A balloon contains gas of density rg and is to lift a mass M, including the balloon but

Why do your ears pop when you drive up a mountain?

Commercial aircraft cabins are usually pressurized to the pressure of the atmosphere at about 2 km above sea level. Why dont you feel the lower pressure on your entire body?

Water pressure at the bottom of the ocean arises from the weight of the overlying water. Does this mean that the water exerts pressure only in the downward direction? Explain.

The three containers in Fig. 15.22 are filled to the same level and are open to the atmosphere. How do the pressures at the bottoms of the three containers compare? Figure 15.22 For Thought and Discussion 4.

Why is it easier to float in the ocean than in fresh water?

Figure 15.23 shows a cork suspended from the bottom of a sealed container of water. The container is on a turntable rotating about a vertical axis, as shown. Explain the position of the cork. v Figure 15.23 For Thought and Discussion 6.

Meteorologists in the United States usually report barometer readings in inches. What are they talking about?

A mountain stream, frothy with entrained air bubbles, presents a serious hazard to hikers who fall into it, for they may sink in the stream where they would float in calm water. Why?

Why are dams thicker at the bottom than at the top?

. Its not possible to breathe through a snorkel from a depth greater than a meter or so. Why not?

A helium-filled balloon stops rising long before it reaches the top of the atmosphere, but a cork released from the bottom of a lake rises all the way to the surface. Why the difference?

A barge filled with steel beams overturns in a lake, spilling its cargo. Does the water level in the lake rise, fall, or remain the same?

Why do airplanes take off into the wind?

Is the flow speed behind a wind turbine greater or less than in front? Is the pressure behind the turbine higher or lower than in front? Is this a violation of Bernoullis principle? Explain.

The density of molasses is 1600 kg/m3 . Find the mass of the molasses in a 0.75-L jar.

Atomic nuclei have densities around 1017 kg/m3 , while waters density is 103 kg/m3 . Roughly what fraction of waters volume is not empty space?

Compressed air with mass 8.8 kg is stored in a 0.050@m3 cylinder. (a) Whats the density of the compressed air? (b) What volume would the same gas occupy at a typical atmospheric density of 1.2 kg/m3 ?

The pressure unit torr is defined as the pressure that will support a column of mercury 1 mm high. Meteorologists often give barometric pressure in inches of mercury, defined analogously. Express each of these in SI units. (Hint: Mercurys density is 1.36*104 kg/m3 .)

Measurement of small pressure differencesfor example, between the interior of a chimney and the ambient atmosphereis often given in inches of water, where 1 in. of water is the pressure that will support a 1-in.-high water column. Express this pressure difference in SI units.

Whats the weight of a column of air with cross-sectional area 1 m2 extending from Earths surface to the top of the atmosphere?

A 4680-kg circus elephant balances on one foot. If the foot is a circle 42.6 cm in diameter, what pressure does it exert on the ground?

You unbend a paper clip made from 1.5-mm-diameter wire and push the end against the wall. What force must you apply to give a pressure of 120 atm?

Whats the density of a fluid whose pressure increases at the rate of 100 kPa for every 6.0 m of depth?

A research submarine can withstand an external pressure of 62 MPa when its internal pressure is 101 kPa. How deep can it dive?

Scuba equipment provides a diver with air at the same pressure as the surrounding water. But at pressures higher than about 1 MPa, the nitrogen in air becomes dangerously narcotic. At what depth does nitrogen narcosis become a hazard?

A vertical tube open at the top contains 5.0 cm of oil with density 0.82 g/cm3 , floating on 5.0 cm of water. Find the gauge pressure at the bottom of the tube.

A child attempts to drink water through a 36-cm-long straw but finds that the water rises only 25 cm. By how much has the child reduced the pressure in her mouth below atmospheric pressure?

Barometric pressure in the eye of a hurricane is 0.91 atm (27.2 in. of mercury). How does the level of the ocean surface under the eye compare with the level under a distant fair-weather region where the pressure is 1.0 atm?

On land, the most massive concrete block you can carry is 25 kg. Given concretes 2200@kg/m3 density, how massive a block could you carry underwater?

