m3⎠ = 970 kg
m3.
Discussion
Her density is less than the fluid density. We expect this because she floats. Body density is one indicator of a person’s percent body fat, of
interest in medical diagnostics and athletic training. (See Figure 11.24.)
Figure 11.24 Subject in a “fat tank,” where he is weighed while completely submerged as part of a body density determination. The subject must completely empty his lungs
and hold a metal weight in order to sink. Corrections are made for the residual air in his lungs (measured separately) and the metal weight. His corrected submerged weight,
his weight in air, and pinch tests of strategic fatty areas are used to calculate his percent body fat.
There are many obvious examples of lower-density objects or substances floating in higher-density fluids—oil on water, a hot-air balloon, a bit of cork
in wine, an iceberg, and hot wax in a “lava lamp,” to name a few. Less obvious examples include lava rising in a volcano and mountain ranges
floating on the higher-density crust and mantle beneath them. Even seemingly solid Earth has fluid characteristics.
More Density Measurements
One of the most common techniques for determining density is shown in Figure 11.25.
Figure 11.25 (a) A coin is weighed in air. (b) The apparent weight of the coin is determined while it is completely submerged in a fluid of known density. These two
measurements are used to calculate the density of the coin.
CHAPTER 11 | FLUID STATICS 377
An object, here a coin, is weighed in air and then weighed again while submerged in a liquid. The density of the coin, an indication of its authenticity,
can be calculated if the fluid density is known. This same technique can also be used to determine the density of the fluid if the density of the coin is
known. All of these calculations are based on Archimedes’ principle.
Archimedes’ principle states that the buoyant force on the object equals the weight of the fluid displaced. This, in turn, means that the object appears
to weigh less when submerged; we call this measurement the object’s apparent weight. The object suffers an apparent weight loss equal to the
weight of the fluid displaced. Alternatively, on balances that measure mass, the object suffers an apparent mass loss equal to the mass of fluid
displaced. That is
(11.44)
apparent weight loss = weight of fluid displaced
or
(11.45)
apparent mass loss = mass of fluid displaced.
The next example illustrates the use of this technique.
Example 11.10 Calculating Density: Is the Coin Authentic?
The mass of an ancient Greek coin is determined in air to be 8.630 g. When the coin is submerged in water as shown in Figure 11.25, its
apparent mass is 7.800 g. Calculate its density, given that water has a density of 1.000 g/cm3 and that effects caused by the wire suspending
the coin are negligible.
Strategy
To calculate the coin’s density, we need its mass (which is given) and its volume. The volume of the coin equals the volume of water displaced.
The volume of water displaced V w can be found by solving the equation for density ρ = m
V for V .
Solution
The volume of water is V w = m w
ρ w where m w is the mass of water displaced. As noted, the mass of the water displaced equals the apparent
mass loss, which is m w = 8.630 g−7.800 g = 0.830 g . Thus the volume of water is V w =
0.830 g
1.000 g/cm3 = 0.830 cm3 . This is also the
volume of the coin, since it is completely submerged. We can now find the density of the coin using the definition of density:
(11.46)
ρ c = m c
V = 8.630 g
c
0.830 cm3 = 10.4 g/cm3.
Discussion
You can see from Table 11.1 that this density is very close to that of pure silver, appropriate for this type of ancient coin. Most modern
counterfeits are not pure silver.
This brings us back to Archimedes’ principle and how it came into being. As the story goes, the king of Syracuse gave Archimedes the task of
determining whether the royal crown maker was supplying a crown of pure gold. The purity of gold is difficult to determine by color (it can be diluted
with other metals and still look as yellow as pure gold), and other analytical techniques had not yet been conceived. Even ancient peoples, however,
realized that the density of gold was greater than that of any other then-known substance. Archimedes purportedly agonized over his task and had
his inspiration one day while at the public baths, pondering the support the water gave his body. He came up with his now-famous principle, saw how
to apply it to determine density, and ran naked down the streets of Syracuse crying “Eureka!” (Greek for “I have found it”). Similar behavior can be
observed in contemporary physicists from time to time!
PhET Explorations: Buoyancy
When will objects float and when will they sink? Learn how buoyancy works with blocks. Arrows show the applied forces, and you can modify the
properties of the blocks and the fluid.
