Chapter 11

The ice on a lake is \(0.010 \mathrm{~m}\) thick. The lake is circular, with a radius of \(480 \mathrm{~m}\). Find the mass of the ice.

One of the concrete pillars that support a house is \(2.2 \mathrm{~m}\) tall and has a radius of \(0.50 \mathrm{~m}\). The density of concrete is about \(2.2 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\). Find the weight of this pillar in pounds \((1 \mathrm{~N}=0.2248 \mathrm{lb})\).

Accomplished silver workers in India can pound silver into incredibly thin sheets, as thin as \(3.00 \times 10^{-7} \mathrm{~m}\) (about onehundredth of the thickness of this sheet of paper). Find the area of such a sheet that can be formed from \(1.00 \mathrm{~kg}\) of silver.

A full can of black cherry soda has a mass of \(0.416 \mathrm{~kg}\). It contains \(3.54 \times 10^{-4} \mathrm{~m}^{3}\) of liquid. Assuming that the soda has the same density as water, find the volume of aluminum used to make the can.

An antifreeze solution is made by mixing ethylene glycol \(\left(\rho=1116 \mathrm{~kg} / \mathrm{m}^{3}\right)\) with water. Suppose the specific gravity of such a solution is \(1.0730 .\) Assuming that the total volume of the solution is the sum of its parts, determine the volume percentage of ethylene glycol in the solution.

United States currency is printed using intaglio presses that generate a printing pressure of \(8.0 \times 10^{4} \mathrm{lb} / \mathrm{in} .2 \mathrm{~A} \$ 20\) bill is 6.1 in. by 2.6 in. Calculate the magnitude of the force that the printing press applies to one side of the bill.

An airtight box has a removable lid of area \(1.3 \times 10^{-2} \mathrm{~m}^{2}\) and negligible weight. The box is taken up a mountain where the air pressure outside the box is \(0.85 \times 10^{5} \mathrm{~Pa}\). The inside of the box is completely evacuated. What is the magnitude of the force required to pull the lid off the box?

A glass bottle of soda is sealed with a screw cap. The absolute pressure of the carbon dioxide inside the bottle is \(1.80 \times 10^{5} \mathrm{~Pa}\). Assuming that the top and bottom surfaces of the cap each have an area of \(4.10 \times 10^{-4} \mathrm{~m}^{2},\) obtain the magnitude of the force that the screw thread exerts on the cap in order to keep it on the bottle. The air pressure outside the bottle is one atmosphere.

A person who weighs \(625 \mathrm{~N}\) is riding a 98-N mountain bike. Suppose the entire weight of the rider and bike is supported equally by the two tires. If the gauge pressure in each tire is \(7.60 \times 10^{5} \mathrm{~Pa}\), what is the area of contact between each tire and the ground?

A person who weighs \(625 \mathrm{~N}\) is riding a \(98-\mathrm{N}\) mountain bike. Suppose the entire weight of the rider and bike is supported equally by the two tires. If the gauge pressure in each tire is \(7.60 \times 10^{5} \mathrm{~Pa}\), what is the area of contact between each tire and the ground?

Interactive Solution \(\underline{11.14}\) at presents a model for solving this problem. A solid concrete block weighs \(169 \mathrm{~N}\) and is resting on the ground. Its dimensions are \(0.400 \mathrm{~m} \times 0.200 \mathrm{~m} \times 0.100 \mathrm{~m}\). A number of identical blocks are stacked on top of this one. What is the smallest number of whole blocks (including the one on the ground) that can be stacked so that their weight creates a pressure of at least two atmospheres on the ground beneath the first block?

A suitcase (mass \(\mathrm{m}=16 \mathrm{~kg}\) ) is resting on the floor of an elevator. The part of the suitcase in contact with the floor measures \(0.50 \mathrm{~m}\) by \(0.15 \mathrm{~m}\). The elevator is moving upward, the magnitude of its acceleration being \(1.5 \mathrm{~m} / \mathrm{s}^{2}\). What pressure (in excess of atmospheric pressure) is applied to the floor beneath the suitcase?