A 5.4-g jewel has apparent weight 32 mN when submerged in water. Could the jewel be a diamond (density 3.51 g/cm3 )?

Styrofoams density is 160 kg/m3 . What percent error is introduced by weighing a Styrofoam block in air (density 1.2 kg/m3 ), which exerts an upward buoyancy force, rather than in vacuum?

A steel drum has volume 0.23 m3 and mass 16 kg. Will it float in water when filled with (a) water or (b) gasoline (density 860 kg/m3 )? Sections 15.4 and 15.5 Fluid Dynamics and Applications

Water flows through a 2.5-cm-diameter pipe at 1.8 m/s. If the pipe narrows to 2.0-cm diameter, whats the flow speed in the constriction?

Show that pressure has the units of energy density.

A typical mass flow rate for the Mississippi River is 1.8*107 kg/s. Find (a) the volume flow rate and (b) the flow speed in a region where the river is 2.0 km wide and averages 6.1 m deep.

A fire hose 10 cm in diameter delivers water at 15 kg/s. The hose terminates in a 2.5-cm-diameter nozzle. What are the flow speeds (a) in the hose and (b) at the nozzle?

A typical human aorta, the main artery from the heart, is 1.8 cm in diameter and carries blood at 35 cm/s. Find the flow speed around a clot that reduces the flow area by 80%.

When a couple with total mass 120 kg lies on a water bed, pressure in the bed increases by 4700 Pa. What surface area of the two bodies is in contact with the bed?

A fully loaded Volvo station wagon has mass 1950 kg. If each of its four tires is inflated to a gauge pressure of 230 kPa, whats the total tire area in contact with the road?

Youre stuck in the exit row on a long flight, and you suddenly worry that your seatmate, whos next to the window, might pull the emergency window inward while youre in flight. The window measures 40 cm by 55 cm. Cabin pressure is 0.77 atm, and atmospheric pressure at the planes altitude is 0.22 atm. Should you worry?

A vertical tube 1.0 cm in diameter and open at the top contains 5.0 g of oil (density 0.82 g/cm3 ) floating on 5.0 g of water. Find the gauge pressure (a) at the oilwater interface and (b) at the bottom.

Dam breaks present a serious risk of widespread property damage and loss of life. Youre asked to assess a 1500-m-wide dam holding back a lake 95 m deep. The dam was built to withstand a force of 100 GN, which is supposed to be at least 50% over the force it actually experiences. Should the dam be reinforced? (Hint: Youll need your calculus skills.)

A U-shaped tube open at both ends contains water and a quantity of oil occupying a 2.0-cm length of the tube, as shown in Fig. 15.24. If the oils density is 82% of waters, whats the height difference h? Water Oil 2.0 cm h = ? Figure 15.24 Problem 43

Youre a robotics engineer designing a hydraulic system to move a robotic arm. The hydraulic cylinder that drives the arm has diameter 5.0 cm and can exert a maximum force of 5.6 kN. Hydraulic tubing comes rated in multiples of 1/2 MPa, and for safety, youre to specify tubing capable of withstanding 50% greater pressure than it will ever experience in use. What pressure rating do you specify?

A garage lift has a 45-cm-diameter piston supporting the load. Compressed air with maximum pressure 500 kPa is applied to a small piston at the other end of the hydraulic system. Whats the maximum mass the lift can support?

Archimedes purportedly used his principle to verify that the kings crown was pure gold by weighing the crown submerged in water. Suppose the crowns actual weight was 25.0 N. What would be its apparent weight if it were made of (a) pure gold and (b) 75% gold and 25% silver, by volume? The densities of gold, silver, and water are 19.3 g/cm3 , 10.5 g/cm3 , and 1.00 g/cm3 , respectively.

Youre testifying in a drunk-driving case for which a blood alcohol measurement is unavailable. The accused weighs 140 lb, and would be legally impaired after consuming 36 oz of beer. The accused was observed at a beach party where a keg with interior diameter 40 cm was floating in the lake to keep it cool. After the accuseds drinking stint, the keg floated 1.2 cm higher than before. Beers density is essentially that of water. Does your testimony help or hurt the accuseds case?

