An 81.5 -kg man stands on a horizontal surface. (a) What is the volume of the man's body if his average density is 985 $\mathrm{kg} / \mathrm{m}^{3}$ ? (b) What average pressure from his weight is exerted on the horizontal surface if the man's two feet have a combined area of $4.50 \times 10^{-2} \mathrm{m}^{2}$ ?

Salamat A.

Numerade Educator

The British gold sovereign coin is an alloy of gold and copper having a total mass of $7.988 \mathrm{g},$ and is 22 -karat gold. (a) Find the mass of gold in the sovereign in kilograms using the fact that the number of karats $=24 \times(\text { mass of gold }) /(\text { total mass })$ . (b) Calculate the volumes of gold and copper, respectively, used to manufacture the coin. (c) Calculate the density of the

British sovereign coin.

Averell H.

Carnegie Mellon University

The weight of Earth's atmosphere exerts an average pressure of $1.01 \times 10^{5}$ Pa on the ground at sea level. Use the definition of pressure to estimate the weight of Earth's atmosphere by

approximating Earth as a sphere of radius $R_{E}=6.38 \times 10^{6} \mathrm{m}$ and surface area $A=4 \pi R_{E}^{2}$

Salamat A.

Numerade Educator

Calculate the mass of a solid gold rectangular bar that has dimensions of 4.50 $\mathrm{cm} \times 11.0 \mathrm{cm} \times 26.0 \mathrm{cm} .$

Averell H.

Carnegie Mellon University

The nucleus of an atom can be modeled as several protons and neutrons closely packed together. Each particle has a mass of $1.67 \times 10^{-27} \mathrm{kg}$ and radius on the order of $10^{-15} \mathrm{m} .$ (a) Use this model and the data provided to estimate the density of the nucleus of an atom. (b) Compare your result with the density of a material such as iron. What do your result and

comparison suggest about the structure of matter?

Salamat A.

Numerade Educator

The four tires of an automobile are inflated to a gauge pressure of $2.0 \times 10^{5}$ Pa. Each tire has an area of 0.024 $\mathrm{m}^{2}$ in contact with the ground. Determine the weight of the automobile.

Averell H.

Carnegie Mellon University

Suppose a distant world with surface gravity of 7.44 $\mathrm{m} / \mathrm{s}^{2}$ has an atmospheric pressure of $8.04 \times 10^{4} \mathrm{Pa}$ at the surface. (a) What force is exerted by the atmosphere on a disk-shaped region 2.00 $\mathrm{m}$ in radius at the surface of a methane ocean? (b) What is the weight of a $10.0-\mathrm{m}$ deep cylindrical column of methane with radius 2.00 $\mathrm{m}$ ? (c) Calculate the pressure at a depth of 10.0 $\mathrm{m}$ in the methane ocean. Note: The density of liquid methane is 415 $\mathrm{kg} / \mathrm{m}^{3} .$

Salamat A.

Numerade Educator

A normal blood pressure reading is less than 120$/ 80$ where both numbers are gauge pressures measured in millimeters of mercury (mmHg). What are the (a) absolute and (b) gauge pressures in pascals at the base of a 0.120 $\mathrm{m}$ column of mercury?

Averell H.

Carnegie Mellon University

(a) Calculate the absolute pressure at the bottom of a freshwater lake at a depth of 27.5 $\mathrm{m}$ . Assume the density of the water is $1.00 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$ and the air above is at a pressure of 101.3 $\mathrm{kPa}$ . (b) What force is exerted by the water on the window of an underwater vehicle at this depth if the window is circular and has a diameter of 35.0 $\mathrm{cm}$ ?

Salamat A.

Numerade Educator

Mercury is poured into a U-tube as shown in Figure P9.10a. The left arm of the tube has cross-sectional area $A_{1}$ of 10.0 $\mathrm{cm}^{2}$ , and the right arm has a cross-sectional area $A_{2}$ of 5.00 $\mathrm{cm}^{2}$ . One hundred grams of water are then poured into the right arm as shown in Figure P9.10b. (a) Determine the length of the water column in the right arm of the U-ube. (b) Given that the density of mercury is $13.6 \mathrm{g} / \mathrm{cm}^{3},$ what distance $h$ does the mercury rise in the left arm?

Averell H.

Carnegie Mellon University

A collapsible plastic bag (Fig. P9.11) contains a glucose solution. If the average gauge pressure in the vein is $1.33 \times 10^{3} \mathrm{Pa}$ what must be the minimum height $h$ of the bag to infuse glucose into the vein? Assume the specific gravity of the solution is 1.02 .

Salamat A.

Numerade Educator

A hydraulic jack has an input piston of area 0.050 $\mathrm{m}^{2}$ and an output piston of

area 0.70 $\mathrm{m}^{2} .$ How much force on the input piston is required to lift a car weighing $1.2 \times 10^{4} \mathrm{N} ?$

Averell H.

Carnegie Mellon University

A container is filled to a depth of 20.0 $\mathrm{cm}$ with water. On top of the water floats a 30.0 -cm-thick layer of oil with specific gravity 0.700 . What is the absolute pressure at the bottom of the container?

Salamat A.

Numerade Educator

Blaise Pascal duplicated Torricelli's barometer using a red Bordeaux wine, of density 984 $\mathrm{kg} / \mathrm{m}^{3}$ as the working liquid (Fig. P9.14). (a) What was the height hof the wine column

for normal atmospheric pressure? (b) Would you expect the vacuum above the column to be as good as for mercury?

