1. | In FIGURE P14.40, how much force does the fluid exert on the end of the cylinder at A?

   FIGURE P14.40

   

   FIGURE P14.40 Full Alternative Text

2. How much force does the fluid exert on the end of the cylinder at B?

| BIO Postural hypotension is the occurrence of low systolic blood pressure when a person stands up too quickly from a reclining position. A brain blood pressure lower than $90\text{mm}$ of Hg can cause fainting or lightheadedness. In a healthy adult, the automatic constriction and expansion of blood vessels keep the brain blood pressure constant while posture is changing, but disease or aging can weaken this response. If the blood pressure in your brain is $118\text{mm of Hg}$ while lying down, what would it be when you stand up if this automatic response failed? Assume your brain is $40\text{cm}$ from your heart and the density of blood is $1060\text{kg}\text{/}{\text{m}}^3$.

|| A friend asks you how much pressure is in your car tires. You know that the tire manufacturer recommends 30 psi, but it’s been a while since you’ve checked. You can’t find a tire gauge in the car, but you do find the owner’s manual and a ruler. Fortunately, you’ve just finished taking physics, so you tell your friend, “I don’t know, but I can figure it out.” From the owner’s manual you find that the car’s mass is 1500 kg. It seems reasonable to assume that each tire supports one-fourth of the weight. With the ruler you find that the tires are 15 cm wide and the flattened segment of the tire in contact with the road is 13 cm long. What answer—in psi—will you give your friend?

1. || The 70 kg student in FIGURE P14.43 balances a 1200 kg elephant on a hydraulic lift. What is the diameter of the piston the student is standing on?

   FIGURE P14.43

   

   FIGURE P14.43 Full Alternative Text

2. When a second student joins the first, the piston sinks $35\text{cm}$. What is the second student’s mass?

||| A 55 kg cheerleader uses an oil-filled hydraulic lift to hold four 110 kg football players at a height of 1.0 m. If her piston is 16 cm in diameter, what is the diameter of the football players’ piston?

|| A U-shaped tube, open to the air on both ends, contains mercury. Water is poured into the left arm until the water column is 10.0 cm deep. How far upward from its initial position does the mercury in the right arm rise?

|| Glycerin is poured into an open U-shaped tube until the height in both sides is 20 cm. Ethyl alcohol is then poured into one arm until the height of the alcohol column is 20 cm. The two liquids do not mix. What is the difference in height between the top surface of the glycerin and the top surface of the alcohol?

|| CALC An aquarium of length $L$, width (front to back) $W$, and depth $D$ is filled to the top with liquid of density $\rho$.

1. Find an expression for the force of the liquid on the bottom of the aquarium.

2. Find an expression for the force of the liquid on the front window of the aquarium.

3. Evaluate the forces for a 100-cm-long, 35-cm-wide, 40-cm-deep aquarium filled with water.

|| CALC It’s possible to use the ideal-gas law to show that the density of the earth’s atmosphere decreases exponentially with height. That is, $\rho ={\rho }_0\text{exp(}-z\text{/}{z}_0)$, where $z$ is the height above sea level, ${\rho }_0$ is the density at sea level (you can use the Table 14.1 value), and $z_0$ is called the scale height of the atmosphere.

1. Determine the value of $z_0$.

   Hint: What is the weight of a column of air?

2. What is the density of the air in Denver, at an elevation of 1600 m? What percent of sea-level density is this?

|| BIO The average density of the body of a fish is $1080\text{kg}\text{/}{\text{m}}^3$. To keep from sinking, a fish increases its volume by inflating an internal air bladder, known as a swim bladder, with air. By what percent must the fish increase its volume to be neutrally buoyant in fresh water? The density of air at $20\text{°}\text{C}$ is $1.19\text{kg}\text{/}{\text{m}}^3$.

|| CALC A cylinder with cross-section area $A$ floats with its long axis vertical in a liquid of density $\rho$.

1. Pressing down on the cylinder pushes it deeper into the liquid. Find an expression for the force needed to push the cylinder distance $x$ deeper into the liquid and hold it there.

