Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 15

Fluid Mechanics - all with Video Answers

Educators

Chapter Questions

Problem 1

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

Vipender Yadav

Vipender Yadav

Numerade Educator

Problem 2

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

Prabhu Ramji

Numerade Educator

Problem 3

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

Surjit Tewari

Numerade Educator

Problem 4

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

Surjit Tewari

Numerade Educator

Problem 5

What is the total mass of the Earth's atmosphere? (The radius of the Earth is $6.37 \times 10^{6} \mathrm{~m}$, and atmospheric pressure at the Earth's surface is $1.013 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2} .$ )

Surjit Tewari

Numerade Educator

Problem 6

(a) Calculate the absolute pressure at an ocean depth of 1 $000 \mathrm{~m}$. Assume the density of seawater is $1024 \mathrm{~kg} / \mathrm{m}^{3}$ and that the air above exerts a pressure of $101.3 \mathrm{kPa}$.
(b) At this depth, what force must the frame around a circular submarine porthole having a diameter of $30.0 \mathrm{~cm}$ exert to counterbalance the force exerted by the water?

Prabhu Ramji

Numerade Educator

Problem 7

The spring of the pressure gauge shown in Figure $15.2$ has a force constant of $1000 \mathrm{~N} / \mathrm{m}$, and the piston has a diameter of $2.00 \mathrm{~cm}$. When the gauge is lowered into water, at what depth does the piston move in by $0.500 \mathrm{~cm} ?$

Prabhu Ramji

Numerade Educator

Problem 8

The small piston of a hydraulic lift has a cross-sectional area of $3.00 \mathrm{~cm}^{2}$, and its large piston has a cross-sectional area of $200 \mathrm{~cm}^{2}$ (see Fig. $\left.15.5 \mathrm{a}\right)$. What force must be applied to the small piston for it to raise a load of $15.0 \mathrm{kN} ?$ (In service stations, this force is usually generated with the use of compressed air.)

Prabhu Ramji

Numerade Educator

Problem 9

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

Penny Riley

Numerade Educator

Problem 10

(a) A very powerful vacuum cleaner has a hose $2.86 \mathrm{~cm}$ in diameter. With no nozzle on the hose, what is the weight of the heaviest brick that the cleaner can lift (Fig. P15.10)? (b) A very powerful octopus uses one sucker of diameter $2.86 \mathrm{~cm}$ on each of the two shells of a clam in an attempt to pull the shells apart. Find the greatest force that the octopus can exert in salt water $32.3 \mathrm{~m}$ in depth. (Caution: Experimental verification can be interesting, but do not drop a brick on your foot. Do not overheat the motor of a vacuum cleaner. Do not get an octopus mad at you.)

Prabhu Ramji

Numerade Educator

Problem 11

For the cellar of a new house, a hole with vertical sides descending $2.40 \mathrm{~m}$ is dug in the ground. 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 that the water causes on the foundation wall. For
comparison, the weight of the water is given by
$$
\begin{aligned}
2.40 \mathrm{~m} & \times 9.60 \mathrm{~m} \times 0.183 \mathrm{~m} \times 1000 \mathrm{~kg} / \mathrm{m}^{3} \\
& \times 9.80 \mathrm{~m} / \mathrm{s}^{2}=41.3 \mathrm{kN}
\end{aligned}
$$

Prabhu Ramji

Numerade Educator

Problem 12

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

Surjit Tewari

Numerade Educator

Problem 13

A sealed spherical shell of diameter $d$ is rigidly attached to a cart that is moving horizontally with an acceleration $a$, as shown in Figure P15.13. The sphere is nearly filled with a fluid having density $\rho$ and also contains one small bubble of air at atmospheric pressure. Find an expression for the pressure $P$ at the center of the sphere.

Prabhu Ramji

Numerade Educator

Problem 14

The tank shown in Figure $\mathrm{P} 15.14$ is filled with water to a depth of $2.00 \mathrm{~m}$. At the bottom of one of the side walls is a rectangular hatch $1.00 \mathrm{~m}$ high and $2.00 \mathrm{~m}$ wide. The hatch is hinged at its top. (a) Determine the force that the water exerts on the hatch. (b) Find the torque exerted about the hinges.