Averell H.

Carnegie Mellon University

A sphygmomanometer is a device used to measure blood pressure, typically consisting of an inflatable cuff and a manometer used to measure air pressure in the cuff. In a mercury sphyg-momanometer, blood pressure is related to the difference in heights between two columns of mercury.

The mercury sphygmomanometer shown in Figure P9.15 contains air at the cuff pressure $P .$ The difference in mercury heights between the left tube and the right tube is $h=$ $115 \mathrm{mm} \mathrm{Hg}=0.115 \mathrm{m},$ a normal systolic reading. What is the gauge systolic blood pressure $P_{\text { gayge }}$ in pascals? The density of mercury is $\rho=13.6 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$ and the ambient pressure is $P_{0}=1.01 \times 10^{5} \mathrm{Pa} .$

Khoobchandra A.

Numerade Educator

Piston 1 in Figure P9.16 has a diameter of 0.25 in. piston 2 has a diameter of 1.5 in. In the absence of friction, determine the force $\overrightarrow{\mathbf{F}}$ necessary to support the $500-\mathrm{lb}$ weight.

Averell H.

Carnegie Mellon University

A table-tennis ball has a diameter of 3.80 $\mathrm{cm}$ and average density of 0.0840 $\mathrm{g} / \mathrm{cm}^{3} .$ What force is required to hold it completely submerged under water?

Salamat A.

Numerade Educator

A 20.0 -kg lead mass rests on the bottom of a pool. (a) What is the volume of the lead? (b) What buoyant force acts on the lead? (c) Find the lead's weight. (d) What is the normal force acting on the lead?

Averell H.

Carnegie Mellon University

A small ferryboat is 4.00 $\mathrm{m}$ wide and 6.00 $\mathrm{m}$ long. When a loaded truck pulls onto it, the boat sinks an additional 4.00 $\mathrm{cm}$ into the river. What is the weight of the truck?

Salamat A.

Numerade Educator

A 62.0 -kg survivor of a cruise line disaster rests atop a block of Styrofoam insulation, using it as a raft. The Styrofoam has dimensions 2.00 $\mathrm{m} \times 2.00 \mathrm{m} \times 0.0900 \mathrm{m} .$ The bottom 0.024 $\mathrm{m}$ of the raft is submerged. (a) Draw a force diagram of the system consisting of the survivor and raft. (b) Write Newton's second law for the system in one dimension, using $B$ for buoyancy, w for the weight of the survivor, and $w_{r}$ for the weight of the raft. (Set $a=0 . )$ (c) Calculate the numeric value for the buoyancy, $B$ . (Seawater has density 1025 $\mathrm{kg} / \mathrm{m}^{3} .$ ) (d) Using the value of $B$ and the weight $w$ of the survivor, calculate the weight $w_{r}$ of the Styrofoam. (e) What is the density of the Styrofoam? (f) What is the maximum buoyant force, corresponding to the raft being submerged up to its top surface? (g) What total mass of survivors can the raft support?

Averell H.

Carnegie Mellon University

A hot-air balloon consists of a basket hanging beneath a large envelope filled with hot air. A typical hot-air balloon has a total mass of 545 $\mathrm{kg}$ , including passengers in its basket, and holds $2.55 \times 10^{3} \mathrm{m}^{3}$ of hot air in its envelope. If the ambient air density is 1.25 $\mathrm{kg} / \mathrm{m}^{3}$ , determine the density of hot air inside the envelope when the balloon is neutrally buoyant. Neglect the volume of air displaced by the basket and passengers.

Vishal G.

Numerade Educator

A large balloon of mass 226 $\mathrm{kg}$ is filled with helium gas until its volume is 325 $\mathrm{m}^{3} .$ Assume the density of air is 1.29 $\mathrm{kg} / \mathrm{m}^{3}$ and the density of helium is 0.179 $\mathrm{kg} / \mathrm{m}^{3} .$ (a) Draw a force diagram for the balloon. (b) Calculate the buoyant force acting on the balloon. (c) Find the net force on the balloon and determine whether the balloon will rise or fall after it is released. (d) What maximum additional mass can the balloon support in equilibrium? (e) What happens to the balloon if the mass of the load is less than the value calculated in part (d)? (f) What limits the height to which the balloon can rise?

Averell H.

Carnegie Mellon University

A spherical weather balloon is filled with hydrogen until its radius is 3.00 $\mathrm{m}$ . Its total mass including the instruments it carries is 15.0 $\mathrm{kg}$ . (a) Find the buoyant force acting on the balloon, assuming the density of air is 1.29 $\mathrm{kg} / \mathrm{m}^{3}$ . (b) What is the net force acting on the balloon and its instruments after the balloon is released from the ground? (c) Why does the radius of the balloon tend to increase as it rises to higher altitude?

Salamat A.

Numerade Educator

The average human has a density of 945 $\mathrm{kg} / \mathrm{m}^{3}$ after inhaling and 1020 $\mathrm{kg} / \mathrm{m}^{3}$ after exhaling. (a) Without making any swimming movements, what percentage of the human body would be above the surface in the Dead Sea (a body of water with a density of about 1230 $\mathrm{kg} / \mathrm{m}^{3}$ ) in each of these cases? (b) Given that bone and muscle are denser than fat, what physical characteristics differentiate sinkers (those who tend to sink in water) from floaters (those who readily float)?

