Chapter 13

Salt water has a greater density than freshwater. A boat floats in both freshwater and salt water. The buoyant force on the boat in salt water is that in freshwater. a) equal to b) smaller than c) larger than

You fill a tall glass with ice and then add water to the level of the glass's rim, so some fraction of the ice floats above the rim. When the ice melts, what happens to the water level? (Neglect evaporation, and assume that the ice and water remain at \(0^{\circ} \mathrm{C}\) during the melting process.) a) The water overflows the rim. b) The water level drops below the rim. c) The water level stays at the top of the rim. d) It depends on the difference in density between water and ice.

You are in a boat filled with large rocks in the middle of a small pond. You begin to drop the rocks into the water. What happens to the water level of the pond? a) It rises. d) It rises momentarily and then b) It falls. falls when the rocks hit bottom. c) It doesn't change. e) There is not enough information to say.

In a horizontal water pipe that narrows to a smaller radius, the velocity of the water in the section with the smaller radius will be larger. What happens to the pressure? a) The pressure will be the same in both the wider and narrower sections of the pipe. b) The pressure will be higher in the narrower section of the pipe. c) The pressure will be higher in the wider section of the pipe d) It is impossible to tell.

Many altimeters determine altitude changes by measuring changes in the air pressure. An altimeter that is designed to be able to detect altitude changes of \(100 \mathrm{~m}\) near sea level should be able to detect pressure changes of a) approximately \(1 \mathrm{~Pa}\). d) approximately \(1 \mathrm{kPa}\). b) approximately 10 Pa. e) approximately \(10 \mathrm{kPa}\). c) approximately \(100 \mathrm{~Pa}\).

Which of the following assumptions is not made in the derivation of Bernoulli's Equation? a) Streamlines do not cross. c) There is negligible friction. b) There is negligible d) There is no turbulence. viscosity. e) There is negligible gravity.

A beaker is filled with water to the rim. Gently placing a plastic toy duck in the beaker causes some of the water to spill out. The weight of the beaker with the duck floating in it is a) greater than the weight before adding the duck. b) less than the weight before adding the duck. c) the same as the weight before adding the duck. d) greater or less than the weight before the duck was added, depending on the weight of the duck.

You know from experience that if a car you are riding in suddenly stops, heavy objects in the rear of the car move toward the front. Why does a helium-filled balloon in such a situation move, instead, toward the rear of the car?

In what direction does a force due to water flowing from a showerhead act on a shower curtain, inward toward the shower or outward? Explain.

Given two springs of identical size and shape, one made of steel and the other made of aluminum, which has the higher spring constant? Why? Does the difference depend more on the shear modulus or the bulk modulus of the material?

Analytic balances are calibrated to give correct mass values for such items as steel objects of density \(\rho_{s}=\) \(8000.00 \mathrm{~kg} / \mathrm{m}^{3}\). The calibration compensates for the buoyant force arising because the measurements are made in air, of density \(\rho_{\mathrm{a}}=1.205 \mathrm{~kg} / \mathrm{m}^{3}\). What compensation must be made to measure the masses of objects of a different material, of density \(\rho\) ? Does the buoyant force of air matter?

If you turn on the faucet in the bathroom sink, you will observe that the stream seems to narrow from the point at which it leaves the spigot to the point at which it hits the bottom of the sink. Why does this occur?

In many problems involving application of Newton's Second Law to the motion of solid objects, friction is neglected for the sake of making the solution easier. The counterpart of friction between solids is viscosity of liquids. Do problems involving fluid flow become simpler if viscosity is neglected? Explain.

You have two identical silver spheres and two unknown fluids, \(A\) and \(B\). You place one sphere in fluid \(A\), and it sinks; you place the other sphere in fluid \(\mathrm{B}\), and it floats. What can you conclude about the buoyant force of fluid \(\mathrm{A}\) versus that of fluid \(\mathrm{B} ?\)

Water flows from a circular faucet opening of radius \(r_{0}\) directed vertically downward, at speed \(v_{0}\). As the stream of water falls, it narrows. Find an expression for the radius of the stream as a function of distance fallen, \(r(y),\) where \(y\) is measured downward from the opening. Neglect the eventual breakup of the stream into droplets, and any resistance due to drag or viscosity.