A glass beaker measures 14 cm high by 5.0 cm in diameter. Empty, it floats in water with one-third of its height submerged. How many 12-g rocks can be placed in the beaker before it sinks?

A typical supertanker has mass 2.0*106 kg and carries twice that much oil. If 9.0 m of the ship is submerged when its empty, whats the minimum water depth needed for it to navigate when full? Assume the sides of the ship are vertical.

A balloon contains gas of density rg and is to lift a mass M, including the balloon but not the gas. Show that the minimum mass of gas required is mg = Mrg/1ra - rg2, where ra is the atmospheric density.

(a) How much helium (density 0.18 kg/m3 ) is needed to lift a balloon carrying two people, if the total mass of people, basket, and balloon (but not gas) is 280 kg? (b) Repeat for a hot-air balloon whose air density is 10% less than that of the surrounding atmosphere.

A 55-kg swimmer climbs onto a Styrofoam block of density 160 kg/m3 . If the water level comes right to the top of the Styrofoam, whats the blocks volume?

If the blood pressure in the unobstructed artery of Exercise 37 is 16 kPa gauge (about 120 mm of mercury, the unit commonly reported by doctors), what will it be at the clot? (Note: Bloods density is 1.06 g/cm3 .)

Youre a consultant for maple syrup producers. They tap maple trees and collect sap with plastic tubing that connects to a common pipe delivering sap to an evaporator. There its boiled to produce thick, tasty syrup. The system can be modeled as a pipe with one end, of cross-sectional area A, exposed to atmospheric pressure. The pipe drops through a vertical distance h1 while its area decreases to A/2, as shown in Fig. 15.25. A small vertical glass tube extends from the lower portion, as shown, and is open to atmospheric pressure. Youre asked to provide a formula for the volume flow rate of the sap as a function of the height h2 of sap in the tube. h2 h1 A 1 2A Figure 15.25 Problem 54

The water in a garden hose is at 140-kPa gauge pressure and is moving at negligible speed. The hose terminates in a sprinkler consisting of many small holes. Find the maximum height reached by the water emerging from the holes.

The venturi flowmeter shown in Fig. 15.26 is used to measure the flow rate of water in a solar collector system. The flowmeter is inserted in a pipe with diameter 1.9 cm; at the venturi the diameter is 0.64 cm. The manometer tube contains oil with density 0.82 times that of water. If the difference in oil levels on the two sides of the manometer tube is 1.4 cm, whats the volume flow rate? Oil Water Flow Figure 15.26 Problem 56

A 1.0-cm-diameter venturi flowmeter is inserted in a 2.0-cmdiameter pipe carrying water (density 1000 kg/m3 ). Find (a) the flow speed in the pipe and (b) the volume flow rate if the pressure difference between venturi and unconstricted pipe is 17 kPa.

A balloons mass is 1.6 g when its empty. Its inflated with helium (density 0.18 kg/m3 ) to form a sphere 28 cm in diameter. How many 0.63-g paper clips can you hang from the balloon before it loses buoyancy?

Blood with density 1.06 g/cm3 and 10-kPa gauge pressure flows through an artery at 30 cm/s. It encounters a plaque deposit where the pressure drops by 5%. What fraction of the arterys area is obstructed?

A venturi flowmeter in an oil pipeline has radius half that of the pipe. Oil flows in the unconstricted pipe at 1.9 m/s. If the pressure difference between unconstricted flow and venturi is 16 kPa, whats the oils density?

A drinking straw 20 cm long and 3.0 mm in diameter stands vertically in a cup of juice 8.0 cm in diameter. A section of straw 6.5 cm long extends above the juice. A child sucks on the straw, and the juice level begins dropping at 2.0 mm/s. (a) By how much does the pressure in the childs mouth differ from atmospheric pressure? (b) Whats the greatest height above the water surface from which the child could drink, assuming this same mouth pressure?