Averell H.

Carnegie Mellon University

On October $21,2001$ , Ian Ashpole of the United Kingdom achieved a record altitude of 3.35 $\mathrm{km}$ (11 000 ft) powered by 600 toy balloons filled with helium. Each filled balloon had a radius of about 0.50 $\mathrm{m}$ and an estimated mass of 0.30 $\mathrm{kg}$ . (a) Estimate the total buoyant force on the 600 balloons. (b) Estimate the net upward force on all 600 balloons. (c) Ashpole parachuted to Earth after the balloons began to burst at the high altitude and the system lost buoyancy. Why did the balloons burst?

Salamat A.

Numerade Educator

The gravitational force exerted on a solid object is 5.00 $\mathrm{N}$ as measured when the object is suspended from a spring scale as in Figure P9.26a. When the suspended object is submerged in water, the scale reads 3.50 $\mathrm{N}(\text { Fig. } \mathrm{P9} .26 \mathrm{b}) .$ Find the density

of the object.

Averell H.

Carnegie Mellon University

A cube of wood having an edge dimension of 20.0 $\mathrm{cm}$ and a density of $650 . \mathrm{kg} / \mathrm{m}^{3}$ floats on water. (a) What is the distance from the horizontal top surface of the cube

to the water level? (b) What mass of lead should be placed on the cube so that the top of the cube will be just level with the water surface?

Salamat A.

Numerade Educator

A light spring of force constant $k=160 \mathrm{N} / \mathrm{m}$ rests vertically on the bottom of a large beaker of water (Fig. $\mathrm{Pg} .28 \mathrm{a} ) . \mathrm{A}$ 5.00 $\mathrm{kg}$ block of wood (density $=650 \mathrm{kg} / \mathrm{m}^{3} )$ is connected to the spring, and the block-spring system is allowed to come to static equilibrium (Fig. P9.28b). What is the elongation $\Delta L$ of

the spring?

Averell H.

Carnegie Mellon University

A sample of an unknown material appears to weigh $300 . \mathrm{N}$ in air and $200 . \mathrm{N}$ when immersed in alcohol of specific gravity $0.700 .$ What are (a) the volume and (b) the density of the

material?

Salamat A.

Numerade Educator

An object weighing 300 $\mathrm{N}$ in air is immersed in water after being tied to a string connected to a balance. The scale now reads 265 $\mathrm{N}$ . Immersed in oil, the object appears to weigh 275 $\mathrm{N}$ . Find (a) the density of the object and (b) the density of the oil.

Averell H.

Carnegie Mellon University

A 1.00 -kg beaker containing 2.00 $\mathrm{kg}$ of oil (density $=916 \mathrm{kg} / \mathrm{m}^{3} )$ rests on a scale. A 2.00 $\mathrm{kg}$ block of iron is suspended from a spring scale and is completely submerged in the oil (Fig. P9.31). Find the equilibrium readings of both scales.

Salamat A.

Numerade Educator

A horizontal pipe narrows from a radius of 0.250 $\mathrm{m}$ to 0.100 $\mathrm{m} .$ If the speed of the water in the pipe is 1.00 $\mathrm{m} / \mathrm{s}$ in the larger-radius pipe, what is the speed in the smaller pipe?

Averell H.

Carnegie Mellon University

A large water tank is 3.00 $\mathrm{m}$ high and filled to the brim, the top of the tank open to the air. A small pipe with a faucet is attached to the side of the tank, 0.800 $\mathrm{m}$ above the ground. If the valve is opened, at what speed will water come out of the pipe?

Salamat A.

Numerade Educator

Water flowing through a garden hose of diameter 2.74 $\mathrm{cm}$ fills a $25.0-\mathrm{L}$ bucket in 1.50 $\mathrm{min}$ . (a) What is the speed of the water leaving the end of the hose? (b) A nozzle is now attached to the end of the hose. If the nozzle diameter is one-third the diameter of the hose, what is the speed of the water leaving the nozzle?

Averell H.

Carnegie Mellon University

(a) Calculate the mass flow rate (in grams per second) of blood $\left(\rho=1.0 \mathrm{g} / \mathrm{cm}^{3}\right)$ in an aorta with a cross-sectional area of 2.0 $\mathrm{cm}^{2}$ if the flow speed is $40 . \mathrm{cm} / \mathrm{s} .$ (b) Assume that the aorta branches to form a large number of capillaries with a combined crosssectional area of $3.0 \times 10^{3} \mathrm{cm}^{2} .$ What is the

flow speed in the capillaries?

Salamat A.

Numerade Educator

A liquid $(\rho=1.65 \mathrm{g} /$ $\mathrm{cm}^{3}$ ) flows through a horizontal pipe of varying cross

section as in Figure $\mathrm{P9} .36$ In the first section, the cross-sectional area is 10.0 $\mathrm{cm}^{2}$ the flow speed is $275 \mathrm{cm} / \mathrm{s},$ and the pressure is $1.20 \times$ $10^{5}$ Pa. In the second section, the cross-sectional area is 2.50 $\mathrm{cm}^{2}$ . Calculate the smaller section's (a) flow speed and (b) pressure.

Averell H.

