Chapter 13

You purchase a rectangular piece of metal that has dimensions 5.0 \(\mathrm{mm} \times 15.0 \mathrm{mm} \times 30.0 \mathrm{mm}\) and mass 0.0158 \(\mathrm{kg}\) . The seller tells you that the metal is gold. To check this, you compute the average density of the piece. What value do you get? Were you cheated?

By how many newtons do you increase the weight of your car when you fill up your 11.5 gal gas tank with gasoline? A gallon is equal to 3.788 \(\mathrm{L}\) and the density of gasoline is 737 \(\mathrm{kg} / \mathrm{m}^{3}\) .

How big is a million dollars? At the time this problem was written, the price of gold was about \(\$ 1239\) per ounce, while that of platinum was about \(\$ 1508\) an ounce. The "ounce" in this case is the troy ounce, which is equal to 31.1035 g. The more familiar avoirdupois ounce is equal to 28.35 g.) The density of gold is 19.3 \(\mathrm{g} / \mathrm{cm}^{3}\) and that of platinum is 21.4 \(\mathrm{g} / \mathrm{cm}^{3} .\) (a) If you find a spherical gold nugget worth 1.00 million dollars, what would be its diameter? (b) How much would a platinum nugget of this size be worth?

A cube 5.0 \(\mathrm{cm}\) on each side is made of a metal alloy. After you drill a cylindrical hole 2.0 \(\mathrm{cm}\) in diameter all the way through and perpendicular to one face, you find that the cube weighs 7.50 \(\mathrm{N}\) (a) What is the density of this metal? (b) What did the cube weigh before you drilled the hole in it?

A cube of compressible material (such as Styrofoam or balsa wood) has a density \(\rho\) and sides of length \(L .\) (a) If you keep its mass the same, but compress each side to half its length, what will be its new density, in terms of \(\rho ?\) (b) If you keep the mass and shape the same, what would the length of each side have to be (in terms of \(L )\) so that the density of the cube was three times its original value?

A uniform lead sphere and a uniform aluminum sphere have the same mass. What is the ratio of the radius of the aluminum sphere to the radius of the lead sphere?

Blood pressure. Systemic blood pressure is defined as the ratio of two pressures, both expressed in millimeters of mercury. Normal blood pressure is about \(\frac{120 \mathrm{mm}}{80 \mathrm{mm}},\) which is usually just stated as \(\frac{120}{80}\) . (See also Problem \(24 . )\) What would normal systemic blood pressure be if, instead of millimeters of mercury, we expressed pressure in each of the following units, but continued to use the same ratio format? (a) atmospheres, (b) torr, (c) Pa, (d) \(\mathrm{N} / \mathrm{m}^{2},\) (e) psi.

Blood. (a) Mass of blood. The human body typically contains 5 L of blood of density 1060 \(\mathrm{kg} / \mathrm{m}^{3} .\) How many kilograms of blood are in the body? (b) The average blood pressure is \(13,000 \mathrm{Pa}\) at the heart. What average force does the blood exert on each square centimeter of the heart? (c) Red blood cells. Red blood cells have a specific gravity of 5.0 and a diameter of about 7.5\(\mu \mathrm{m}\) . If they are spherical in shape (which is not quite true), what is the mass of such a cell?

Landing on Venus. One of the great difficulties in landing on Venus is dealing with the crushing pressure of the atmosphere, which is 92 times the earth's atmospheric pressure. (a) If you are designing a lander for Venus in the shape of a hemisphere 2.5 \(\mathrm{m}\) in diameter, how many newtons of inward force must it be prepared to withstand due to the Venusian atmosphere? (Don't forget about the bottom!) (b) How much force would the lander have to withstand on the earth?

You are designing a diving bell to withstand the pressure of seawater at a depth of 250 \(\mathrm{m}\) (a) What is the gauge pressure at this depth? (You can ignore the small changes in the density of the water with depth.) (b) At the 250 \(\mathrm{m}\) depth, what is the net force due to the water outside and the air inside the bell on a circular glass window 30.0 \(\mathrm{cm}\) in diameter if the pressure inside the diving bell equals the pressure at the surface of the water? (You may ignore the small variation in pressure over the surface of the window.)

