Chapter 14

Fresh water flows horizontally from pipe section 1 of cross-sectional area \(A_{1}\) into pipe section 2 of cross-sec tional area \(A_{2}\). Figure \(14-52\) gives a plot of the pressure difference \(p_{2}-p_{1}\) versus the inverse area squared \(A_{1}^{-2}\) that would be expected for a volume flow rate of a certain value if the water flow were laminar under all circumstances. The scale on the vertical axis is set by \(\Delta p_{s}=300 \mathrm{kN} / \mathrm{m}^{2} .\) For the conditions of the figure, what are the values of (a) \(A_{2}\) and (b) the volume flow rate? \({ }^{\infty} 69\) A liquid of density \(900 \mathrm{~kg} / \mathrm{m}^{3}\) flows through a horizontal pipe that has a cross-sectional area of \(1.90 \times 10^{-2} \mathrm{~m}^{2}\) in region \(A\) and a cross-sectional area of \(9.50 \times 10^{-2} \mathrm{~m}^{2}\) in region \(B\). The pressure difference between the two regions is \(7.20 \times 10^{3} \mathrm{~Pa}\). What are (a) the volume flow rate and (b) the mass flow rate?

A liquid of density \(900 \mathrm{~kg} / \mathrm{m}^{3}\) flows through a horizontal pipe that has a cross-sectional area of \(1.90 \times 10^{-2} \mathrm{~m}^{2}\) in region \(A\) and a cross-sectional area of \(9.50 \times 10^{-2} \mathrm{~m}^{2}\) in region \(B\). The pressure difference between the two regions is \(7.20 \times 10^{3} \mathrm{~Pa}\). What are (a) the volume flow rate and (b) the mass flow rate?

About one-third of the body of a person floating in the Dead Sea will be above the waterline. Assuming that the human body density is \(0.98 \mathrm{~g} / \mathrm{cm}^{3}\), find the density of the water in the Dead Sea. (Why is it so much greater than \(1.0 \mathrm{~g} / \mathrm{cm}^{3} ?\) )

A simple open U-tube contains mercury. When \(11.2 \mathrm{~cm}\) of water is poured into the right arm of the tube, how high above its initial level does the mercury rise in the left arm?

Suppose that your body has a uniform density of \(0.95\) times that of water. (a) If you float in a swimming pool, what fraction of your body's volume is above the water surface? Quicksand is a fluid produced when water is forced up into sand, moving the sand grains away from one another so they are no longer locked together by friction. Pools of quicksand can form when water drains underground from hills into valleys where there are sand pockets. (b) If you float in a deep pool of quicksand that has a density \(1.6\) times that of water, what fraction of your body's volume is above the quicksand surface? (c) Are you unable to breathe?

A glass ball of radius \(2.00 \mathrm{~cm}\) sits at the bottom of a container of milk that has a density of \(1.03 \mathrm{~g} / \mathrm{cm}^{3} .\) The normal force on the ball from the container's lower surface has magnitude \(9.48 \times 10^{-2} \mathrm{~N}\). What is the mass of the ball?

Caught in an avalanche, a skier is fully submerged in flowing snow of density \(96 \mathrm{~kg} / \mathrm{m}^{3}\). Assume that the average density of the skier, clothing, and skiing equipment is \(1020 \mathrm{~kg} / \mathrm{m}^{3} .\) What percentage of the gravitational force on the skier is offset by the buoyant force from the snow?

An object hangs from a spring balance. The balance registers \(30 \mathrm{~N}\) in air, \(20 \mathrm{~N}\) when this object is immersed in water, and \(24 \mathrm{~N}\) when the object is immersed in another liquid of unknown density. What is the density of that other liquid?