Find the minimum diameter of a \(50.0-\mathrm{m}\) -long nylon string that will stretch no more than \(1.00 \mathrm{~cm}\) when a load of \(70.0 \mathrm{~kg}\) is suspended from its lower end. Assume that \(Y_{\text {nylon }}=3.51 \cdot 10^{8} \mathrm{~N} / \mathrm{m}^{2}\)

Blood pressure is usually reported in millimeters of mercury (mmHg) or the height of a column of mercury producing the same pressure value. Typical values for an adult human are \(130 / 80\); the first value is the systolic pressure, during the contraction of the ventricles of the heart, and the second is the diastolic pressure, during the contraction of the auricles of the heart. The head of an adult male giraffe is \(6.0 \mathrm{~m}\) above the ground; the giraffe's heart is \(2.0 \mathrm{~m}\) above the ground. What is the minimum systolic pressure (in \(\mathrm{mmHg}\) ) required at the heart to drive blood to the head (neglect the additional pressure required to overcome the effects of viscosity)? The density of giraffe blood is \(1.00 \mathrm{~g} / \mathrm{cm}^{3},\) and that of mercury is \(13.6 \mathrm{~g} / \mathrm{cm}^{3}\)

A scuba diver must decompress after a deep dive to allow excess nitrogen to exit safely from his bloodstream. The length of time required for decompression depends on the total change in pressure that the diver experienced. Find this total change in pressure for a diver who starts at a depth of \(d=20.0 \mathrm{~m}\) in the ocean (density of seawater \(\left.=1024 \mathrm{~kg} / \mathrm{m}^{3}\right)\) and then travels aboard a small plane (with an unpressurized cabin) that rises to an altitude of \(h=5000 . \mathrm{m}\) above sea level.

The atmosphere of Mars exerts a pressure of only 600\. Pa on the surface and has a density of only \(0.0200 \mathrm{~kg} / \mathrm{m}^{3}\). a) What is the thickness of the Martian atmosphere, assuming the boundary between atmosphere and outer space to be the point where atmospheric pressure drops to \(0.0100 \%\) of its yalue at surface level? b) What is the atmospheric pressure at the bottom of Mars's Hellas Planitia canyon, at a depth of \(7.00 \mathrm{~km} ?\) c) What is the atmospheric pressure at the top of Mars's Olympus Mons volcano, at a height of \(27.0 \mathrm{~km} ?\) d) Compare the relative change in air pressure, \(\Delta p / p\), between these two points on Mars and between the equivalent extremes on Earth-the Dead Sea shore, at \(400 . \mathrm{m}\) below sea level, and Mount Everest, at an altitude of \(8850 \mathrm{~m}\).

A sealed vertical cylinder of radius \(R\) and height \(h=0.60 \mathrm{~m}\) is initially filled halfway with water, and the upper half is filled with air. The air is initially at standard atmospheric pressure, \(p_{0}=1.01 \cdot 10^{5} \mathrm{~Pa}\). A small valve at the bottom of the cylinder is opened, and water flows out of the cylinder until the reduced pressure of the air in the upper part of the cylinder prevents any further water from escaping. By what distance is the depth of the water lowered? (Assume that the temperature of water and air do not change and that no air leaks into the cylinder.)

A square pool with \(100 .-\mathrm{m}\) -long sides is created in a concrete parking lot. The walls are concrete \(50.0 \mathrm{~cm}\) thick and have a density of \(2.50 \mathrm{~g} / \mathrm{cm}^{3}\). The coefficient of static friction between the walls and the parking lot is \(0.450 .\) What is the maximum possible depth of the pool?

The calculation of atmospheric pressure at the summit of Mount Everest carried out in Example 13.3 used the model known as the isothermal atmosphere, in which gas pressure is proportional to density: \(p=\gamma \rho\), with \(\gamma\) constant. Consider a spherical cloud of gas supporting itself under its own gravitation and following this model. a) Write the equation of hydrostatic equilibrium for the cloud, in terms of the gas density as a function of radius, \(\rho(r) .\) b) Show that \(\rho(r)=A / r^{2}\) is a solution of this equation, for an appropriate choice of constant \(A\). Explain why this solution is not suitable as a model of a star.