In 2012, film producer James Cameron (Terminator, Titanic, Avatar) rode his submersible Deepsea Challenger to the bottom of the 11-km-deep Marianas Trench, the deepest spot in Earths oceans. Cameron could barely fit into Deepsea Challengers crew compartment, a steel sphere with inside diameter 109 cm and walls 6.4 cm thick. Find the total pressure force exerted on the sphere at the bottom of the trench. (The total force is the sum of all pressure forces without regard to direction; its not the same as the buoyancy force, which is the net pressure forcea vectorial sum.)

A probe descending through Mars atmosphere records pressure as a function of altitude; the data are in the table below. Plot the natural logarithm of the pressure versus altitude and fit a line to your plotted points. Marss atmospheric pressure is governed by the same equation that describes Earths; see Problem 70. Use your fitted line, in connection with that equation, to determine (a) Marss surface pressure and (b) the scale height h0. Altitude (km) 10 20 30 40 50 60 Pressure (Pa) 242 98.7 37.6 16.2 7.21 2.38

Water emerges from a faucet of diameter d0 in steady, nearvertical flow with speed v0. Show that the diameter of the falling water column is given by d = d0 3v 2 0 /1v 2 0 + 2gh24 1/4, where h is the distance below the faucet (Fig. 15.27). Figure 15.27 Problem 64

Assuming normal atmospheric pressure, how massive an object can a 5.0-cm-diameter suction cup support on a vertical wall, if the coefficient of friction between cup and wall is 0.72?

Figure 15.28 shows a simplified diagram of a Pitot tube, used for measuring aircraft speeds. The tube is mounted on the aircraft with opening A at right angles to the flow and opening B pointing into the flow. The gauge prevents airflow through the tube. Use Bernoullis equation to show that the planes speed relative to the air is v = 12 p/r, where p is the pressure difference between the tubes and r is the density of air. (Hint: The flow must be stopped at B, but continues past A with its normal speed.) Pressure difference indicator Fuselage Airow A B Figure 15.28 Problem 66

At a hearing on a proposed wind farm, a wind-energy advocate says an installation of 800 turbines, with blade diameter 95 m, could displace a 1-GW nuclear power plant. Youre asked if thats really possible. How do you answer, given an average wind speed of 12 m/s and a turbine power output that averages 30% of the theoretical maximum?

A pencil is weighted so it floats vertically with length L submerged. Its pushed vertically downward without being totally submerged, then released. Show that it undergoes simple harmonic motion with period T = 2p1L/g.

A can of height h and cross-sectional area A0 is initially full of water. A small hole of area A1 66 A0 is cut in the bottom of the can. Find an expression for the time it takes all the water to drain from the can. (Hint: Call the water depth y, use the continuity equation, and integrate.)

Density and pressure in Earths atmosphere are proportional: r = p/h0g, where h0 = 8.2 km is a constant called the scale height and g is the gravitational acceleration. (a) Integrate Equation 15.2 for this case to show that atmospheric pressure as a function of height h above the surface is given by p = p0 e -h/h0 , where p0 is the surface pressure. (b) At what height will the pressure have dropped to half its surface value?

(a) Use the result of Problem 70 to express Earths atmospheric density as a function of height. (b) Use your result from (a) to find the height below which half of Earths atmospheric mass lies (this will require integration).

A circular pan of liquid with density r is centered on a horizontal turntable rotating with angular speed v, as shown in Fig. 15.29. Atmospheric pressure is pa. Find expressions for (a) the pressure at the bottom of the pan and (b) the height of the liquid surface, both as functions of the distance r from the axis, given that the height at the center is h0. v h0 Figure 15.29 Problem 72

A solid sphere of radius R and mass M has density r that varies with distance r from the center: r = r0 e-r/R. Find an expression for the central density r0 in terms of M and R.

The difference in air pressure between the inside and outside of a ball is a constant p. Show by direct integration that the net pressure force on one hemisphere is pR2 p, with R the balls radius.

Find the torque that the water exerts about the bottom edge of the dam in Problem 42.

One vertical wall of a swimming pool is a regular trapezoid, with its bottom 15 m long and its top 22 m long. The pool is 3.3 m deep, and its full to the brim with water. Find the pressure force the water exerts on this side of the pool.