Figure 11.26 Buoyancy (http://cnx.org/content/m42196/1.7/buoyancy_en.jar)
11.8 Cohesion and Adhesion in Liquids: Surface Tension and Capillary Action
Cohesion and Adhesion in Liquids
Children blow soap bubbles and play in the spray of a sprinkler on a hot summer day. (See Figure 11.27.) An underwater spider keeps his air supply
in a shiny bubble he carries wrapped around him. A technician draws blood into a small-diameter tube just by touching it to a drop on a pricked finger.
378 CHAPTER 11 | FLUID STATICS
A premature infant struggles to inflate her lungs. What is the common thread? All these activities are dominated by the attractive forces between
atoms and molecules in liquids—both within a liquid and between the liquid and its surroundings.
Attractive forces between molecules of the same type are called cohesive forces. Liquids can, for example, be held in open containers because
cohesive forces hold the molecules together. Attractive forces between molecules of different types are called adhesive forces. Such forces cause
liquid drops to cling to window panes, for example. In this section we examine effects directly attributable to cohesive and adhesive forces in liquids.
Cohesive Forces
Attractive forces between molecules of the same type are called cohesive forces.
Adhesive Forces
Attractive forces between molecules of different types are called adhesive forces.
Figure 11.27 The soap bubbles in this photograph are caused by cohesive forces among molecules in liquids. (credit: Steve Ford Elliott)
Surface Tension
Cohesive forces between molecules cause the surface of a liquid to contract to the smallest possible surface area. This general effect is called
surface tension. Molecules on the surface are pulled inward by cohesive forces, reducing the surface area. Molecules inside the liquid experience
zero net force, since they have neighbors on all sides.
Surface Tension
Cohesive forces between molecules cause the surface of a liquid to contract to the smallest possible surface area. This general effect is called
surface tension.
Making Connections: Surface Tension
Forces between atoms and molecules underlie the macroscopic effect called surface tension. These attractive forces pull the molecules closer
together and tend to minimize the surface area. This is another example of a submicroscopic explanation for a macroscopic phenomenon.
The model of a liquid surface acting like a stretched elastic sheet can effectively explain surface tension effects. For example, some insects can walk
on water (as opposed to floating in it) as we would walk on a trampoline—they dent the surface as shown in Figure 11.28(a). Figure 11.28(b) shows
another example, where a needle rests on a water surface. The iron needle cannot, and does not, float, because its density is greater than that of
water. Rather, its weight is supported by forces in the stretched surface that try to make the surface smaller or flatter. If the needle were placed point
down on the surface, its weight acting on a smaller area would break the surface, and it would sink.
CHAPTER 11 | FLUID STATICS 379
Figure 11.28 Surface tension supporting the weight of an insect and an iron needle, both of which rest on the surface without penetrating it. They are not floating; rather, they are supported by the surface of the liquid. (a) An insect leg dents the water surface. F ST is a restoring force (surface tension) parallel to the surface. (b) An iron needle
similarly dents a water surface until the restoring force (surface tension) grows to equal its weight.
Surface tension is proportional to the strength of the cohesive force, which varies with the type of liquid. Surface tension γ is defined to be the force
F per unit length L exerted by a stretched liquid membrane:
(11.47)
γ = FL.
Table 11.3 lists values of γ for some liquids. For the insect of Figure 11.28(a), its weight w is supported by the upward components of the surface tension force: w = γL sin θ , where L is the circumference of the insect’s foot in contact with the water. Figure 11.29 shows one way to measure surface tension. The liquid film exerts a force on the movable wire in an attempt to reduce its surface area. The magnitude of this force depends on
the surface tension of the liquid and can be measured accurately.
Surface tension is the reason why liquids form bubbles and droplets. The inward surface tension force causes bubbles to be approximately spherical
and raises the pressure of the gas trapped inside relative to atmospheric pressure outside. It can be shown that the gauge pressure P inside a
spherical bubble is given by
(11.48)
P = 4γ r,
where r is the radius of the bubble. Thus the pressure inside a bubble is greatest when the bubble is the smallest. Another bit of evidence for this is
balloon.
Figure 11.29 Sliding wire device used for measuring surface tension; the device exerts a force to reduce the film’s surface area. The force needed to hold the wire in place is
F = γL = γ(2 l) , since there are two liquid surfaces attached to the wire. This force remains nearly constant as the film is stretched, until the film approaches its breaking point.