A racquetball with a diameter of \(5.6 \mathrm{~cm}\) and a mass of \(42 \mathrm{~g}\) is cut in half to make a boat for American pennies made after \(1982 .\) The mass and volume of an American penny made after 1982 are \(2.5 \mathrm{~g}\) and \(0.36 \mathrm{~cm}^{3} .\) How many pennies can be placed in the racquetball boat without sinking it?

A supertanker filled with oil has a total mass of \(10.2 \cdot 10^{8} \mathrm{~kg}\). If the dimensions of the ship are those of a rectangular box \(250 . \mathrm{m}\) long, \(80.0 \mathrm{~m}\) wide, and \(80.0 \mathrm{~m}\) high, determine how far the bottom of the ship is below sea level \(\left(\rho_{\mathrm{sea}}=1020 \mathrm{~kg} / \mathrm{m}^{3}\right)\)

A box with a volume \(V=0.0500 \mathrm{~m}^{3}\) lies at the bottom of a lake whose water has a density of \(1.00 \cdot 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\). How much force is required to lift the box, if the mass of the box is (a) \(1000 . \mathrm{kg},\) (b) \(100 . \mathrm{kg},\) and \((\mathrm{c}) 55.0 \mathrm{~kg} ?\)

A block of cherry wood that is \(20.0 \mathrm{~cm}\) long, \(10.0 \mathrm{~cm}\) wide, and \(2.00 \mathrm{~cm}\) thick has a density of \(800 . \mathrm{kg} / \mathrm{m}^{3}\). What is the volume of a piece of iron that, if glued to the bottom of the block makes the block float in water with its top just at the surface of the water? The density of iron is \(7860 \mathrm{~kg} / \mathrm{m}^{3},\) and the density of water is \(1000 . \mathrm{kg} / \mathrm{m}^{3}\).

The average density of the human body is \(985 \mathrm{~kg} / \mathrm{m}^{3}\) and the typical density of seawater is about \(1020 \mathrm{~kg} / \mathrm{m}^{3}\) a) Draw a free-body diagram of a human body floating in seawater and determine what percentage of the body's volume is submerged. b) The average density of the human body, after maximum inhalation of air, changes to \(945 \mathrm{~kg} / \mathrm{m}^{3}\). As a person floating in seawater inhales and exhales slowly, what percentage of his volume moves up out of and down into the water? c) The Dead Sea (a saltwater lake between Israel and Jordan ) is the world's saltiest large body of water. Its average salt content is more than six times that of typical seawater, which explains why there is no plant and animal life in it. Two-thirds of the volume of the body of a person floating in the Dead Sea is observed to be submerged. Determine the density (in \(\mathrm{kg} / \mathrm{m}^{3}\) ) of the seawater in the Dead Sea.

A tourist of mass \(60.0 \mathrm{~kg}\) notices a chest with a short chain attached to it at the bottom of the ocean. Imagining the riches it could contain, he decides to dive for the chest. He inhales fully, thus setting his average body density to \(945 \mathrm{~kg} / \mathrm{m}^{3}\), jumps into the ocean (with saltwater density = \(1020 \mathrm{~kg} / \mathrm{m}^{3}\) ), grabs the chain, and tries to pull the chest to the surface. Unfortunately, the chest is too heavy and will not move. Assume that the man does not touch the bottom. a) Draw the man's free-body diagram, and determine the tension on the chain. b) What mass (in kg) has a weight that is equivalent to the tension force in part (a)? c) After realizing he cannot free the chest, the tourist releases the chain. What is his upward acceleration (assuming that he simply allows the buoyant force to lift him up to the surface)?

A very large balloon with mass \(M=10.0 \mathrm{~kg}\) is inflated to a volume of \(20.0 \mathrm{~m}^{3}\) using a gas of density \(\rho_{\text {eas }}=\) \(0.20 \mathrm{~kg} / \mathrm{m}^{3}\). What is the maximum mass \(m\) that can be tied to the balloon using a \(2.00 \mathrm{~kg}\) piece of rope without the balloon falling to the ground? (Assume that the density of air is \(1.30 \mathrm{~kg} / \mathrm{m}^{3}\) and that the volume of the gas is equal to the volume of the inflated balloon).