Prabhu Ramji

Numerade Educator

Problem 15

Review Problem. A solid copper ball with a diameter of $3.00 \mathrm{~m}$ at sea level is placed at the bottom of the ocean (at a depth of $10.0 \mathrm{~km})$. If the density of seawater is $1030 \mathrm{~kg} / \mathrm{m}^{3}$, by how much (approximately) does the diameter of the ball decrease when it reaches bottom? Take the bulk modulus of copper as $14.0 \times 10^{10} \mathrm{~N} / \mathrm{m}^{2}$.

Prabhu Ramji

Numerade Educator

Problem 16

Normal atmospheric pressure is $1.013 \times 10^{5} \mathrm{~Pa}$. The approach of a storm causes the height of a mercury barometer to drop by $20.0 \mathrm{~mm}$ from the normal height. What is the atmospheric pressure? (The density of mercury is $13.59 \mathrm{~g} / \mathrm{cm}^{3} .$ )

Kon Aoki

Numerade Educator

Problem 17

Blaise Pascal duplicated Torricelli's barometer, using a red Bordeaux wine, of density $984 \mathrm{~kg} / \mathrm{m}^{3}$, as the working liquid (Fig. P15.17). What was the height $h$ of the wine column for normal atmospheric pressure? Would you expect the vacuum above the column to be as good as that for mercury?

Surjit Tewari

Numerade Educator

Problem 18

Mercury is poured into a U-tube, as shown in Figure P15.18a. The left arm of the tube has a 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 P15.18b. (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?

Prabhu Ramji

Numerade Educator

Problem 19

A U-tube of uniform cross-sectional area and open to the atmosphere is partially filled with mercury. Water is then poured into both arms. If the equilibrium configuration of the tube is as shown in Figure $\mathrm{P} 15.19$, with $h_{2}=1.00 \mathrm{~cm}$, determine the value of $h_{1}$.

Surjit Tewari

Numerade Educator

Problem 20

(a) A light balloon is filled with $400 \mathrm{~m}^{3}$ of helium. At $0{ }^{\circ} \mathrm{C}$, what is the mass of the payload that the balloon can lift? (b) In Table 15.1, note that the density of hydrogen is nearly one-half the density of helium. What load can the balloon lift if it is filled with hydrogen?

Prabhu Ramji

Numerade Educator

Problem 21

A Styrofoam slab has a thickness of $10.0 \mathrm{~cm}$ and a density of $300 \mathrm{~kg} / \mathrm{m}^{3}$. When a $75.0$ -kg swimmer is resting on it, the slab floats in fresh water with its top at the same level as the water's surface. Find the area of the slab.

Prabhu Ramji

Numerade Educator

Problem 22

A Styrofoam slab has thickness $h$ and density $\rho_{S}$. What is the area of the slab if it floats with its upper surface just awash in fresh water, when a swimmer of mass $m$ is on top?

Prabhu Ramji

Numerade Educator

Problem 23

A piece of aluminum with mass $1.00 \mathrm{~kg}$ and density $2700 \mathrm{~kg} / \mathrm{m}^{3}$ is suspended from a string and then completely immersed in a container of water (Fig. P15.23). Calculate the tension in the string (a) before and
(b) after the metal is immersed.

Surjit Tewari

Numerade Educator

Problem 24

A $10.0-\mathrm{kg}$ block of metal measuring $12.0 \mathrm{~cm} \times$ $10.0 \mathrm{~cm} \times 10.0 \mathrm{~cm}$ is suspended from a scale and immersed in water, as shown in Figure $\mathrm{P} 15.23 \mathrm{~b}$. The 12.0-cm dimension is vertical, and the top of the block is $5.00 \mathrm{~cm}$ from the surface of the water. (a) What are the forces acting on the top and on the bottom of the block? (Take $P_{0}=1.0130 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}$.) (b) What is the reading of the spring scale? (c) Show that the buoyant force equals the difference between the forces at the top and bottom of the block.

