Exercise No. 7 1. A \( 60 \mathrm{~kg} \) woman walking through an alley accidentally broke the heel of her left high-heeled shoes. Now the woman tries to balance on one heel of a pair of high-heeled shoes. If the heel is circular and has a radius of \( 0.500 \mathrm{~cm} \), what pressure does she exert on the floor? 2. A barrel contains a \( 0.120 \mathrm{~m} \) layer of oil floating on water that is \( 0.250 \mathrm{~m} \) deep. The density of the oil is \( 600 \mathrm{~kg} / \mathrm{m}^{3} \). What is the gauge pressure at the oil- water interface and at the bottom of the barrel? 3. A \( 950-\mathrm{kg} \) cylindrical can buoy floats vertically in salt water. The diameter of the buoy is \( 0.900 \mathrm{~m} \). Calculate the additional distance the buoy will sink when a \( 70.0-\mathrm{kg} \) man stands on top of it. 4. A frog in a hemispherical pod, as shown in the figure on the right, just floats without sinking into a sea of blue-green ooze with density \( 1.35 \mathrm{~g} / \mathrm{cm} 3 \). If the pod has radius \( 6.00 \mathrm{~cm} \) and negligible mass, what is the mass of the frog? 5. A Ping-Pong 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? 6. A piece of aluminum with mass \( 1.00 \mathrm{~kg} \) and density \( 2700 \mathrm{~kg} / \mathrm{m}^{3} \) is suspended from a string and then completely immersed in a container of water. Calculate the tension in the string (a) before and (b) after the metal is immersed. 7. Water runs into a fountain, filling all the pipes, at a steady rate of \( 0.750 \mathrm{~m}^{3} / \mathrm{s} \). (a) How fast will it shoot out of a hole \( 4.50 \mathrm{~cm} \) in diameter? (b) At what speed will it shoot out if the diameter of the hole is three times as large? 8. You need to extend a 2.50-inch-diameter pipe, but you have only a 1.00-inch-diameter pipe on hand. You make a fitting to connect these pipes end to end. If the water is flowing at \( 6.00 \mathrm{~cm} \) s in the wide pipe, how fast will it be flowing through the narrow one? 9. A sealed tank containing seawater to a height of \( 11.0 \mathrm{~m} \) also contains air above the water at a gauge pressure of \( 3.00 \mathrm{~atm} \). Water flows out from the bottom through a small hole. How fast is this water moving? 10. A U-shaped tube open to the air at both ends contains some mercury. A quantity of water is carefully poured into the left arm of the U-shaped tube until the vertical height of the water column is \( 15.0 \mathrm{~cm} \) as shown in the figure. (a) What is the gauge pressure at the water-mercury interface? (b) Calculate the vertical distance \( h \) from the top of the mercury in the righthand arm of the tube to the top of the water in the lefthand arm. Answers: 1. \( 7.5 \times 10^{6} \mathrm{~N} / \mathrm{m}^{2} \) 2. (a) \( 760 \mathrm{~Pa} \), (b) \( 3.6 \times 10^{3} \mathrm{~Pa} \) 3. \( 0.107 \mathrm{~m} \) 4. \( 0.611 \mathrm{~kg} \) 5. \( 0.258 \mathrm{~N} \) 6. (a) \( 9.80 \mathrm{~N} \), (b) \( 6.17 \mathrm{~N} \) 7. (a) \( 472 \mathrm{~m} / \mathrm{s} \), (b) \( 52.4 \mathrm{~m} / \mathrm{s} \) 8. \( 37.5 \mathrm{~cm} / \mathrm{s} \) 9. \( 28.4 \mathrm{~m} / \mathrm{s} \) 10. (a) \( 1470 \mathrm{~Pa} \), (b) \( 13.9 \mathrm{~cm} \)
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A $siphon$ ($\textbf{Fig. P12.88}$) is a convenient device for removing liquids from containers. To establish the flow, the tube must be initially filled with fluid. Let the fluid have density $\rho$, and let the atmospheric pressure be $p$$_a$$_t$$_m$. Assume that the cross-sectional area of the tube is the same at all points along it. (a) If the lower end of the siphon is at a distance $h$ below the surface of the liquid in the container, what is the speed of the fluid as it flows out the lower end of the siphon? (Assume that the container has a very large diameter, and ignore any effects of viscosity.) (b) A curious feature of a siphon is that the fluid initially flows "uphill." What is the greatest height $H$ that the high point of the tube can have if flow is still to occur? $\textbf{ELEPHANTS UNDER PRESSURE.