2. A 4.0-cm-diameter cylinder floats in water. How much work must be done to push the cylinder 10 cm deeper into the water?

|| A 30-cm-tall, 4.0-cm-diameter plastic tube has a sealed bottom. 250 g of lead pellets are poured into the bottom of the tube, whose mass is 30 g, then the tube is lowered into a liquid. The tube floats with 5.0 cm extending above the surface. What is the density of the liquid?

|| One day when you come into physics lab you find several plastic hemispheres floating like boats in a tank of fresh water. Each lab group is challenged to determine the heaviest rock that can be placed in the bottom of a plastic boat without sinking it. You get one try. Sinking the boat gets you no points, and the maximum number of points goes to the group that can place the heaviest rock without sinking. You begin by measuring one of the hemispheres, finding that it has a mass of 21 g and a diameter of 8.0 cm. What is the mass of the heaviest rock that, in perfectly still water, won’t sink the plastic boat?

|| INT A spring with spring constant $35\text{N}\text{/}\text{m}$ is attached to the ceiling, and a 5.0-cm-diameter, 1.0 kg metal cylinder is attached to its lower end. The cylinder is held so that the spring is neither stretched nor compressed, then a tank of water is placed underneath with the surface of the water just touching the bottom of the cylinder. When released, the cylinder will oscillate a few times but, damped by the water, quickly reach an equilibrium position. When in equilibrium, what length of the cylinder is submerged?

|| A plastic “boat” with a $25{\text{cm}}^2$ square cross section floats in a liquid. One by one, you place $50\text{g}$ masses inside the boat and measure how far the boat extends below the surface. Your data are as follows:

Draw an appropriate graph of the data and, from the slope and intercept of the best-fit line, determine the mass of the boat and the density of the liquid.

|| A 355 mL soda can is 6.2 cm in diameter and has a mass of 20 g.

Such a soda can half full of water is floating upright in water. What length of the can is above the water level?

|| A nuclear power plant draws $3.0\times {10}^6\text{L}\text{/}\text{min}$ of cooling water from the ocean. If the water is drawn in through two parallel, 3.0-m-diameter pipes, what is the water speed in each pipe?

1. || A nonviscous liquid of density $\rho$ flows at speed $v_0$ through a horizontal pipe that expands smoothly from diameter $d_0$ to a larger diameter $d_1$. The pressure in the narrower section is $p_0$. Find an expression for the pressure $p_1$ in the wider section.

2. A pressure gauge reads $50\text{kPa}$ as water flows at $10.0\text{m}\text{/}\text{s}$ through a $16.8\text{-}\text{cm}\text{-}\text{diameter}$ horizontal pipe. What is the reading of a pressure gauge after the pipe has expanded to $20.0\text{cm}$ in diameter?

||| BIO A tree loses water to the air by the process of transpiration at the rate of $110\text{g}\text{/}\text{h}$. This water is replaced by the upward flow of sap through vessels in the trunk. If the trunk contains 2000 vessels, each $100\mu \text{m}$ in diameter, what is the upward speed in mm/s of the sap in each vessel? The density of tree sap is $1040\text{kg}\text{/}{\text{m}}^3$.

|| Water flows from the pipe shown in FIGURE P14.59 with a speed of $4.0\text{m}\text{/}\text{s}$.

1. What is the water pressure as it exits into the air?

2. What is the height $h$ of the standing column of water?

FIGURE P14.59

||| Water from a vertical pipe emerges as a 10-cm-diameter cylinder and falls straight down 7.5 m into a bucket. The water exits the pipe with a speed of $2.0\text{m}\text{/}\text{s}$. What is the diameter of the column of water as it hits the bucket?

| A hurricane wind blows across a $6.0\text{m}\times 15.0\text{m}$ flat roof at a speed of $130\text{km}\text{/}\text{h}$.

1. Is the air pressure above the roof higher or lower than the pressure inside the house? Explain.

2. What is the pressure difference?

3. How much force is exerted on the roof? If the roof cannot withstand this much force, will it “blow in” or “blow out”?

||| Air flows through the tube shown in FIGURE P14.62 at a rate of $1200{\text{cm}}^3\text{/}\text{s}$. Assume that air is an ideal fluid. What is the height $h$ of mercury in the right side of the U-tube?

FIGURE P14.62

|| Air flows through the tube shown in FIGURE P14.63. Assume that air is an ideal fluid.

FIGURE P14.63

1. What are the air speeds $v_1$ and $v_2$ at points 1 and 2?

2. What is the volume flow rate?

|| INT A water tank of height $h$ has a small hole at height y. The water is replenished to keep $h$ from changing. The water squirting from the hole has range x. The range approaches zero as $y\to 0$ because the water squirts right onto the ground. The range also approaches zero as $y\to h$ because the horizontal velocity becomes zero. Thus there must be some height $y$ between 0 and $h$ for which the range is a maximum.