Carnegie Mellon University

A hypodermic syringe contains a medicine with the density of water (Fig. P9.37). The barrel of the syringe has a cross-sectional area of $2.50 \times 10^{-5} \mathrm{m}^{2} .$ In the absence of a force on the plunger, the pressure everywhere is 1.00 $\mathrm{atm} . \mathrm{A}$ force $\overrightarrow{\mathbf{F}}$ of magnitude 2.00 $\mathrm{N}$ is exerted on the plunger, making medicine squirt from the needle. Determine the medicine's flow speed through the needle. Assume the pressure in the needle remains equal to 1.00 $\mathrm{atm}$ and that the syringe is horizontal.

Salamat A.

Numerade Educator

When a person inhales, air moves down the bronchus (windpipe) at 15 $\mathrm{cm} / \mathrm{s}$ . The average flow speed of the air doubles through a constriction in the bronchus. Assuming incompressible flow, determine the pressure drop in the constriction.

Averell H.

Carnegie Mellon University

A jet airplane in level fight has a mass of $8.66 \times$ $10^{4} \mathrm{kg},$ and the two wings have an estimated total area of 90.0 $\mathrm{m}^{2}$ . (a) What is the pressure difference between the lower and upper surfaces of the wings? (b) If the speed of air under the wings is $225 \mathrm{m} / \mathrm{s},$ what is the speed of the air over the wings? Assume air has a density of 1.29 $\mathrm{kg} / \mathrm{m}^{3}$ . (c) Explain why all aircraft have a ceiling, a maximum operational altitude.

Salamat A.

Numerade Educator

A man attaches a divider to an outdoor faucet so that water flows through a single pipe of radius 9.00 $\mathrm{mm}$ into two pipes, each with a radius of 6.00 $\mathrm{mm}$ . If water flows through the single pipe at 1.25 $\mathrm{m} / \mathrm{s}$ , calculate the speed of the water in the narrower pipes.

Averell H.

Carnegie Mellon University

In a water pistol, a piston drives water through a larger tube of radius 1.00 $\mathrm{cm}$ into a smaller tube of radius 1.00 $\mathrm{mm}$ as in Figure $\mathrm{P9} .41 .$ (a) If the pistol is fired horizontally at a height of $1.50 \mathrm{m},$ use ballistics to determine the time it takes water to travel from the nozale to the ground. (Neglect air resistance and assume atmospheric pressure is 1.00 atm. (b) If the range of the stream is to be 8.00 $\mathrm{m}$ , with what speed must the stream leave the nozale? (c) Given the areas of the nozzle and cylinder, use the equation of continuity to calculate the speed at which the plunger must be moved. (d) What is the pressure at the nozzle? (e) Use Bernoulli's equation to find the pressure needed in the larger cylinder. Can gravity terms be neglected? (f) Calculate the force that must be exerted on the trigger to achieve the desired range. (The force that must be exerted is due to pressure over and above atmospheric pressure.)

Salamat A.

Numerade Educator

Water moves through a constricted pipe in steady, ideal flow. At the lower point shown in Figure $P 9.42,$ the pressure is $1.75 \times 10^{5}$ Pa and the pipe radius is 3.00 $\mathrm{cm} .$ At

the higher point located at $y=2.50 \mathrm{m},$ the pressure is $1.20 \times 10^{5}$ Pa and the pipe

radius is 1.50 $\mathrm{cm} .$ Find the speed of flow (a) in the lower section and (b) in the upper

section. (c) Find the volume flow rate through the pipe.

Averell H.

Carnegie Mellon University

A jet of water squirts out horizontally from a hole near the bottom of the tank shown in Figure P9.43. If the hole has a diameter of 3.50 $\mathrm{mm}$ , what is the height $h$ of the water level in the tank?

Salamat A.

Numerade Educator

A large storage tank, open to the atmosphere at the top and filled with water, develops a small hole in its side at a point 16.0 $\mathrm{m}$ below the water level. If the rate of flow from the leak is $2.50 \times 10^{-3} \mathrm{m}^{3} / \mathrm{min}$ , determine (a) the speed at which the water leaves the hole and (b) the diameter of the hole.

Averell H.

Carnegie Mellon University

The inside diameters of the larger portions of the horizontal pipe depicted in Figure $\mathrm{P9} .45$ are 2.50 $\mathrm{cm} .$ Water flows to the right at a rate of $1.80 \times 10^{-4} \mathrm{m}^{3} / \mathrm{s}$ . Determine the inside diameter of the constriction.

Salamat A.

Numerade Educator

Water is pumped through a pipe of diameter 15.0 $\mathrm{cm}$ from the Colorado River up to Grand Canyon Village, on the rim of the canyon. The river is at 564 $\mathrm{m}$ elevation and the village is at 2096 $\mathrm{m} .$ (a) At what minimum pressure must the water be pumped to arrive at the village? (b) If 4500 $\mathrm{m}^{3}$ are pumped per day, what is the speed of the water in the pipe? (c) What

additional pressure is necessary to deliver this flow? Note: You may assume the free-fall acceleration and the density of air are constant over the given range of elevations.

Averell H.

Carnegie Mellon University

Old Faithful geyser in Yellowstone Park erupts at approximately 1 -hour intervals, and the height of the fountain reaches 40.0 $\mathrm{m}$ (Fig. P9.47). (a) Consider the rising stream as a series of separate drops. Analyze the free-fall motion of one of the drops to determine the speed at which the water leaves the ground. (b) Treat the rising stream as an ideal fluid in streamline flow. Use Bernoulli's equation to determine the speed of the water as it leaves ground level. (c) What is the pressure (above atmospheric pressure) in the heated underground chamber 175 $\mathrm{m}$ below the vent? You may assume the

chamber is large compared with the geyser vent.