}$ An elephant can swim or walk with its chest several meters underwater while the animal breathes through its trunk, which remains above the water surface and acts like a snorkel. The elephant’s tissues are at an increased pressure due to the surrounding water, but the lungs are at atmospheric pressure because they are connected to the air through the trunk. The figure shows the gauge pressures in an elephant's lungs and abdomen when the elephant’s chest is submerged to a particular depth in a lake. In this situation, the elephant's diaphragm, which separates the lungs from the abdomen, must sustain the difference in pressure between the lungs and the abdomen. The diaphragm of an elephant is typically 3.0 cm thick and 120 cm in diameter. (See "Why Doesn't the Elephant Have a Pleural Space?" by John B. West, $Physiology$, Vol. 17:47–50, April 1, 2002.) Figure P12.88 (FIGURE CAN'T COPY)
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A siphon (Fig. P12.88) is a convenient device for removing liquids from containers. To establish the flow, the tube must be initially filled with fluid. Let the fluid have density $\rho,$ and let the atmospheric pressure be $p_{\text {atm }}$. Assume that the cross-sectional area of the tube is the same at all points along it. (a) If the lower end of the siphon is at a distance $h$ below the surface of the liquid in the container, what is the speed of the fluid as it flows out the lower end of the siphon? (Assume that the container has a very large diameter, and ignore any effects of viscosity.) (b) A curious feature of a siphon is that the fluid initially flows "uphill." What is the greatest height $H$ that the high point of the tube can have if flow is still to occur? An elephant can swim or walk with its chest several meters underwater while the animal breathes through its trunk, which remains above the water surface and acts like a snorkel. The elephant's tissues are at an increased pressure due to the surrounding water, but the lungs are at atmospheric pressure because they are connected to the air through the trunk. The figure shows the gauge pressures in an elephant's lungs and abdomen when the elephant's chest is submerged to a particular depth in a lake. In this situation, the elephant's diaphragm, which separates the lungs from the abdomen, must sustain the difference in pressure between the lungs and the abdomen. The diaphragm of an elephant is typically $3.0 \mathrm{~cm}$ thick and $120 \mathrm{~cm}$ in diameter. (See "Why Doesn't the Elephant Have a Pleural Space?" by John B. West, Physiology, Vol. $17: 47-50,$ April $1,2002 .)$
A cylindrical diving bell has a radius of $750 \mathrm{~cm}$ and a height of $2.50 \mathrm{~m}$. The bell includes a top compartment that holds an undersea adventurer. A bottom compartment separated from the top by a sturdy grating holds a tank of compressed air with a valve to release air into the bell, a second valve that can release air from the bell into the sea, a third valve that regulates the entry of seawater for ballast, a pump that removes the ballast to increase buoyancy, and an electric heater that maintains a constant temperature of $20.0^{\circ} \mathrm{C}$. The total mass of the bell and all of its apparatuses is $4350 \mathrm{~kg}$. The density of seawater is $1025 \mathrm{~kg} / \mathrm{m}^{3}$. (a) An $80.0 \mathrm{~kg}$ adventurer enters the bell. How many liters of seawater should be moved into the bell so that it is neutrally buoyant? (b) By carefully regulating ballast, the bell is made to descend into the sea at a rate of $1.0 \mathrm{~m} / \mathrm{s}$. Compressed air is released from the tank to raise the pressure in the bell to match the pressure of the seawater outside the bell. As the bell descends, at what rate should air be released through the first valve? (Hint: Derive an expression for the number of moles of air in the bell $n$ as a function of depth $y ;$ then differentiate this to obtain $d n / d t$ as a function of $d y / d t .)$ (c) If the compressed air tank is a fully loaded, specially designed, $600 \mathrm{ft}^{3}$ tank, which means it contains that volume of air at standard temperature and pressure ( $0^{\circ} \mathrm{C}$ and $1 \mathrm{~atm}$ ), how deep can the bell descend?
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