Glaucoma. Under normal circumstances, the vitreous humor, a jelly-like substance in the main part of the eye, exerts a pressure of up to 24 \(\mathrm{mm}\) of mercury that maintains the shape of the eye. If blockage of the drainage duct for aqueous humor causes this pressure to increase to about 50 \(\mathrm{mm}\) of mercury, the condition is called glaucoma. What is the increase in the total force (in newtons) on the walls of the eye if the pressure increases from 24 \(\mathrm{mm}\) to 50 \(\mathrm{mm}\) of mercury? We can quite accurately model the eye as a sphere 2.5 \(\mathrm{cm}\) in diameter.

By means of physiological adaptations that are still not very well understood, sperm whales are thought to be able to hunt for their food at depths of between 400 \(\mathrm{m}\) and 3000 \(\mathrm{m.}\) (a) What range of gauge pressures (in Pa and atm) do the whales withstand at these depths? (b) Estimate the total inward force of water pressure on the surface of a sperm whale at a depth of \(3000 \mathrm{m},\) modeling the whale as a cylinder 16 \(\mathrm{m}\) long and 4 \(\mathrm{m}\) in diameter.

What gauge pressure must a pump produce to pump water from the bottom of the Grand Canyon (elevation 730 \(\mathrm{m} )\) to Indian Gardens (elevation 1370 \(\mathrm{m} ) ?\) Express your result in pascals and in atmospheres.

An electrical short cuts off all power to a submersible diving vehicle when it is 30 m below the surface of the ocean. The crew must push out a hatch of area 0.75 \(\mathrm{m}^{2}\) and weight 300 \(\mathrm{N}\) on the bottom to escape. If the pressure inside is 1.0 atm, what downward force must the crew exert on the hatch to open it?

Standing on your head. (a) When you stand on your head, what is the difference in pressure of the blood in your brain compared with the pressure when you stand on your feet if you are 1.85 m tall? The density of blood is 1060 \(\mathrm{kg} / \mathrm{m}^{3}\) . (b) What effect does the increased pressure have on the blood vessels in your brain?

You are designing a machine for a space exploration vehicle. It contains an enclosed column of oil that is 1.50 m tall, and you need the pressure difference between the top and the bottom of this column to be 0.125 atm. (a) What must be the density of the oil? (b) If the vehicle is taken to Mars, where the acceleration due to gravity is \(0.379 g,\) what will be the pressure difference (in earth atmospheres) between the top and bottom of the oil column?

A barrel contains a 0.120 \(\mathrm{m}\) layer of oil of density 600 \(\mathrm{kg} / \mathrm{m}^{3}\) floating on water that is 0.250 \(\mathrm{m}\) deep. (a) What is the gauge pressure at the oil-water interface? (b) What is the gauge pressure at the bottom of the barrel?

Blood pressure. Systemic blood pressure is expressed as the ratio of the systolic pressure (when the heart first ejects blood into the arteries) to the diastolic pressure (when the heart is relaxed): systemic blood pressure \(=\frac{\text { systolic pressure }}{\text { diastolic pressure }}\) Both pressures are measured at the level of the heart and are expressed in millimeters of mercury (or torr), although the units are not written. Normal systemic blood pressure is \(\frac{120}{80}\) . (a) What are the maximum and minimum forces (in newtons) that the blood exerts against each square centimeter of the heart for a person with normal blood pressure? (b) As pointed out in the text, blood pressure is normally measured on the upper arm at the same height as the heart. Due to therapy for an injury, a patient's upper arm is extended 30.0 \(\mathrm{cm}\) above his heart. In that position, what should be his systemic blood pressure reading, expressed in the standard way, if he has normal blood pressure? The density of blood is 1060 \(\mathrm{kg} / \mathrm{m}^{3}\) .