Prabhu Ramji

Numerade Educator

Problem 25

A cube of wood having a side dimension of $20.0 \mathrm{~cm}$ and a density of $650 \mathrm{~kg} / \mathrm{m}^{3}$ floats on water. (a) What is the distance from the horizontal top surface of the cube to the water level? (b) How much lead weight must be placed on top of the cube so that its top is just level with the water?

Prabhu Ramji

Numerade Educator

Problem 26

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 27

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 the glycerin and the sphere.

Surjit Tewari

Numerade Educator

Problem 28

A frog in a hemispherical pod finds that he just floats without sinking into a sea of blue-green ooze having a density of $1.35 \mathrm{~g} / \mathrm{cm}^{3}$ (Fig. $\mathrm{P} 15.28$ ). If the pod has a radius of $6.00 \mathrm{~cm}$ and a negligible mass, what is the mass of the frog?

Prabhu Ramji

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 $\rho_{\mathrm{He}}=0.180 \mathrm{~kg} / \mathrm{m}^{3}$.) Assume that the balloon maintains a constant volume and that the density of air decreases with the altitude $z$ according to the expression $\rho_{\text {air }}=\rho_{0} e^{-z / 8000}$, where $z$ is in meters and $\rho_{0}=1.25 \mathrm{~kg} / \mathrm{m}^{3}$ is the density of air at sea level.

Prabhu Ramji

Numerade Educator

Problem 30

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

Prabhu Ramji

Numerade Educator

Problem 31

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

Surjit Tewari

Numerade Educator

Problem 32

The United States possesses the eight largest warships in the world-aircraft carriers of the Nimitz class - and it is building one more. Suppose that one of the ships bobs up to float $11.0 \mathrm{~cm}$ higher in the water when 50 fighters take off from it at a location where $g=$ $9.78 \mathrm{~m} / \mathrm{s}^{2}$. The planes have an average mass of $29000 \mathrm{~kg} .$ Find the horizontal area enclosed by the waterline of the ship. (By comparison, its flight deck has an area of $18000 \mathrm{~m}^{2}$.)

Prabhu Ramji

Numerade Educator

Problem 33

(a) A water hose $2.00 \mathrm{~cm}$ in diameter is used to fill a 20.0-L bucket. If it takes $1.00$ min to fill the bucket, what is the speed $v$ at which water moves through the hose? $\left(\right.$ Note $\left.: 1 \mathrm{~L}=1000 \mathrm{~cm}^{3} .\right)$ (b) If 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 34

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

Surjit Tewari

Numerade Educator

Problem 35

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

Surjit Tewari

Numerade Educator

Problem 36

Through a pipe of diameter $15.0 \mathrm{~cm}$, water is pumped from the Colorado River up to 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 $2096 \mathrm{~m}$. (a) What is the minimum pressure at which the water must be pumped if it is to arrive at the village? (b) If $4500 \mathrm{~m}^{3}$ are pumped per day, what is the speed of the water in the pipe? (c) What additional pressure is necessary to deliver this flow? (Note: You may assume that the acceleration due to gravity and the density of air are constant over this range of elevations.)

Surjit Tewari

Numerade Educator

Problem 37

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 with an inner diameter of $2.20 \mathrm{~cm}$. What is the speed at which the water exits the nozzle?

Surjit Tewari

Numerade Educator

Problem 38

Old Faithful Geyser in Yellowstone National Park erupts at approximately 1 -h intervals, and the height of the water column reaches $40.0 \mathrm{~m}$ (Fig. P15.38). (a) Consider the rising stream as a series of separate drops. Analyze the free-fall motion of one of these drops to determine the speed at which the water leaves the ground.
(b) Treating the rising stream as an ideal fluid in streamline flow, use Bernoulli's equation to determine the speed of the water as it leaves ground level.
(c) What is the pressure (above atmospheric) in the heated underground chamber if its depth is $175 \mathrm{~m}$ ? You may assume that the chamber is large compared with the geyser's vent.

Surjit Tewari

Numerade Educator

Problem 39

An airplane has a mass of $1.60 \times 10^{4} \mathrm{~kg}$, and each wing has an area of $40.0 \mathrm{~m}^{2}$. During level flight, the pressure on the lower wing surface is $7.00 \times 10^{4} \mathrm{~Pa}$. Determine the pressure on the upper wing surface.

