Chapter 14

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?

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?

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 \(1.1 \mathrm{~cm}\).

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.)

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?

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?

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.

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 \(0.87 \mathrm{~kg} / \mathrm{m}^{3}\).

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?

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.)

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 \(1.5 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\) )?

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.)

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 \(1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\).

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 \(1.5 \mathrm{~m} ?\)

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.

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 \(1.00 \mathrm{~atm} .\)

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.)

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 \(4.00 \mathrm{~m} ?\)

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.

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 \(1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\).

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.)

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)?

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?

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?

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.

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?

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?

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?

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 \(800 \mathrm{~kg} / \mathrm{m}^{3}\).

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.

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.

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.)

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?

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?

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.

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 \(7.87 \mathrm{~g} / \mathrm{cm}^{3}\).

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.)

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?

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?

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?

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?

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 \(26 \mathrm{~L} / \mathrm{min}\) ?

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 \(1.0 \mathrm{~atm}\) ?

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?

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?

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?

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?

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?

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?

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?