Salamat A.

Numerade Educator

The Venturi tube shown in Figure $\mathrm{P} 9.48$ may be used as a fluid flowmeter. Suppose the device is used at a service station to measure the flow rate of gasoline $(\rho=7.00 \times$ $10^{2} \mathrm{kg} / \mathrm{m}^{3}$ ) through a hose having an outlet radius of 1.20 $\mathrm{cm} .$ If the difference in pressure is measured to be $P_{1}-P_{2}=1.20 \mathrm{kPa}$ and the radius of the inlet tube to the meter is $2.40 \mathrm{cm},$ find $(\mathrm{a})$ the speed of the gasoline as it leaves the hose and (b) the fluid flow rate in cubic meters per second.

Averell H.

Carnegie Mellon University

A square metal sheet 3.0 $\mathrm{cm}$ on a side and of negligible thickness is attached to a balance and inserted into a container of fluid. The contact angle is found to be zero, as shown in Figure P9.49a, and the balance to which the metal sheet is attached reads 0.40 N. A thin veneer of oil is then spread over the sheet, and the contact angle becomes $180^{\circ},$ as shown in Figure P9.49b. The balance now reads 0.39 $\mathrm{N}$ . What is the surface tension of the fluid?

Salamat A.

Numerade Educator

To lift a wire ring of radius 1.75 $\mathrm{cm}$ from the surface of a container of blood plasma, a vertical force of $1.61 \times 10^{-2} \mathrm{N}$ greater than the weight of the ring is required. Calculate the surface tension of blood plasma from this information.

Averell H.

Carnegie Mellon University

A certain fluid has a density of 1080 $\mathrm{kg} / \mathrm{m}^{3}$ and is observed to rise to a height of 2.1 $\mathrm{cm}$ in a $1.0-\mathrm{mm}$ -diameter tube. The contact angle between the wall and the fluid is zero. Calculate the surface tension of the fluid.

Salamat A.

Numerade Educator

Whole blood has a surface tension of 0.058 $\mathrm{N} / \mathrm{m}$ and a density of 1050 $\mathrm{kg} / \mathrm{m}^{3} .$ To what height can whole blood rise in a capillary blood vessel that has a radius of $2.0 \times 10^{-6} \mathrm{m}$ if the contact angle is zero?

Averell H.

Carnegie Mellon University

The block of ice (temperature $0^{\circ} \mathrm{C} )$ shown in Figure $\mathrm{P9.53}$ is drawn over a level surface lubricated by a layer of water 0.10 $\mathrm{mm}$ thick. Determine the magnitude of the force $\overrightarrow{\mathrm{F}}$ needed to pull the block with a constant speed of 0.50 $\mathrm{m} / \mathrm{s} .$ At $0^{\circ} \mathrm{C},$ the viscosity of water has the value $\eta=1.79 \times 10^{-3} \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}$

Salamat A.

Numerade Educator

A thin $1.5-\mathrm{mm}$ coating of glycerine has been placed between two microscope slides of width 1.0 $\mathrm{cm}$ and length 4.0 $\mathrm{cm} .$ Find the force required to pull one of the microscope slides at a constant speed of 0.30 $\mathrm{m} / \mathrm{s}$ relative to the other slide.

Averell H.

Carnegie Mellon University

A straight horizontal pipe with a diameter of 1.0 $\mathrm{cm}$ and a length of 50 $\mathrm{m}$ carries oil with a coefficient of viscosity of 0.12 $\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2} .$ At the output of the pipe, the flow rate is $8.6 \times$ $10^{-5} \mathrm{m}^{3} / \mathrm{s}$ and the pressure is 1.0 $\mathrm{atm}$ . Find the gauge pressure at the pipe input.

Salamat A.

Numerade Educator

The pulmonary artery, which connects the heart to the lungs, has an inner radius of 2.6 $\mathrm{mm}$ and is 8.4 $\mathrm{cm}$ long. If the pressure drop between the heart and lungs is 400 $\mathrm{Pa}$ , what is the average speed of blood in the pulmonary artery?

Averell H.

Carnegie Mellon University

Spherical particles of a protein of density 1.8 $\mathrm{g} / \mathrm{cm}^{3}$ are shaken up in a solution of $20^{\circ} \mathrm{C}$ water. The solution is allowed to stand for 1.0 $\mathrm{h}$ . If the depth of water in the tube is $5.0 \mathrm{cm},$ find the radius of the largest particles that remain in solution at the end of the hour.

Salamat A.

Numerade Educator

A hypodermic needle is 3.0 $\mathrm{cm}$ in length and 0.30 $\mathrm{mm}$ in diameter. What pressure difference between the input and output of the needle is required so that the flow rate of water

through it will be 1 $\mathrm{g} / \mathrm{s} ?$ (Use $1.0 \times 10^{-3} \mathrm{Pa} \cdot \mathrm{s}$ as the viscosity of water.)

Averell H.

Carnegie Mellon University

What radius needle should be used to inject a volume of $500 . \mathrm{cm}^{3}$ of a solution into a patient in 30.0 $\mathrm{min}$ ? Assume the length of the needle is 2.5 $\mathrm{cm}$ and the solution is elevated 1.0 $\mathrm{m}$ above the point of injection. Further, assume the viscosity and density of the solution are those of pure water, and that the pressure inside the vein is atmospheric.