1. Find an algebraic expression for the flow speed $v$ with which the water exits the hole at height y.

2. Find an algebraic expression for the range of a particle shot horizontally from height $y$ with speed v.

3. Combine your expressions from parts a and b. Then find the maximum range $x_{\text{max}}$ and the height $y$ of the hole. “Real” water won’t achieve quite this range because of viscosity, but it will be close.

1. ||| A cylindrical tank of radius $R$, filled to the top with a liquid, has a small hole in the side, of radius $r$, at distance $d$ below the surface. Find an expression for the volume flow rate through the hole.

2. A 4.0-mm-diameter hole is $1.0\text{m}$ below the surface of a 2.0-m-diameter tank of water. What is the rate, in mm/min, at which the water level will initially drop if the water is not replenished?

|| 20°C water flows through a 2.0-m-long, 6.0-mm-diameter pipe. What is the maximum flow rate in L/min for which the flow is laminar?

|| 20°C water flows at 1.5 m/s through a 10-m-long, 1.0-mm-diameter horizontal tube and then exits into the air. What is the gauge pressure in kPa at the point where the water enters the tube?

|| BIO The 30-cm-long left coronary artery is 4.6 mm in diameter. Blood pressure drops by 3.0 mm of mercury over this distance. What are the (a) average blood speed and (b) volume flow rate in L/min through this artery?

||| FIGURE P14.69 shows a tank of 20°C water with a nozzle 25 cm below the surface. Bernoulli’s equation applies to the flow between points 1 and 2 on the streamline, but not to the flow between points 2 and 3. What is the volume flow rate through the nozzle in L/min? Assume that the water level in the tank is held constant.

FIGURE P14.69

|| BIO There is a disk of cartilage between each pair of vertebrae in your spine. Young’s modulus for cartilage is $1.0\times {10}^6\text{N}\text{/}{\text{m}}^2$. Suppose a relaxed disk is $4.0\text{cm}$ in diameter and $5.0\text{mm}$ thick. If a disk in the lower spine supports half the weight of a $66\text{kg}$ person, by how many mm does the disk compress?

||| The 1.0-m-tall cylinder shown in FIGURE CP14.71 contains air at a pressure of 1 atm. A very thin, frictionless piston of negligible mass is placed at the top of the cylinder, to prevent any air from escaping, then mercury is slowly poured into the cylinder until no more can be added without the cylinder overflowing. What is the height $h$ of the column of compressed air?

FIGURE CP14.71

Hint: Boyle’s law, which you learned in chemistry, says $p_1{V}_1=p_2{V}_2$ for a gas compressed at constant temperature, which we will assume to be the case.

||| The bottom of a steel “boat” is a $5.0\text{m}\times 10\text{m}\times 2.0\text{cm}$ piece of steel $({\rho }_{\text{steel}}=7900\text{k}\text{g}\text{/}{\text{m}}^3)$. The sides are made of 0.50-cm-thick steel. What minimum height must the sides have for this boat to float in perfectly calm water?

||| CALC The tank shown in FIGURE CP14.73 is open at the top and completely filled with a liquid of density $\rho$. The right face is not permanently attached to the tank but, instead, is held against a rubber seal by the tension in a spring. To prevent leakage, the spring must both pull with sufficient strength and prevent a torque from pushing the bottom of the right face out.

1. What minimum spring tension is needed?

2. If the spring has the minimum tension, at what height $d$ from the bottom must it be attached?

FIGURE CP14.73

||| In FIGURE CP14.74, a cone of density ${\rho }_{\text{o}}$ and total height $l$ floats in a liquid of density ${\rho }_{\text{f}}$. The height of the cone above the liquid is h. What is the ratio $h\text{/}l$ of the exposed height to the total height?

FIGURE CP14.74

||| Disk brakes, such as those in your car, operate by using pressurized oil to push outward on a piston. The piston, in turn, presses brake pads against a spinning rotor or wheel, as seen in FIGURE CP14.75. Consider a 15 kg industrial grinding wheel, 26 cm in diameter, spinning at 900 rpm. The brake pads are actuated by 2.0-cm-diameter pistons, and they contact the wheel an average distance 12 cm from the axis. If the coefficient of kinetic friction between the brake pad and the wheel is 0.60, what oil pressure is needed to stop the wheel in 5.0 s?

FIGURE CP14.75