Surjit Tewari

Numerade Educator

Problem 40

A Venturi tube may be used as a fluid flow meter (see Fig. $15.21$ ). If the difference in pressure is $P_{1}-P_{2}=21.0 \mathrm{kPa}$, find the fluid flow rate in cubic meters per second, given that the radius of the outlet tube is $1.00 \mathrm{~cm}$, the radius of the inlet tube is $2.00 \mathrm{~cm}$, and the fluid is gasoline $\left(\rho=700 \mathrm{~kg} / \mathrm{m}^{3}\right)$.

Kon Aoki

Numerade Educator

Problem 41

A Pitot tube can be used to determine the velocity of air flow by measuring the difference between the total pressure and the static pressure (Fig. P15.41). If the fluid in the tube is mercury, whose density is $\rho_{\mathrm{Hg}}=$ $13600 \mathrm{~kg} / \mathrm{m}^{3}$, and if $\Delta h=5.00 \mathrm{~cm}$, find the speed of air flow. (Assume that the air is stagnant at point $A$, and take $\rho_{\text {air }}=1.25 \mathrm{~kg} / \mathrm{m}^{3}$.)

Prabhu Ramji

Numerade Educator

Problem 42

An airplanc is cruising at an altitudc of $10 \mathrm{~km}$. Thc pressure outside the craft is $0.287 \mathrm{~atm}$; within the passenger compartment, the pressure is $1.00 \mathrm{~atm}$ and the temperature is $20^{\circ} \mathrm{C}$. A small leak occurs in one of the window seals in the passenger compartment. Model the air as an ideal fluid to find the speed of the stream of air flowing through the leak.

Matthew Baker

Numerade Educator

Problem 43

A siphon is used to drain water from a tank, as illustrated in Figure $\mathrm{P} 15.43 .$ The siphon has a uniform diameter. Assume steady flow without friction. (a) If the distance $h=1.00 \mathrm{~m}$, find the speed of outflow at the end of the siphon. (b) What is the limitation on the height of the top of the siphon above the water surface? (For the flow of liquid to be continuous, the pressure must not drop below the vapor pressure of the liquid.)

Dading Chen

Numerade Educator

Problem 44

A hypodermic syringe contains a medicine with the density of water (Fig. P15.44). The barrel of the syringe has a cross-sectional area $A=2.50 \times 10^{-5} \mathrm{~m}^{2}$, and the needle has a cross-sectional area $a=1.00 \times 10^{-8} \mathrm{~m}^{2}$. In the absence of a force on the plunger, the pressure everywhere is $1 \mathrm{~atm}$. A force $\mathbf{F}$ of magnitude $2.00 \mathrm{~N}$ acts on the plunger, making the medicine squirt horizontally from the needle. Determine the speed of the medicine as it leaves the needle's tip.

BP

Bilas Pal

Numerade Educator

Problem 45

A large storage tank is filled to a height $h_{0}$. The tank is punctured at a height $h$ above the bottom of the tank (Fig. P15.45). Find an expression for how far from the tank the exiting stream lands.

Prabhu Ramji

Numerade Educator

Problem 46

A hole is punched at a height $h$ in the side of a container of height $h_{0}$. The container is full of water, as shown in Figure $\mathrm{P} 15.45$. If the water is to shoot as far as possible horizontally, (a) how far from the bottom of the container should the hole be punched? (b) Neglecting frictional losses, how far (initially) from the side of the container will the water land?

Ummatul Choudary

Ummatul Choudary

Numerade Educator

Problem 47

A Ping-Pong ball has a diameter of $3.80 \mathrm{~cm}$ and an average density of $0.0840 \mathrm{~g} / \mathrm{cm}^{3}$. What force would be required to hold it completely submerged under water?

Matthew Baker

Numerade Educator

Problem 48

Figure $\mathrm{P} 15.48$ shows a tank of water with a valve at the bottom. If this valve is opened, what is the maximum height attained by the water stream exiting the right side of the tank? Assume that $h=10.0 \mathrm{~m}, L=2.00 \mathrm{~m}$, and $\theta=30.0^{\circ}$, and that the cross-sectional area at point $A$ is very large compared with that at point $B$.