The piston of a hydraulic automobile lift is 0.30 \(\mathrm{m}\) in diameter. What gauge pressure, in pascals, is required to lift a car with a mass of 1200 \(\mathrm{kg}\) ? Now express this pressure in atmospheres.

There is a maximum depth at which a diver can breathe through a snorkel tube (Fig. 13.42\()\) , because as the depth increases, so does the pressure difference, which the difference, if any.tends to collapse the diver's lungs. since the snorkel connects the air in the lungs to the atmosphere at the surface, the pressure inside the lungs is atmospheric pressure. What is the external-internal pressure difference when the diver's lungs are at a depth of 6.1 \(\mathrm{m}\) (about 20 \(\mathrm{ft}\) ? Assume that the diver is in fresh- water. (A scuba diver breathing from compressed air tanks can operate at greater depths than can a snorkeler, since the pressure of the air inside the scuba diver's lungs increases to match the external pressure of the water.)

Fish navigation. (a) As you can tell by watching them in an aquarium, fish are able to remain at any depth in water with no effort. What does this ability tell you about their density? (b) Fish are able to inflate themselves using a sac (called the swim bladder) located under their spinal column. These sacs can be filled with an oxygen-nitrogen mixture that comes from the blood. If a 2.75 \(\mathrm{kg}\) fish in fresh water inflates itself and increases its volume by \(10 \%,\) find the net force that the water exerts on it. (c) What is the net external force on it? Does the fish go up or down when it inflates itself?

When an open-faced boat has a mass of 5750 \(\mathrm{kg}\) , including its cargo and passengers, it floats with the water just up to the top of its gunwales (sides) on a freshwater lake. (a) What is the volume of this boat? (b) The captain decides that it is too dangerous to float with his boat on the verge of sinking, so he decides to throw some cargo overboard so that 20\(\%\) of the boat's volume will be above water. How much mass should he throw out?

An ore sample weighs 17.50 \(\mathrm{N}\) in air. When the sample is suspended by a light cord and totally immersed in water, the tension in the cord is 11.20 \(\mathrm{N}\) . Find the total volume and the density of the sample.

A slab of ice floats on a freshwater lake. What minimum volume must the slab have for a \(45.0 \mathrm{~kg}\) woman to be able to stand on it without getting her feet wet?

A hollow plastic sphere is held below the surface of a fresh- water lake by a cord anchored to the bottom of the lake. The sphere has a volume of 0.650 \(\mathrm{m}^{3}\) and the tension in the cord is 900 \(\mathrm{N} .\) (a) Calculate the buoyant force exerted by the water on the sphere. (b) What is the mass of the sphere? (c) The cord breaks and the sphere rises to the surface. When the sphere comes to rest, what fraction of its volume will be submerged?

(a) Calculate the buoyant force of air (density 1.20 \(\mathrm{kg} / \mathrm{m}^{3} )\) on a spherical party balloon that has a radius of 15.0 \(\mathrm{cm}\) . (b) If the rubber of the balloon itself has a mass of 2.00 \(\mathrm{g}\) and the balloon is filled with helium (density 0.166 \(\mathrm{kg} / \mathrm{m}^{3}\) ), calculate the net upward force (the "lift") that acts on it in air.

At \(20^{\circ} \mathrm{C},\) the surface tension of water is 72.8 dynes/cm. Find the excess pressure inside of (a) an ordinary-size water drop of radius 1.50 \(\mathrm{mm}\) and (b) a fog droplet of radius 0.0100 \(\mathrm{mm} .\)

Find the gauge pressure in pascals inside a soap bubble 7.00 \(\mathrm{cm}\) in diameter. The surface tension of this soap is 25.0 dynes/cm.

What radius must a water drop have for the difference between the inside and outside pressures to be 0.0200 atm? The surface tension of water is 72.8 dynes/cm.

At \(20^{\circ} \mathrm{C}\) , the surface tension of water is 72.8 dynes/cm and that of carbon tetrachloride \(\left(\mathrm{CCl}_{4}\right)\) is 26.8 dynes/cm. If the gauge pressure is the same in two drops of these liquids, what is the ratio of the volume of the water drop to that of the \(\mathrm{CCl}_{4}\) drop?