Youre a private investigator assisting a large food manufacturer in tracking down counterfeit salad dressing. The genuine dressing is by volume one part vinegar (density 1.0 g/cm3 ) to three parts olive oil (density 0.92 g/cm3 ). The counterfeit dressing is diluted with water (density 1.0 g/cm3 ). You measure the density of a dressing sample and find it to be 0.97 g/cm3 . Has the dressing been altered?

A plumber comes to your ancient apartment building where you have a part-time job as caretaker. Hes checking the hot-water heating system, and notes that the water pressure in the basement is 18 psi. He asks, How high is the building? Three stories, each about 11 feet, you reply. OK, about 33 feet, he says, pausing to do some calculations in his head. The pressure is fine, he declares. On what basis did he come to that conclusion?

Your class in naval architecture is working on the design for a ship with a V-shaped cross section, as shown in Fig. 15.30. The ship has total length L and keel-to-deck height h0. When empty, the distance from waterline to keel is h1. Youre asked for the maximum load the ship can carry below deck if water is not to come over the deck. Answer in terms of h0, h1, L, u, and the water density r.

Arterial stenosis is a constriction of an artery, often due to plaque buildup on the arterys inner walls. Serious medical conditions can result, depending on the affected artery. Stenosis of the carotid arteries that supply blood to the brain is a leading cause of stroke, while stenosis of the renal arteries can lead to kidney failure. Pulmonary artery stenosis results from birth defects, and can result in insufficient oxygen supply. Because the heart has to work harder to get blood through a constricted artery, stenosis can contribute to high blood pressure. In answering the questions below, assume steady flow (which is true in arteries only on short timescales). How does the volume flow rate of blood at a stenosis compare with the rate in the surrounding artery? a. lower b. the same c. higher

Arterial stenosis is a constriction of an artery, often due to plaque buildup on the arterys inner walls. Serious medical conditions can result, depending on the affected artery. Stenosis of the carotid arteries that supply blood to the brain is a leading cause of stroke, while stenosis of the renal arteries can lead to kidney failure. Pulmonary artery stenosis results from birth defects, and can result in insufficient oxygen supply. Because the heart has to work harder to get blood through a constricted artery, stenosis can contribute to high blood pressure. In answering the questions below, assume steady flow (which is true in arteries only on short timescales). How does the blood flow speed at a stenosis compare with the speed in the surrounding artery? a. lower b. the same c. higher

Arterial stenosis is a constriction of an artery, often due to plaque buildup on the arterys inner walls. Serious medical conditions can result, depending on the affected artery. Stenosis of the carotid arteries that supply blood to the brain is a leading cause of stroke, while stenosis of the renal arteries can lead to kidney failure. Pulmonary artery stenosis results from birth defects, and can result in insufficient oxygen supply. Because the heart has to work harder to get blood through a constricted artery, stenosis can contribute to high blood pressure. In answering the questions below, assume steady flow (which is true in arteries only on short timescales). Which of the following medical problems is more likely to occur? a. An artery might collapse because of lower blood pressure at the stenosis. b. An artery might burst because of higher blood pressure at the stenosis. c. Neither; pressure at the stenosis is the same as in the surrounding artery.

Arterial stenosis is a constriction of an artery, often due to plaque buildup on the arterys inner walls. Serious medical conditions can result, depending on the affected artery. Stenosis of the carotid arteries that supply blood to the brain is a leading cause of stroke, while stenosis of the renal arteries can lead to kidney failure. Pulmonary artery stenosis results from birth defects, and can result in insufficient oxygen supply. Because the heart has to work harder to get blood through a constricted artery, stenosis can contribute to high blood pressure. In answering the questions below, assume steady flow (which is true in arteries only on short timescales). If the artery has circular cross section even at the stenosis, but the diameter at the stenosis is half that in the surrounding artery, the blood flow speed in the stenosis will be a. one-fourth that in the surrounding artery. b. one-half that in the surrounding artery. c. the same as in the surrounding artery. d. 12 times that in the surrounding artery. e. four times that in the surrounding artery