Chapter 9

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?

Four acrobats of mass \(75.0 \mathrm{~kg}, 68.0 \mathrm{~kg}, 62.0 \mathrm{~kg}\), and \(55.0 \mathrm{~kg}\) form a human tower, with each acrobat standing on the shoulders of another acrobat. The \(75.0-\mathrm{kg}\) acrobat is at the bottom of the tower. (a) What is the normal force acting on the \(75-\mathrm{kg}\) acrobat? (b) If the area of each of the 75.0-kg acrobat's shoes is \(425 \mathrm{~cm}^{2}\), what average pressure (not including atmospheric pressure) does the column of acrobats exert on the floor? (c) Will the pressure be the same if a different acrobat is on the bottom?

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

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 \(415 \mathrm{~kg} / \mathrm{m}^{3}\).

A \(200-\mathrm{kg}\) load is hung on a wire of length \(4.00 \mathrm{~m}\), cross-sectional 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?

Comic-book superheroes are sometimes able to punch holes through steel walls. (a) If the ultimate shear strength of steel is taken to be \(2.50 \times 10^{8} \mathrm{~Pa}\), what force is required to punch through a steel plate \(2.00 \mathrm{~cm}\) thick? Assume the superhero's fist has crosssectional area of \(1.00 \times 10^{2} \mathrm{~cm}^{2}\) and is approximately circular. (b) Qualitatively, what would happen to the superhero on delivery of the punch? What physical law applies?

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.

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 \(90-\mathrm{kg}\) climber, the rope elongates \(1.6 \mathrm{~m}\). Find its Young's modulus.

A stainless-steel orthodontic wire is applied to a tooth, as in Figure P9.14. 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 \(18 \times 10^{10} \mathrm{~Pa}\).

Bone has a Young's modulus of \(18 \times 10^{9} \mathrm{~Pa}\). 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.

A high-speed lifting mechanism supports an \(800-\mathrm{kg}\) 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 \(3.0 \mathrm{~m} / \mathrm{s}^{2} ?\) (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 \(2.2 \times 10^{8} \mathrm{~Pa}\) ?

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?

(a) Calculate the absolute pressure at the bottom of a fresh-water 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 \(35.0 \mathrm{~cm}\) ?

A collapsible plastic bag (Fig. P9.23) 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 in order to infuse glucose into the vein? Assume the specific gravity of the solution is \(1.02\).

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)?

A \(62.0-\mathrm{kg}\) 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?

A wooden block of volume \(5.24 \times 10^{-4} \mathrm{~m}^{3}\) floats in water, and a small steel object of mass \(m\) is placed on top of the block. When \(m=0.310 \mathrm{~kg}\), the system is in equilibrium, and the top of the wooden block is at the level of the water. (a) What is the density of the wood? (b) What happens to the block when the steel object is replaced by a second steel object with a mass less than \(0.310 \mathrm{~kg}\) ? What happens to the block when the steel object is replaced by yet another steel object with a mass greater than \(0.310 \mathrm{~kg}\) ?

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?

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?

On October 21, 2001, Ian Ashpole of the United Kingdom achieved a record altitude of \(3.35 \mathrm{~km}\) (11 \(000 \mathrm{ft}\) ) 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?

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?

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?

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.

A \(1.00-\mathrm{kg}\) 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. P9.43). Find the equilibrium readings of both scales.

Water flowing through a garden hose of diameter \(2.74 \mathrm{~cm}\) fills a \(25.0\) - 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?

(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 crosssectional 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 cross-sectional area of \(3.0 \times 10^{3} \mathrm{~cm}^{2}\). What is the flow speed in the capillaries?

A hypodermic syringe contains a medicine with the density of water (Fig. P9.47). 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 \(1.00 \mathrm{~atm}\). 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.

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 \(\mathrm{P} 9.51\). (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 nozzle to the ground. (Neglect air resistance and assume atmospheric pressure is \(1.00 \mathrm{~atm}\).) (b) If the range of the stream is to be \(8.00 \mathrm{~m}\), with what speed must the stream leave the nozzle? (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.)

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.

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.

Old Faithful geyser in Yellowstone Park erupts at approximately 1-hour intervals, and the height of the fountain reaches \(40.0 \mathrm{~m}\) (Fig. P9.57). (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.

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.

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 \(1.0\)-mm-diameter tube. The contact angle between the wall and the fluid is zero. Calculate the surface tension of the fluid.

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?

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.

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.)

What radius needle should be used to inject a volume of \(500 \mathrm{~cm}^{3}\) of a solution into a patient in \(30 \mathrm{~min}\) ? 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.

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 \(2.7 \times 10^{-3} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}\).

Sucrose is allowed to diffuse along a \(10-\mathrm{cm}\) length of tubing filled with water. The tube is \(6.0 \mathrm{~cm}^{2}\) in crosssectional 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?

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.

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?

The true weight of an object can be measured in a vacuum, where buoyant forces are absent. A measurement in air, however, is disturbed by buoyant forces. An object of volume \(V\) is weighed in air on an equal-arm balance with the use of counterweights of density \(\rho\). Representing the density of air as \(\rho_{\text {air }}\) and the balance reading as \(F_{g}^{\prime}\), show that the true weight \(F_{g}\) is $$ F_{g}=F_{g}^{\prime}+\left(V-\frac{F_{g}^{\prime}}{\rho g}\right) \rho_{\text {air }} g $$

As a first approximation, Earth's continents may be thought of as granite blocks floating in a denser rock (called peridotite) in the same way that ice floats in water. (a) Show that a formula describing this phenomenon is $$ \rho_{g} t=\rho_{p} d $$ where \(\rho_{g}\) is the density of granite \(\left(2.8 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\right), \rho_{p}\) is the density of peridotite \(\left(3.3 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\right), t\) is the thickness of a continent, and \(d\) is the depth to which a continent floats in the peridotite. (b) If a continent sinks \(5.0 \mathrm{~km}\) into the peridotite layer (this surface may be thought of as the ocean floor), what is the thickness of the continent?

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.

The approximate diameter of the aorta is \(0.50 \mathrm{~cm}\); that of a capillary is \(10 \mu \mathrm{m}\). The approximate average blood flow 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.

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?

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 \(B\).

In about 1657 , Otto von Guericke, inventor of the air pump, evacuated a sphere made of two brass hemispheres (Fig. P9.89). Two teams of eight horses each could pull the hemispheres apart only on some trials and then "with greatest difficulty," with the resulting sound likened to a cannon 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 \(P_{0}\).