Salamat A.

Numerade Educator

The aorta in humans has a diameter of about 2.0 $\mathrm{cm}$ and at certain times the blood speed through it is about 55 $\mathrm{cm} / \mathrm{s}$ . Is the blood flow turbulent? The density of whole

blood is $1050 \mathrm{kg} / \mathrm{m}^{3},$ and its coefficient of viscosity is $2.7 \times$ $10^{-3} \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}$

Averell H.

Carnegie Mellon University

Sucrose is allowed to diffuse along a 10 -cm length of tubing filled with water. The tube is 6.0 $\mathrm{cm}^{2}$ in cross-sectional area. The diffusion coefficient is equal to $5.0 \times 10^{-10} \mathrm{m}^{2} / \mathrm{s},$ and $8.0 \times 10^{-14} \mathrm{kg}$ is transported along the tube in 15 $\mathrm{s}$ . What is the difference in the concentration levels of sucrose at the two ends of the tube?

Salamat A.

Numerade Educator

Glycerin in water diffuses along a horizontal column that has a cross-sectional area of 2.0 $\mathrm{cm}^{2} .$ The concentration gradient is $3.0 \times 10^{-2} \mathrm{kg} / \mathrm{m}^{4},$ and the diffusion rate is found to be $5.7 \times 10^{-15} \mathrm{kg} / \mathrm{s}$ . Determine the diffusion coefficient.

Averell H.

Carnegie Mellon University

The viscous force on an oil drop is measured to be equal to $3.0 \times 10^{-13} \mathrm{N}$ when the drop is falling through air with a speed of $4.5 \times 10^{-4} \mathrm{m} / \mathrm{s} .$ If the radius of the drop is $2.5 \times$ $10^{-6} \mathrm{m},$ what is the viscosity of air?

Salamat A.

Numerade Educator

Small spheres of diameter 1.00 $\mathrm{mm}$ fall through $20^{\circ} \mathrm{C}$ water with a terminal speed of 1.10 $\mathrm{cm} / \mathrm{s}$ . Calculate the density of the spheres.

Averell H.

Carnegie Mellon University

A 200 . -kg load is hung on a wire of length $4.00 \mathrm{m},$ crosssectional area $0.200 \times 10^{-4} \mathrm{m}^{2},$ and Young's modulus $8.00 \times$ $10^{10} \mathrm{N} / \mathrm{m}^{2} .$ What is its increase in length?

Salamat A.

Numerade Educator

A 25.0 -m long steel cable with a cross-sectional area of $2.03 \times$ $10^{-3} \mathrm{m}^{2}$ is used to suspend a $3.50 \times 10^{3}-\mathrm{kg}$ container. By how much will the cable stretch once bearing the load?

Averell H.

Carnegie Mellon University

A plank 2.00 $\mathrm{cm}$ thick and 15.0 $\mathrm{cm}$ wide is firmly attached to the railing of a ship by clamps so that the rest of the board extends 2.00 $\mathrm{m}$ horizontally over the sea below. A man of mass 80.0 $\mathrm{kg}$ is forced to stand on the very end. If the end of the board drops by 5.00 $\mathrm{cm}$ because of the man's weight, find the shear modulus of the wood.

Salamat A.

Numerade Educator

Artificial diamonds can be made using high-pressure, high-temperature presses. Suppose an artificial diamond of volume $1.00 \times 10^{-6} \mathrm{m}^{3}$ is formed under a pressure of 5.00 GPa. Find the change in its volume when it is released from the press and brought to atmospheric pressure. Take the diamond's bulk modulus to be $B=194 \mathrm{GPa}$ .

Averell H.

Carnegie Mellon University

For safety in climbing, a mountaineer uses a nylon rope that is $50 . \mathrm{m}$ long and 1.0 $\mathrm{cm}$ in diameter. When supporting a 90 . -kg climber, the rope elongates 1.6 $\mathrm{m} .$ Find its Young's modulus.

Salamat A.

Numerade Educator

Assume that if the shear stress in steel exceeds about $4.00 \times 10^{8} \mathrm{N} / \mathrm{m}^{2},$ the steel ruptures. Determine the shearing force necessary to (a) shear a steel bolt 1.00 $\mathrm{cm}$ in diameter and (b) punch a $1.00-\mathrm{cm}$ -diameter hole in a steel plate

0.500 $\mathrm{cm}$ thick.

Averell H.

Carnegie Mellon University

Bone has a Young's modulus of $18 \times 10^{9}$ Pa. Under compression, it can withstand a stress of about $160 \times 10^{6} \mathrm{Pa}$ before breaking. Assume that a femur (thigh bone) is 0.50 $\mathrm{m}$ long, and calculate the amount of compression this bone can withstand before breaking.

Salamat A.

Numerade Educator

A stainless-steel orthodontic wire is applied to a tooth, as in Figure P9.72. The wire has an unstretched length of 3.1 $\mathrm{cm}$ and a radius of 0.11 $\mathrm{mm}$ . If the wire is stretched 0.10 $\mathrm{mm}$ , find the magnitude and direction of the force on the tooth. Disregard the width of the tooth and assume Young's modulus for stainless steel is $18 \times 10^{10} \mathrm{Pa}$ .

Averell H.