A suitcase (mass \(m=16 \mathrm{~kg}\) ) is resting on the floor of an elevator. The part of the suitcase in contact with the floor measures \(0.50 \mathrm{~m}\) by \(0.15 \mathrm{~m} .\) The elevator is moving upward, the magnitude of its acceleration being \(1.5 \mathrm{~m} / \mathrm{s}^{2}\). What pressure (in excess of atmospheric pressure) is applied to the floor beneath the suitcase?

A 58 -kg skier is going down a slope oriented \(35^{\circ}\) above the horizontal. The area of each ski in contact with the snow is \(0.13 \mathrm{~m}^{2}\). Determine the pressure that each ski exerts on the snow.

A cylinder (with circular ends) and a hemisphere are solid throughout and made from the same material. They are resting on the ground, the cylinder on one of its ends and the hemisphere on its flat side. The weight of each causes the same pressure to act on the ground. The cylinder is \(0.500 \mathrm{~m}\) high. What is the radius of the hemisphere?

The main water line enters a house on the first floor. The line has a gauge pressure of \(1.90 \times 10^{5} \mathrm{~Pa}\). (a) A faucet on the second floor, \(6.50 \mathrm{~m}\) above the first floor, is turned off. What is the gauge pressure at this faucet? (b) How high could a faucet be before no water would flow from it, even if the faucet were open?

At a given instant, the blood pressure in the heart is \(1.60 \times 10^{4} \mathrm{~Pa}\). If an artery in the brain is \(0.45 \mathrm{~m}\) above the heart, what is the pressure in the artery? Ignore any pressure changes due to blood flow.

The Mariana trench is located in the floor of the Pacific Ocean at a depth of about 11000 \(\mathrm{m}\) below the surface of the water. The density of seawater is \(1025 \mathrm{~kg} / \mathrm{m}^{3}\). (a) If an underwater vehicle were to explore such a depth, what force would the water exert on the vehicle's observation window (radius \(=0.10 \mathrm{~m}\) )? (b) For comparison, determine the weight of a jetliner whose mass is \(1.2 \times 10^{5} \mathrm{~kg}\).

Some researchers believe that the dinosaur Barosaurus held its head erect on a long neck, much as a giraffe does. If so, fossil remains indicate that its heart would have been about \(12 \mathrm{~m}\) below its brain. Assume that the blood has the density of water, and calculate the amount by which the blood pressure in the heart would have exceeded that in the brain. Size estimates for the single heart needed to withstand such a pressure range up to two tons. Alternatively, Barosaurus may have had a number of smaller hearts.

The human lungs can function satisfactorily up to a limit where the pressure difference between the outside and inside of the lungs is one-twentieth of an atmosphere. If a diver uses a snorkel for breathing, how far below the water can she swim? Assume the diver is in salt water whose density is \(1025 \mathrm{~kg} / \mathrm{m}^{3}\)

The vertical surface of a reservoir dam that is in contact with the water is \(120 \mathrm{~m}\) wide and \(12 \mathrm{~m}\) high. The air pressure is one atmosphere. Find the magnitude of the total force acting on this surface in a completely filled reservoir. (Hint: The pressure varies linearly with depth, so you must use an average pressure.)

Two identical containers are open at the top and are connected at the bottom via a tube of negligible volume and a valve that is closed. Both containers are filled initially to the same height of \(1.00 \mathrm{~m}\), one with water, the other with mercury, as the drawing indicates. The valve is then opened. Water and mercury are immiscible. Determine the fluid level in the left container when equilibrium is reestablished.

A 1.00-m-tall container is filled to the brim, partway with mercury and the rest of the way with water. The container is open to the atmosphere. What must be the depth of the mercury so that the absolute pressure on the bottom of the container is twice the atmospheric pressure?

A \(1.00-\mathrm{m}\) -tall container is filled to the brim, partway with mercury and the rest of the way with water. The container is open to the atmosphere. What must be the depth of the mercury so that the absolute pressure on the bottom of the container is twice the atmospheric pressure?

The atmospheric pressure above a swimming pool changes from 755 to \(765 \mathrm{~mm}\) of mercury. The bottom of the pool is a rectangle \((12 \mathrm{~m} \times 24 \mathrm{~m})\). By how much does the force on the bottom of the pool increase?

In the process of changing a flat tire, a motorist uses a hydraulic jack. She begins by applying a force of \(45 \mathrm{~N}\) to the input piston, which has a radius \(r_{1}\). As a result, the output plunger, which has a radius \(r_{2}\), applies a force to the car. The ratio \(r_{2} / r_{1}\) has a value of 8.3. Ignore the height difference between the input piston and output plunger and determine the force that the output plunger applies to the car.