Kon Aoki

Numerade Educator

Problem 49

A helium-filled balloon is tied to a 2.00-m-long, $0.0500$ -kg uniform string. The balloon is spherical with a radius of $0.400 \mathrm{~m}$. When released, the balloon lifts a length $h$ of string and then remains in equilibrium, as shown in Figure $\mathrm{P} 15.49$. Determine the value of $h$. The envelope of the balloon has a mass of $0.250 \mathrm{~kg}$.

Surjit Tewari

Numerade Educator

Problem 50

Water is forced out of a fire extinguisher by air pressure, as shown in Figure $\mathrm{P} 15.50 .$ How much gauge air pressure in the tank (above atmospheric) is required for the water jet to have a speed of $30.0 \mathrm{~m} / \mathrm{s}$ when the water level is $0.500 \mathrm{~m}$ below the nozzle?

Surjit Tewari

Numerade Educator

Problem 51

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

Kon Aoki

Numerade Educator

Problem 52

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

Surjit Tewari

Numerade Educator

Problem 53

A wooden dowel has a diameter of $1.20 \mathrm{~cm} .$ It floats in water with $0.400 \mathrm{~cm}$ of its diameter above water level (Fig. P15.53). Determine the density of the dowel.

Vishal Gupta

Numerade Educator

Problem 54

A light spring of constant $k=90.0 \mathrm{~N} / \mathrm{m}$ rests vertically on a table (Fig. P15.54a). A $2.00$ -g balloon is filled with helium (density $\left.=0.180 \mathrm{~kg} / \mathrm{m}^{3}\right)$ to a volume of $5.00 \mathrm{~m}^{3}$ and is then connected to the spring, causing it to stretch as shown in Figure $\mathrm{P} 15.54 \mathrm{~b}$. Determine the extension distance $L$ when the balloon is in equilibrium.

Surjit Tewari

Numerade Educator

Problem 55

A 1.00-kg beaker containing $2.00 \mathrm{~kg}$ of oil (density = $916.0 \mathrm{~kg} / \mathrm{m}^{3}$ ) rests on a scale. A $2.00-\mathrm{kg}$ block of iron is suspended from a spring scale and completely submerged in the oil, as shown in Figure $\mathrm{P} 15.55 .$ Determine the equilibrium readings of both scales.

Mayukh Banik

Numerade Educator

Problem 56

A beaker of mass $m_{b}$ containing oil of mass $m_{0}$ (density $=\rho_{0}$ ) rests on a scale. A block of iron of mass $m_{\mathrm{Fe}}$ is suspended from a spring scale and completely submerged in the oil, as shown in Figure P15.55. Determine the equilibrium readings of both scales.

Vipender Yadav

Numerade Educator

Problem 57

With reference to Figure 15.7, show that the total torque exerted by the water behind the dam about an axis through $O$ is $\frac{1}{6} \rho g w H^{3}$. Show that the effective line of action of the total force exerted by the water is at a distance $\frac{1}{3} H$ above $O$.

Surjit Tewari

Numerade Educator

Problem 58

In about 1657 Otto von Guericke, inventor of the air pump, evacuated a sphere made of two brass hemispheres. Two teams of eight horses each could pull the hemispheres apart only on some trials, and then "with greatest difficulty," with the resulting sound likened to a cannon firing (Fig. P15.58). (a) Show that the force $F$ required to pull the evacuated hemispheres apart is $\pi R^{2}\left(P_{0}-P\right)$, where $R$ is the radius of the hemispheres and $P$ is the pressure inside the hemispheres, which is much less than $P_{0}$. (b) Determine the force if $P=0.100 P_{0}$ and $R=0.300 \mathrm{~m} .$

Mayukh Banik

Numerade Educator

Problem 59

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

Surjit Tewari

Numerade Educator

Problem 60

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

Surjit Tewari

Numerade Educator

Problem 61

An incompressible, nonviscous fluid initially rests 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 speed of the fluid when all of it is in the horizontal section, as in Figure $\mathrm{P} 15.61 \mathrm{~b}$ ? Assume that the cross-sectional area of the entire pipe is constant.

