume that during the expulsion the volume flow rate is $7.0 \times 10^{-3} \mathrm{~m}^{3} / \mathrm{s}$. What multiple of $343 \mathrm{~m} / \mathrm{s}$ (the speed of sound $v_{s}$ ) is the airspeed through the trachea if the trachea diameter
(a) remains its normal value of $14 \mathrm{~mm}$ and $(\mathrm{b})$ contracts to $5.2 \mathrm{~mm}$ ?

Averell Hause
Averell Hause
Carnegie Mellon University
01:25

Problem 85

A tin can has a total volume of $1200 \mathrm{~cm}^{3}$ and a mass of $130 \mathrm{~g}$. How many grams of lead shot of density $11.4 \mathrm{~g} / \mathrm{cm}^{3}$ could it carry without sinking in water?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:04

Problem 86

The tension in a string holding a solid block below the surface of a liquid (of density greater than the block) is $T_{0}$ when the container (Fig. $14-57$ ) is at rest. When the container is given an upward acceleration of $0.250 \mathrm{~g}$, what multiple of $T_{0}$ gives the tension in the string?

Averell Hause
Averell Hause
Carnegie Mellon University
02:16

Problem 87

What is the minimum area (in square meters) of the top surface of an ice slab $0.441 \mathrm{~m}$ thick floating on fresh water that will hold up a $938 \mathrm{~kg}$ automobile? Take the densities of ice and fresh water to be $917 \mathrm{~kg} / \mathrm{m}^{3}$ and $998 \mathrm{~kg} / \mathrm{m}^{3}$, respectively.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
07:52

Problem 88

A $8.60 \mathrm{~kg}$ sphere of radius $6.22 \mathrm{~cm}$ is at a depth of $2.22 \mathrm{~km}$ in seawater that has an average density of $1025 \mathrm{~kg} / \mathrm{m}^{3}$. What are the
(a) gauge pressure, (b) total pressure, and (c) corresponding total force compressing the sphere's surface? What are (d) the magnitude of the buoyant force on the sphere and (e) the magnitude of the sphere's acceleration if it is free to move? Take atmospheric pressure to be $1.01 \times 10^{5} \mathrm{~Pa}$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
02:02

Problem 89

For seawater of density $1.03 \mathrm{~g} / \mathrm{cm}^{3}$, find the weight of water on top of a submarine at a depth of $255 \mathrm{~m}$ if the horizontal cross-sectional hull area is $2200.0 \mathrm{~m}^{2} .(\mathrm{b})$ In atmospheres, what water pressure would a diver experience at this depth?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:52

Problem 90

The sewage outlet of a house constructed on a slope is $6.59 \mathrm{~m}$ below street level. If the sewer is $2.16 \mathrm{~m}$ below street level, find the minimum pressure difference that must be created by the sewage pump to transfer waste of average density $1000.00 \mathrm{~kg} / \mathrm{m}^{3}$ from outlet to sewer.

Matthew Baker
Matthew Baker
Numerade Educator