A dentist's chair with a patient in it weighs \(2100 \mathrm{~N}\). The output plunger of a hydraulic system begins to lift the chair when the dentist's foot applies a force of \(55 \mathrm{~N}\) to the input piston. Neglect any height difference between the plunger and the piston. What is the ratio of the radius of the plunger to the radius of the piston?

A duck is floating on a lake with \(25 \%\) of its volume beneath the water. What is the average density of the duck?

A \(0.10-m \times 0.20-m \times 0.30-m\) block is suspended from a wire and is completely under water. What buoyant force acts on the block?

The density of ice is \(917 \mathrm{~kg} / \mathrm{m}^{3},\) and the density of sea water is \(1025 \mathrm{~kg} / \mathrm{m}^{3}\). A swimming polar bear climbs onto a piece of floating ice that has a volume of \(5.2 \mathrm{~m}^{3}\). What is the weight of the heaviest bear that the ice can support without sinking completely beneath the water?

A paperweight, when weighed in air, has a weight of \(\mathrm{W}=6.9 \mathrm{~N}\). When completely immersed in water, however, it has a weight of \(W_{\text {in water }}=4.3 \mathrm{~N}\). Find the volume of the paperweight.

An \(81-\mathrm{kg}\) person puts on a life jacket, jumps into the water, and floats. The jacket has a volume of \(3.1 \times 10^{-2} \mathrm{~m}^{3}\) and is completely submerged under the water. The volume of the person's body that is underwater is \(6.2 \times 10^{-2} \mathrm{~m}^{3}\). What is the density of the life jacket?

A person can change the volume of his body by taking air into his lungs. The amount of change can be determined by weighing the person under water. Suppose that under water a person weighs \(20.0 \mathrm{~N}\) with partially full lungs and \(40.0 \mathrm{~N}\) with empty lungs. Find the change in body volume.

An object is solid throughout. When the object is completely submerged in ethyl alcohol, its apparent weight is \(15.2 \mathrm{~N}\). When completely submerged in water, its apparent weight is \(13.7 \mathrm{~N}\). What is the volume of the object?

A hollow cubical box is \(0.30 \mathrm{~m}\) on an edge. This box is floating in a lake with one-third of its height beneath the surface. The walls of the box have a negligible thickness. Water is poured into the box. What is the depth of the water in the box at the instant the box begins to sink?

An object is solid throughout. When the object is completely submerged in ethyl alcohol, its apparent weight is \(15.2 \mathrm{~N}\). When completely submerged in water, its apparent weight is \(13.7 \mathrm{~N}\). What is the volume of the object?

A solid cylinder (radius \(=0.150 \mathrm{~m},\) height \(=0.120 \mathrm{~m}\) ) has a mass of \(7.00 \mathrm{~kg}\). This cylinder is floating in water. Then oil \(\left(\rho=725 \mathrm{~kg} / \mathrm{m}^{3}\right)\) is poured on top of the water until the situation shown in the drawing results. How much of the height of the cylinder is in the oil?

One kilogram of glass \(\left(\rho=2.60 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\right)\) is shaped into a hollow spherical shell that just barely floats in water. What are the inner and outer radii of the shell? Do not assume the shell is thin.

Oil is flowing with a speed of \(1.22 \mathrm{~m} / \mathrm{s}\) through a pipeline with a radius of \(0.305 \mathrm{~m} .\) How many gallons of oil \(\left(1 \mathrm{gal}=3.79 \times 10^{-3} \mathrm{~m}^{3}\right)\) flow in one day?

A patient recovering from surgery is being given fluid intravenously. The fluid has a density of \(1030 \mathrm{~kg} / \mathrm{m}^{3}\), and \(9.5 \times 10^{-4} \mathrm{~m}^{3}\) of it flows into the patient every six hours. Find the mass flow rate in \(\mathrm{kg} / \mathrm{s}\).