In an experiment, a rectangular block with height \(h\) is allowed to float in four separate liquids. In the first liquid, which is water, it floats fully submerged. In liquids \(A, B\), and \(C\), it floats with heights \(h / 2,2 h / 3\), and \(h / 4\) above the liquid surface, respectively. What are the relative densities (the densities relative to that of water) of (a) \(A,(\mathrm{~b}) B\), and \((\mathrm{c}) C ?\)

Shows a modified U-tube: the right arm is shorter than the left arm. The open end of the right arm is height \(d=10.0 \mathrm{~cm}\) above the laboratory bench. The radius throughout the tube is \(1.50 \mathrm{~cm}\). Water is gradually poured into the open end of the left arm until the water begins to flow out the open end of the right arm. Then a liquid of density \(0.80 \mathrm{~g} / \mathrm{cm}^{3}\) is gradually added to the left arm until its height in that arm is \(8.0 \mathrm{~cm}\) (it does not mix with the water). How much water flows out of the right arm?

Shows a siphon, which is a device for removing liquid from a container. Tube \(A B C\) must initially be filled, but once this has been done, liquid will flow through the tube until the liquid surface in the container is level with the tube opening at \(A\). The liquid has density \(1000 \mathrm{~kg} / \mathrm{m}^{3}\) and negligible viscosity. The distances shown are \(h_{1}=25 \mathrm{~cm}, d=\) \(12 \mathrm{~cm}\), and \(h_{2}=40 \mathrm{~cm} .\) (a) With what speed does the liquid emerge from the tube at \(C ?(b)\) If the atmospheric pressure is \(1.0 \times 10^{5} \mathrm{~Pa}\), what is the pressure in the liquid at the topmost point \(B ?\) (c) Theoretically, what is the greatest possible height \(h_{1}\) that a siphon can lift water?

When you cough, you expel air at high speed through the trachea and upper bronchi so that the air will remove excess mucus lining the pathway. You produce the high speed by this procedure: You breathe in a large amount of air, trap it by closing the glottis (the narrow opening in the larynx), increase the air pressure by contracting the lungs, partially collapse the trachea and upper bronchi to narrow the pathway, and then expel the air through the pathway by suddenly reopening the glottis. Assume that during the expulsion the volume flow rate is \(7.0 \times 10^{-3} \mathrm{~m}^{3} / \mathrm{s}\). What multiple of \(343 \mathrm{~m} / \mathrm{s}\) (the speed of sound \(v_{s}\) ) is the airspeed through the trachea if the trachea diameter (a) remains its normal value of \(14 \mathrm{~mm}\) and \((\mathrm{b})\) contracts to \(5.2 \mathrm{~mm}\) ?

A tin can has a total volume of \(1200 \mathrm{~cm}^{3}\) and a mass of \(130 \mathrm{~g}\). How many grams of lead shot of density \(11.4 \mathrm{~g} / \mathrm{cm}^{3}\) could it carry without sinking in water?

What is the minimum area (in square meters) of the top surface of an ice slab \(0.441 \mathrm{~m}\) thick floating on fresh water that will hold up a \(938 \mathrm{~kg}\) automobile? Take the densities of ice and fresh water to be \(917 \mathrm{~kg} / \mathrm{m}^{3}\) and \(998 \mathrm{~kg} / \mathrm{m}^{3}\), respectively.

A \(8.60 \mathrm{~kg}\) sphere of radius \(6.22 \mathrm{~cm}\) is at a depth of \(2.22 \mathrm{~km}\) in seawater that has an average density of \(1025 \mathrm{~kg} / \mathrm{m}^{3}\). What are the (a) gauge pressure, (b) total pressure, and (c) corresponding total force compressing the sphere's surface? What are (d) the magnitude of the buoyant force on the sphere and (e) the magnitude of the sphere's acceleration if it is free to move? Take atmospheric pressure to be \(1.01 \times 10^{5} \mathrm{~Pa}\).

For seawater of density \(1.03 \mathrm{~g} / \mathrm{cm}^{3}\), find the weight of water on top of a submarine at a depth of \(255 \mathrm{~m}\) if the horizontal cross-sectional hull area is \(2200.0 \mathrm{~m}^{2} .(\mathrm{b})\) In atmospheres, what water pressure would a diver experience at this depth?

The sewage outlet of a house constructed on a slope is \(6.59 \mathrm{~m}\) below street level. If the sewer is \(2.16 \mathrm{~m}\) below street level, find the minimum pressure difference that must be created by the sewage pump to transfer waste of average density \(1000.00 \mathrm{~kg} / \mathrm{m}^{3}\) from outlet to sewer.