At a point where an irrigation canal having a rectangular cross section is 18.5 \(\mathrm{m}\) wide and 3.75 \(\mathrm{m}\) deep, the water flows at 2.50 \(\mathrm{cm} / \mathrm{s} .\) At a point downstream, but on the same level, the canal is 16.5 \(\mathrm{m}\) wide, but the water flows at 11.0 \(\mathrm{cm} / \mathrm{s} .\) How deep is the canal at this point?

Water is flowing in a pipe with a varying cross-sectional area, and at all points the water completely fills the pipe. At area, \(1,\) the cross-sectional area of the pipe is 0.070 \(\mathrm{m}^{2}\) and the magnitude of the fluid velocity is 3.50 \(\mathrm{m} / \mathrm{s} .\) What is the fluid speed at points in the pipe where the cross-sectional area is \((a) 0.105 \mathrm{m}^{2},\) (b) 0.047 \(\mathrm{m}^{2}\) ?

Water is flowing in a cylindrical pipe of varying circular cross-sectional area, and at all points the water completely fills the pipe. (a) At one point in the pipe, the radius is 0.150 \(\mathrm{m} .\) What is the speed of the water at this point if the volume flow rate in the pipe is 1.20 \(\mathrm{m}^{3} / \mathrm{s} ?\) (b) At a second point in the pipe, the water speed is 3.80 \(\mathrm{m} / \mathrm{s}\) . What is the radius of the pipe at this point?

A shower head has 20 circular openings, each with radius 1.0 \(\mathrm{mm} .\) The shower head is connected to a pipe with radius 0.80 \(\mathrm{cm} .\) If the speed of water in the pipe is \(3.0 \mathrm{m} / \mathrm{s},\) what is its speed as it exits the shower-head openings?

A small circular hole 6.00 \(\mathrm{mm}\) in diameter is cut in the side of a large water tank, 14.0 \(\mathrm{m}\) below the water level in the tank. The top of the tank is open to the air. Find the speed at which the water shoots out of the tank.

A sealed tank containing seawater to a height of 11.0 \(\mathrm{m}\) also contains air above the water at a gauge pressure of 3.00 atm. Water flows out from the bottom through a small hole. Calculate the speed with which the water comes out of the tank.

What gauge pressure is required in the city water mains for a stream from a fire hose connected to the mains to reach a vertical height of 15.0 \(\mathrm{m}\) ? (Assume that the mains have a much larger diameter than the fire hose.)

Lift on an airplane. Air streams horizontally past a small airplane's wings such that the speed is 70.0 \(\mathrm{m} / \mathrm{s}\) over the top surface and 60.0 \(\mathrm{m} / \mathrm{s}\) past the bottom surface. If the plane has a mass of 1340 \(\mathrm{kg}\) and a wing area of \(16.2 \mathrm{m}^{2},\) what is the net vertical force (including the effects of gravity) on the airplane? The density of the air is 1.20 \(\mathrm{kg} / \mathrm{m}^{3}\) .

A golf course sprinkler system discharges water from a horizontal pipe at the rate of 7200 \(\mathrm{cm}^{3} / \mathrm{s}\) . At one point in the pipe, where the radius is \(4.00 \mathrm{cm},\) the water's absolute pressure is \(2.40 \times 10^{5}\) Pa. At a second point in the pipe, the water passes through a constriction where the radius is 2.00 \(\mathrm{cm} .\) What is the water's absolute pressure as it flows through this constriction?

Water discharges from a horizontal cylindrical pipe at the rate of 465 \(\mathrm{cm}^{3} / \mathrm{s}\) . At a point in the pipe where the radius is \(2.05 \mathrm{cm},\) the absolute pressure is \(1.60 \times 10^{5} \mathrm{Pa} .\) What is the pipe's radius at a constriction if the pressure there is reduced to \(1.20 \times 10^{5} \mathrm{Pa}\) ?