Concept Simulation \(11.1\) at reviews the concept that plays the central role in this problem. (a) The volume flow rate in an artery supplying the brain is \(3.6 \times 10^{-6} \mathrm{~m}^{3} / \mathrm{s}\). If the radius of the artery is \(5.2 \mathrm{~mm}\), determine the average blood speed. (b) Find the average blood speed at a constriction in the artery if the constriction reduces the radius by a factor of 3 . Assume that the volume flow rate is the same as that in part (a).

At reviews the concept that plays the central role in this problem. (a) The volume flow rate in an artery supplying the brain is \(3.6 \times 10^{-6} \mathrm{~m}^{3} / \mathrm{s}\). If the radius of the artery is \(5.2 \mathrm{~mm}\), determine the average blood speed. (b) Find the average blood speed at a constriction in the artery if the constriction reduces the radius by a factor of \(3 .\) Assume that the volume flow rate is the same as that in part (a).

A room has a volume of \(120 \mathrm{~m}^{3}\). An air-conditioning system is to replace the air in this room every twenty minutes, using ducts that have a square cross section. Assuming that air can be treated as an incompressible fluid, find the length of a side of the square if the air speed within the ducts is (a) \(3.0 \mathrm{~m} / \mathrm{s}\) and \((\mathrm{b}) 5.0 \mathrm{~m} / \mathrm{s}\).

The aorta carries blood away from the heart at a speed of about \(40 \mathrm{~cm} / \mathrm{s}\) and has a radius of approximately \(1.1 \mathrm{~cm}\). The aorta branches eventually into a large number of tiny capillaries that distribute the blood to the various body organs. In a capillary, the blood speed is approximately \(0.07 \mathrm{~cm} / \mathrm{s},\) and the radius is about \(6 \times 10^{-4} \mathrm{~cm} .\) Treat the blood as an incompressible fluid, and use these data to determine the approximate number of capillaries in the human body.

A water line with an internal radius of \(6.5 \times 10^{-3} \mathrm{~m}\) is connected to a shower head that has 12 holes. The speed of the water in the line is \(1.2 \mathrm{~m} / \mathrm{s}\). (a) What is the volume flow rate in the line? (b) At what speed does the water leave one of the holes (effective hole radius \(=4.6 \times 10^{-4} \mathrm{~m}\) ) in the head?

Prairie dogs are burrowing rodents. They do not suffocate in their burrows, because the effect of air speed on pressure creates sufficient air circulation. The animals maintain a difference in the shapes of two entrances to the burrow, and because of this difference, the air \(\left(\rho=1.29 \mathrm{~kg} / \mathrm{m}^{3}\right)\) blows past the openings at different speeds, as the drawing indicates. Assuming that the openings are at the same vertical level, find the difference in air pressure between the openings and indicate which way the air circulates.

An airplane wing is designed so that the speed of the air across the top of the wing is \(251 \mathrm{~m} / \mathrm{s}\) when the speed of the air below the wing is \(225 \mathrm{~m} / \mathrm{s}\). The density of the air is \(1.29 \mathrm{~kg} / \mathrm{m}^{3} .\) What is the lifting force on a wing of area \(24.0 \mathrm{~m}^{2} ?\)

Water is circulating through a closed system of pipes in a two-floor apartment. On the first floor, the water has a gauge pressure of \(3.4 \times 10^{5} \mathrm{~Pa}\) and a speed of \(2.1 \mathrm{~m} / \mathrm{s}\). However, on the second floor, which is \(4.0 \mathrm{~m}\) higher, the speed of the water is \(3.7 \mathrm{~m} / \mathrm{s}\). The speeds are different because the pipe diameters are different. What is the gauge pressure of the water on the second floor?

Interactive LearningWare 11.2 at reviews the approach taken in problems such as this one. A small crack occurs at the base of a 15.0 -m-high dam. The effective crack area through which water leaves is \(1.30 \times 10^{-3} \mathrm{~m}^{2}\). (a) Ignoring viscous losses, what is the speed of water flowing through the crack? (b) How many cubic meters of water per second leave the dam?

A fountain sends a stream of water straight up into the air to a maximum height of 5.00 \(\mathrm{m} .\) The effective cross-sectional area of the pipe feeding the fountain is \(5.00 \times 10^{-4} \mathrm{~m}^{2}\) Neglecting air resistance and any viscous effects, determine how many gallons per minute are being used by the fountain. (Note: \(1 \mathrm{gal}=3.79 \times 10^{-3} \mathrm{~m}^{3}\).)