Carnegie Mellon University

A high-speed lifting mechanism supports an 800 .-kg object with a steel cable that is 25.0 $\mathrm{m}$ long and 4.00 $\mathrm{cm}^{2}$ in cross-sectional area. (a) Determine the elongation of the cable. (b) By what additional amount does the cable increase in length if the object is accelerated upward at a rate of 3.0 $\mathrm{m} / \mathrm{s}^{22}$ . (c) What is the greatest mass that can be accelerated upward at 3.0 $\mathrm{m} / \mathrm{s}^{2}$ if the stress in the cable is not to exceed the elastic limit of the cable, which is $2.2 \times 10^{8} \mathrm{Pa}$ ?

Salamat A.

Numerade Educator

The deepest point in the ocean is in the Mariana Trench, about 11 $\mathrm{km}$ deep. The pressure at the ocean floor is huge, about $1.13 \times 10^{8} \mathrm{N} / \mathrm{m}^{2} .(\mathrm{a})$ Calculate the change in volume of 1.00 $\mathrm{m}^{3}$ of water carried from the surface to the bottom of the Pacific. (b) The density of water at the surface is $1.03 \times 10^{3}$ $\mathrm{kg} / \mathrm{m}^{3}$ . Find its density at the bottom.

Averell H.

Carnegie Mellon University

Determine the elongation of the rod in Figure $P 9.75$ if it is under a tension of $5.8 \times 10^{3} \mathrm{N} .$

Salamat A.

Numerade Educator

The total cross-sectional area of the load-bearing calcified portion of the two forearm bones (radius and ulna) is approximately 2.4 $\mathrm{cm}^{2} .$ During a car crash, the forearm is slammed against the dashboard. The arm comes to rest from an initial speed of 80 $\mathrm{km} / \mathrm{h}$ in 5.0 $\mathrm{ms}$ . If the arm has an effective mass of 3.0 $\mathrm{kg}$ and bone material can withstand a maximum compressional stress of $16 \times 10^{7} \mathrm{Pa},$ is the arm likely to

withstand the crash?

Averell H.

Carnegie Mellon University

An iron block of volume 0.20 $\mathrm{m}^{3}$ is suspended from a spring scale and immersed in a flask of water. Then the iron block is removed, and an aluminum block of the same volume replaces it. (a) In which case is the buoyant force the greatest, for the iron block or the aluminum block? (b) In which case does the spring scale read the largest value? (c) Use the known densities of these materials to calculate the quantities requested in parts (a) and (b). Are your calculations consistent with your previous answers to parts (a) and (b)?

Salamat A.

Numerade Educator

Suppose two worlds, each having mass $M$ and radius $R$ coalesce into a single world. Due to gravitational contraction, the combined world has a radius of only $\frac{3}{4} R .$ What is the average density of the combined world as a multiple of $\rho_{0},$ the average density of the original two worlds?

Averell H.

Carnegie Mellon University

In most species of clingfish (family Gobiesocidae), pelvic and pectoral fins converge to form a suction cup edged by hairy structures that allow a good seal even on rough surfaces. Experiments have shown that a clingfish's suction cup can support up to 230 times the fish's body weight. Suppose a $30.0 \mathrm{g}$ northern clingfish has a suction cup disk area of 15.0 $\mathrm{cm}^{2}$ and the ambient water pressure is $1.10 \times 10^{5} \mathrm{Pa}$ . What ratio $P_{\text { cupt }} / P_{\text { ambient }}$ of the pressure inside the suction cup to the ambient pressure allows the fish to support 230 times its body weight?

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Take the density of blood to be $\rho$ and the distance between the feet and the heart to be $h_{H}$ . Ignore the flow of blood. (a) Show that the difference in blood pressure between the feet and the heart is given by $P_{F}-P_{H}=\rho g h_{H}$ (b) Take the density of blood to be $1.05 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$ and the distance between the heart and the feet to be 1.20 $\mathrm{m} .$ Find the difference in blood pressure between these two points. This problem indicates that pumping blood from the extremities is very difficult for the heart. The veins in the legs have valves in them that open when blood is pumped toward the heart and close when blood flows away from the heart. Also, pumping action produced by physical activities such as walking and breathing assists the heart.

Averell H.

Carnegie Mellon University

The approximate diameter of the aorta is $0.50 \mathrm{cm} ;$ that of a capillary is $10 . \mu \mathrm{m}$ . The approximate average blow speed is 1.0 $\mathrm{m} / \mathrm{s}$ in the aorta and 1.0 $\mathrm{cm} / \mathrm{s}$ in the capillaries. If all the blood in the aorta eventually flows through the capillaries, estimate the number of capillaries in the circulatory system.

Salamat A.

Numerade Educator

Superman attempts to drink water through a very long vertical straw as in Figure P9.82.

With his great strength, he achieves maximum possible suction. The walls of the straw don't collapse. (a) Find the maximum height through which he can lift the water. (b) Still thirsty, the Man of Steel repeats his attempt on the Moon, which has no atmosphere. Find the difference between the water levels inside and

outside the straw.

Averell H.