Artery blockage. A medical technician is trying to determine what percentage of a patient's artery is blocked by plaque. To do this, she measures the blood pressure just before the region of blockage and finds that it is \(1.20 \times 10^{4}\) Pa, while in the region of blockage it is \(1.15 \times 10^{4}\) Pa. Furthermore, she knows that blood flowing through the normal artery just before the point of blockage is traveling at 30.0 \(\mathrm{cm} / \mathrm{s}\) , and the specific gravity of this patient's blood is \(1.06 .\) What percentage of the cross-sectional area of the patient's artery is blocked by the plaque?

At a certain point in a horizontal pipeline, the water's speed is 2.50 \(\mathrm{m} / \mathrm{s}\) and the gauge pressure is \(1.80 \times 10^{4}\) Pa. Find the gauge pressure at a second point in the line if the cross-sectional area at the second point is twice that at the first.

What speed must a gold sphere of radius 3.00 \(\mathrm{mm}\) have in castor oil for the viscous drag force to be one-fourth of the weight of the sphere? The density of gold is \(19,300 \mathrm{kg} / \mathrm{m}^{3}\) and the viscosity of the oil is 0.986 \(\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\)

A copper sphere with a mass of 0.20 \(\mathrm{g}\) and a density of 8900 \(\mathrm{kg} / \mathrm{m}^{3}\) is observed to fall with a terminal speed of 6.0 \(\mathrm{cm} / \mathrm{s}\) in an unknown liquid. Find the viscosity of the unknown liquid if its buoyancy can be neglected.

Clogged artery. Viscous blood is flowing through an artery partially clogged by cholesterol. A surgeon wants to remove enough of the cholesterol to double the flow rate of blood through this artery. If the original diameter of the artery is \(D,\) what should be the new diameter (in terms of \(D )\) to accomplish this for the same pressure gradient?

Advertisements for a certain small car claim that it floats in water. (a) If the car's mass is 900 \(\mathrm{kg}\) and its interior volume is \(3.0 \mathrm{m}^{3},\) what fraction of the car is immersed when it floats? You can ignore the volume of steel and other materials. (b) Water gradually leaks in and displaces the air in the car. What fraction of the interior volume is filled with water when the car sinks?

A piece of wood is 0.600 \(\mathrm{m}\) long, 0.250 \(\mathrm{m}\) wide, and 0.080 \(\mathrm{m}\) thick. Its density is 600 \(\mathrm{kg} / \mathrm{m}^{3} .\) What volume of lead must be fastened underneath it to sink the wood in calm water so that its top is just even with the water level? What is the mass of this volume of lead?

A hot-air balloon has a volume of 2200 \(\mathrm{m}^{3} .\) The balloon fabric (the envelope) weighs 900 \(\mathrm{N} .\) The basket with gear and full propane tanks weighs 1700 \(\mathrm{N}\) . If the balloon can barely lift an additional 3200 \(\mathrm{N}\) of passengers, breakfast, and champagne when the outside air density is \(1.23 \mathrm{kg} / \mathrm{m}^{3},\) what is the average density of the heated gases in the envelope?

In seawater, a life preserver with a volume of 0.0400 \(\mathrm{m}^{3}\) will support a 75.0 \(\mathrm{kg}\) person (average density 980 \(\mathrm{kg} / \mathrm{m}^{3} )\) with 20\(\%\) of the person's volume above water when the life preserver is fully submerged. What is the density of the material composing the life preserver?

A liquid is used to make a mercury-type barometer, as described in Section \(13.2 .\) The barometer is intended for spacefaring astronauts. At the surface of the earth, the column of liquid rises to a height of 2185 \(\mathrm{mm}\) , but on the surface of Planet \(\mathrm{X},\) where the acceleration due to gravity is one-fourth of its value on earth, the column rises to only 725 \(\mathrm{mm} .\) Find (a) the density of the liquid and (b) the atmospheric pressure at the surface of Planet \(\mathrm{X}\) .