The Hindenburg, the German zeppelin that caught fire in 1937 while docking in Lakehurst, New Jersey, was a rigid duralumin-frame balloon filled with \(2.000 \cdot 10^{5} \mathrm{~m}^{3}\) of hydrogen. The Hindenburg's useful lift (beyond the weight of the zeppelin structure itself) is reported to have been \(1.099 \cdot 10^{6} \mathrm{~N}(\) or \(247,000 \mathrm{lb}) .\) Use \(\rho_{\text {air }}=1.205 \mathrm{~kg} / \mathrm{m}^{3}, \rho_{\mathrm{H}}=\) \(0.08988 \mathrm{~kg} / \mathrm{m}^{3}\) and \(\rho_{\mathrm{He}}=0.1786 \mathrm{~kg} / \mathrm{m}^{3}\) a) Calculate the weight of the zeppelin structure (without the hydrogen gas). b) Compare the useful lift of the (highly flammable) hydrogen-filled Hindenburg with the useful lift the Hindenburg would have had had it been filled with (nonflammable) helium, as originally planned.

Brass weights are used to weigh an aluminum object on an analytical balance. The weighing is done one time in dry air and another time in humid air, with a water vapor pressure of \(P_{\mathrm{h}}=2.00 \cdot 10^{3} \mathrm{~Pa}\). The total atmospheric pressure \(\left(P=1.00 \cdot 10^{5} \mathrm{~Pa}\right)\) and the temperature \(\left(T=20.0^{\circ} \mathrm{C}\right)\) are the same in both cases. What should the mass of the object be to be able to notice a difference in the balance readings, provided the balance's sensitivity is \(m_{0}=0.100 \mathrm{mg}\) ? (The density of aluminum is \(\rho_{\mathrm{A}}=2.70 \cdot 10^{3} \mathrm{~kg} / \mathrm{m}^{3} ;\) the density of brass is \(\left.\rho_{\mathrm{B}}=8.50 \cdot 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\right)\)

A fountain sends water to a height of \(100 . \mathrm{m}\). What is the difference between the pressure of the water just before it is released upward and the atmospheric pressure?

An open-topped tank completely filled with water has a release valve near its bottom. The valve is \(1.0 \mathrm{~m}\) below the water surface. Water is released from the valve to power a turbine, which generates electricity. The area of the top of the tank, \(A_{\mathrm{p}}\) is 10 times the cross-sectional area, \(A_{\mathrm{y}}\) of the valve opening. Calculate the speed of the water as it exits the valve. Neglect friction and viscosity, In addition, calculate the speed of a drop of water released from rest at \(h=1.0 \mathrm{~m}\) when it reaches the elevation of the valve, Compare the two speeds.

A water-powered backup sump pump uses tap water at a pressure of \(3.00 \mathrm{~atm}\left(p_{1}=3 p_{\mathrm{atm}}=\right.\) \(3.03 \cdot 10^{5} \mathrm{~Pa}\) ) to pump water out of a well, as shown in the figure \(\left(p_{\text {well }}=p_{\text {ttm }}\right)\). This system allows water to be pumped out of a basement sump well when the electric pump stops working during an electrical power outage. Using water to pump water may sound strange at first, but these pumps are quite efficient, typically pumping out \(2.00 \mathrm{~L}\) of well water for every \(1.00 \mathrm{~L}\) of pressurized tap water. The supply water moves to the right in a large pipe with cross-sectional area \(A_{1}\) at a speed \(v_{1}=2.05 \mathrm{~m} / \mathrm{s}\). The water then flows into a pipe of smaller diameter with a cross-sectional area that is ten times smaller \(\left(A_{2}=A_{1} / 10\right)\). a) What is the speed \(v_{2}\) of the water in the smaller pipe, with area \(A_{2} ?\) b) What is the pressure \(p_{2}\) of the water in the smaller pipe, with area \(A_{2} ?\) c) The pump is designed so that the vertical pipe, with cross-sectional area \(A_{3}\), that leads to the well water also has a pressure of \(p_{2}\) at its top. What is the maximum height, \(h,\) of the column of water that the pump can support (and therefore act on ) in the vertical pipe?