Surjit Tewari

Numerade Educator

Problem 62

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

Mayukh Banik

Numerade Educator

Problem 63

Figure $\mathrm{P} 15.63$ shows Superman attempting to drink water through a very long straw. With his great strength, he achieves maximum possible suction. The walls of the tubular straw do not collapse. (a) Find the maximum height through which he can lift the water. (b) Still thirsty, the Man of Steel repeats his attempt on the Moon, which has no atmosphere. Find the difference between the water levels inside and outside the straw.

Surjit Tewari

Numerade Educator

Problem 64

Show that the variation of atmospheric pressure with altitude is given by $P=P_{0} e^{-\alpha h}$, where $\alpha=\rho_{0} g / P_{0}, P_{0}$ is atmospheric pressure at some reference level $y=0$, and $\rho_{0}$ is the atmospheric density at this level. Assume that the decrease in atmospheric pressure with increasing altitude is given by Equation 15.4, so that $d P / d y=-\rho g$, and assume that the density of air is proportional to the pressure.

Dominador Tan

Numerade Educator

Problem 65

A cube of ice whose edge measures $20.0 \mathrm{~mm}$ is floating in a glass of ice-cold water with one of its faces parallel to the water's surface. (a) How far below the water surface is the bottom face of the block? (b) Ice-cold ethyl alcohol is gently poured onto the water's 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 laver of ethvl alcohol?

Oliver Mcneely

Numerade Educator

Problem 66

Review Problem. A light balloon filled with helium with a density of $0.180 \mathrm{~kg} / \mathrm{m}^{3}$ is tied to a light string of length $L=3.00 \mathrm{~m}$. The string is tied to the ground, forming an "inverted" simple pendulum, as shown in Figure $\mathrm{P} 15.66 \mathrm{a}$. If the balloon is displaced slightly from its equilibrium position as shown in Figure $\mathrm{P} 15.66 \mathrm{~b}$,
(a) show that the ensuing motion is simple harmonic and (b) determine the period of the motion. Take the density of air to be $1.29 \mathrm{~kg} / \mathrm{m}^{3}$ and ignore any energy loss due to air friction.

Averell Hause

Carnegie Mellon University

Problem 67

The water supply of a building is fed through a main 6.00-cm-diameter pipe. 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 -cm main pipe? (Assume that the faucet is the only "leak" in the building.)

Mayukh Banik

Numerade Educator

Problem 68

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. P15.68). At sufficiently low temperatures, all the spheres float, but as the temperature rises, the spheres sink one after the other. The device is a crude but interesting tool for measuring temperature. Suppose that the tube is filled with ethyl alcohol, whose density is $0.78945 \mathrm{~g} / \mathrm{cm}^{3}$ at $20.0^{\circ} \mathrm{C}$ and decreases to $0.78097 \mathrm{~g} / \mathrm{cm}^{3}$ at $30.0^{\circ} \mathrm{C}$. (a) If one of the spheres has a radius of $1.000 \mathrm{~cm}$ and is in equilibrium halfway up the tube at $20.0^{\circ} \mathrm{C}$, determine its mass.
(b) When the temperature increases to $30.0^{\circ} \mathrm{C}$, what mass must a second sphere of the same radius have to be in equilibrium at the halfway point? (c) At $30.0^{\circ} \mathrm{C}$ the first sphere has fallen to the bottom of the tube. What upward force does the bottom of the tube exert on this sphere?

Prashant Bana

Numerade Educator

Problem 69

A U-tube open at both ends is partially filled with water (Fig. P15.69a). Oil having a density of $750 \mathrm{~kg} / \mathrm{m}^{3}$ is then poured into the right arm and forms a column $L=5.00 \mathrm{~cm}$ in height (Fig. P15.69b). (a) Determine the difference $h$ in the heights of the two liquid surfaces. (b) The right arm is 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. $\mathrm{P} 15.69 \mathrm{c}$ ). Determine the speed of the air being blown across the left arm. (Take the density of air as $\left.1.29 \mathrm{~kg} / \mathrm{m}^{3} .\right)$

Surjit Tewari

Numerade Educator