Carnegie Mellon University

The human brain and spinal cord are immersed in the cerebrospinal fluid. The fluid is normally continuous between the cranial and spinal cavities and exerts a pressure of 100 to 200 $\mathrm{mm}$ of $\mathrm{H}_{2} \mathrm{O}$ above the prevailing atmospheric pressure. In medical work, pressures are often measured in units of $\mathrm{mm}$ of $\mathrm{H}_{2} \mathrm{O}$ because body fluids, including the cerebrospinal fluid, typically have nearly the same density as water. The pressure of the cerebrospinal fluid can be measured by means of a spinal tap. A hollow tube is inserted into the spinal column, and the height to which the fluid rises is observed, as shown in Figure P9.83. If the fluid rises to a height of $160 . \mathrm{mm}$ , we write its gauge pressure as $160 . \mathrm{mm} \mathrm{H}_{2} \mathrm{O} .$ (a) Express this pressure in pascals, in atmospheres, and in millimeters of mercury. (b) Sometimes it is necessary to determine whether

an accident victim has suffered a crushed vertebra that is blocking the flow of cerebrospinal fluid in the spinal column. In other cases, a physician may suspect that a tumor or other growth is blocking the spinal column and inhibiting the flow of cerebrospinal fluid. Such conditions can be investigated by means of the Queckensted test. In this procedure, the veins in the patient's neck are compressed to make the blood pressure rise in the brain. The increase in pressure in the blood vessels is transmitted to the cerebrospinal fluid. What should be the normal effect on the height of the fluid in the spinal tap? (c) Suppose compressing the veins had no effect on the level of the fluid. What might account for this phenomenon?

Salamat A.

Numerade Educator

A hydrometer is an instrument used to determine liquid density. A simple one is sketched in Figure P9.84. The bulb of a syringe is squeezed and released to lift a sample of the liquid of interest into a tube containing a calibrated rod of known density. (Assume the rod is cylindrical.) The rod, of length $L$ and average density $\rho_{0},$ floats partially immersed in the liquid of density $\rho .$ A length $h$ of the rod protrudes above the surface of the liquid. Show that the density of the liquid is given by

Averell H.

Carnegie Mellon University

Figure P9.85 shows a water tank with a valve. If the valve is opened, what is the maximum height attained by the stream of water coming out of the right side of the tank? Assume $h=$ $10.0 \mathrm{m}, L=2.00 \mathrm{m},$ and $\theta=30.0^{\circ},$ and that the cross-sectional area at $A$ is very large compared with that at $B$ .

Salamat A.

Numerade Educator

A helium-filled balloon, whose envelope has a mass of $0.25 \mathrm{kg},$ is tied to a $2.0-\mathrm{m-long}$ , 0.050 -kg string. The balloon is spherical with a radius of 0.40 $\mathrm{m}$ . When released, it lifts a length $h$ of the string and then remains in equilibrium, as in Figure P9.86. Determine the value of $h .$ Hint: Only that part of the string above the floor contributes to the

load being supported by the balloon.

Averell H.

Carnegie Mellon University

A light spring of constant $k=90.0 \quad \mathrm{N} / \mathrm{m}$ is attached vertically to a table

(Fig. P9.87a). A 2.00-g balloon is filled with helium (density = 0.179 $\mathrm{kg} / \mathrm{m}^{3}$ ) to a volume of 5.00 $\mathrm{m}^{3}$ and is then connected to the spring, causing the spring to stretch as shown in Figure $\mathrm{P} 9.87 \mathrm{b}$ . Determine the extension distance $L$ when the

balloon is in equilibrium.

Salamat A.

Numerade Educator

A U-tube open at both ends is partially filled with water (Fig. $\quad \mathrm{P} 9.8 \mathrm{a}$ ). Oil $(\rho=$ 750 $\mathrm{kg} / \mathrm{m}^{3}$ ) is then poured into the right arm and forms a column $L=5.00 \mathrm{cm}$ high (Fig. $\mathrm{P9} .88 \mathrm{b} )$ . (a) Determine the difference $h$ in the heights of the two liquid surfaces. (b) The right arm is then shielded from any air motion while air is blown across the top of the left arm until the surfaces of the two liquids are at the same height (Fig. P9. $\mathrm{Pg} .88 \mathrm{c} ) .$ Determine the speed of the air being blown across the left arm. Assume the density of air is 1.29 $\mathrm{kg} / \mathrm{m}^{3}$

Averell H.

Carnegie Mellon University

In about $1657,$ Otto von Guericke, inventor of the airpump, evacuated a sphere made of two brass hemispheres (Fig. $\mathrm{Pg} .89 ) .$ Two teams of eight horses each could pull the hemispheres apart only on some trials and then with great- est difficulty, with the resulting sound likened to a can non firing. Find the force $F$ required to pull the thin-walled evacuated hemispheres apart in terms of $R,$ the radius of the hemispheres, $P$ the pressure inside the hemispheres, and atmospheric pressure $P_{0}$ .

Salamat A.

Numerade Educator

Oil having a density of 930 $\mathrm{kg} / \mathrm{m}^{3}$ floats on water. A rectangular block of wood 4.00 $\mathrm{cm}$ high and with a density of 960 $\mathrm{kg} / \mathrm{m}^{3}$ floats partly in the oil and partly in the water. The oil completely covers the block. How far below the interface between the two liquids is the bottom of the block?

Averell H.

Carnegie Mellon University

A water tank open to the atmosphere at the top has two small holes punched in its side, one above the other. The holes are 5.00 $\mathrm{cm}$ and 12.0 $\mathrm{cm}$ above the floor. How high does water stand in the tank if the two streams of water hit the floor at the same place?

Salamat A.

Numerade Educator