A basketball of circumference \(75.5 \mathrm{~cm}\) and mass \(598 \mathrm{~g}\) is forced to the bottom of a swimming pool and then released. After initially accelerating upward, it rises at a constant velocity, a) Calculate the buoyant force on the basketball. b) Calculate the drag force the basketball experiences while it is moving upward at constant velocity.

Calculate the ratio of the lifting powers of helium (He) gas and hydrogen (H \(_{2}\) ) gas under identical circumstances. Assume that the molar mass of air is \(29.5 \mathrm{~g} / \mathrm{mol}\).

A water pipe narrows from a radius of \(r_{1}=5.00 \mathrm{~cm}\) to a radius of \(r_{2}=2.00 \mathrm{~cm} .\) If the speed of the water in the wider part of the pipe is \(2.00 \mathrm{~m} / \mathrm{s}\), what is the speed of the water in the narrower part?

Donald Duck and his nephews manage to sink Uncle Scrooge's yacht \((m=4500 \mathrm{~kg}),\) which is made of steel \(\left(\rho=7800 \mathrm{~kg} / \mathrm{m}^{3}\right)\). In typical comic-book fashion, they decide to raise the yacht by filling it with ping-pong balls. A pingpong ball has a mass of \(2.7 \mathrm{~g}\) and a volume of \(3.35 \cdot 10^{-5} \mathrm{~m}^{3}\) a) What is the buoyant force on one ping-pong ball in water? b) How many balls are required to float the ship?

A wooden block floating in seawater has two thirds of its volume submerged. When the block is placed in mineral oil, \(80.0 \%\) of its volume is submerged. Find the density of the (a) wooden block, and (b) the mineral oil.

An approximately round tendon that has an average diameter of \(8.5 \mathrm{~mm}\) and is \(15 \mathrm{~cm}\) long is found to stretch \(3.7 \mathrm{~mm}\) when acted on by a force of \(13.4 \mathrm{~N}\). Calculate Young's modulus for the tendon.

An airplane is moving through the air at a velocity \(v=200 . \mathrm{m} / \mathrm{s} .\) Streamlines just over the top of the wing are compressed to \(80.0 \%\) of their original area, and those under the wing are not compressed at all. a) Determine the velocity of the air just over the wing. b) Find the difference in the pressure between the air just over the wing, \(P\), and that under the wing, \(P\). c) Find the net upward force on both wings due to the pressure difference, if the area of the wing is \(40.0 \mathrm{~m}^{2}\) and the density of the air is \(1.30 \mathrm{~kg} / \mathrm{m}^{3}\).

Water of density \(998.2 \mathrm{~kg} / \mathrm{m}^{3}\) is moving at negligible speed under a pressure of \(101.3 \mathrm{kPa}\) but is then accelerated to high speed by the blades of a spinning propeller. The vapor pressure of the water at the initial temperature of \(20.0^{\circ} \mathrm{C}\) is \(2.3388 \mathrm{kPa}\). At what flow speed will the water begin to boil? This effect, known as cavitation, limits the performance of propellers in water

In many locations, such as Lake Washington in Seattle, floating bridges are preferable to conventional bridges. Such a bridge can be constructed out of concrete pontoons, which are essentially concrete boxes filled with air, Styrofoam, or another extremely low-density material. Suppose a floating bridge pontoon is constructed out of concrete and Styrofoam, which have densities of \(2200 \mathrm{~kg} / \mathrm{m}^{3}\) and \(50.0 \mathrm{~kg} / \mathrm{m}^{3}\). What must the volume ratio of concrete to Styrofoam be if the pontoon is to float with \(35.0 \%\) of its overall volume above water?

A \(1.0-g\) balloon is filled with helium gas. When a mass of \(4.0 \mathrm{~g}\) is attached to the balloon, the combined mass hangs in static equilibrium in midair. Assuming that the balloon is spherical, what is its diameter?