380 CHAPTER 11 | FLUID STATICS
Figure 11.30 With the valve closed, two balloons of different sizes are attached to each end of a tube. Upon opening the valve, the smaller balloon decreases in size with the
air moving to fill the larger balloon. The pressure in a spherical balloon is inversely proportional to its radius, so that the smaller balloon has a greater internal pressure than the larger balloon, resulting in this flow.
Table 11.3 Surface Tension of Some Liquids[1]
Liquid
Surface tension γ(N/m)
Water at 0ºC
0.0756
Water at 20ºC
0.0728
Water at 100ºC
0.0589
Soapy water (typical)
0.0370
Ethyl alcohol
0.0223
Glycerin
0.0631
Mercury
0.465
Olive oil
0.032
Tissue fluids (typical)
0.050
Blood, whole at 37ºC 0.058
Blood plasma at 37ºC 0.073
Gold at 1070ºC
1.000
Oxygen at −193ºC
0.0157
Helium at −269ºC
0.00012
Example 11.11 Surface Tension: Pressure Inside a Bubble
Calculate the gauge pressure inside a soap bubble 2.00×10−4 m in radius using the surface tension for soapy water in Table 11.3. Convert
this pressure to mm Hg.
Strategy
The radius is given and the surface tension can be found in Table 11.3, and so P can be found directly from the equation P = 4γ
r .
Solution
Substituting r and g into the equation P = 4γ
r , we obtain
(11.49)
P = 4γ r = 4(0.037 N/m) = 740 N/m2 = 740 Pa.
2.00×10−4 m
We use a conversion factor to get this into units of mm Hg:
(11.50)
P = ⎛
1.00 mm Hg
⎝740 N/m2⎞⎠ 133 N/m2 = 5.56 mm Hg.
Discussion
Note that if a hole were to be made in the bubble, the air would be forced out, the bubble would decrease in radius, and the pressure inside
would increase to atmospheric pressure (760 mm Hg).
1. At 20ºC unless otherwise stated.
CHAPTER 11 | FLUID STATICS 381
Our lungs contain hundreds of millions of mucus-lined sacs called alveoli, which are very similar in size, and about 0.1 mm in diameter. (See Figure
11.31.) You can exhale without muscle action by allowing surface tension to contract these sacs. Medical patients whose breathing is aided by a
positive pressure respirator have air blown into the lungs, but are generally allowed to exhale on their own. Even if there is paralysis, surface tension
in the alveoli will expel air from the lungs. Since pressure increases as the radii of the alveoli decrease, an occasional deep cleansing breath is
needed to fully reinflate the alveoli. Respirators are programmed to do this and we find it natural, as do our companion dogs and cats, to take a
cleansing breath before settling into a nap.
Figure 11.31 Bronchial tubes in the lungs branch into ever-smaller structures, finally ending in alveoli. The alveoli act like tiny bubbles. The surface tension of their mucous lining aids in exhalation and can prevent inhalation if too great.
The tension in the walls of the alveoli results from the membrane tissue and a liquid on the walls of the alveoli containing a long lipoprotein that acts
as a surfactant (a surface-tension reducing substance). The need for the surfactant results from the tendency of small alveoli to collapse and the air
to fill into the larger alveoli making them even larger (as demonstrated in Figure 11.30). During inhalation, the lipoprotein molecules are pulled apart and the wall tension increases as the radius increases (increased surface tension). During exhalation, the molecules slide back together and the
surface tension decreases, helping to prevent a collapse of the alveoli. The surfactant therefore serves to change the wall tension so that small
alveoli don’t collapse and large alveoli are prevented from expanding too much. This tension change is a unique property of these surfactants, and is
not shared by detergents (which simply lower surface tension). (See Figure 11.32.)
Figure 11.32 Surface tension as a function of surface area. The surface tension for lung surfactant decreases with decreasing area. This ensures that small alveoli don’t
collapse and large alveoli are not able to over expand.
If water gets into the lungs, the surface tension is too great and you cannot inhale. This is a severe problem in resuscitating drowning victims. A
similar problem occurs in newborn infants who are born without this surfactant—their lungs are very difficult to inflate. This condition is known as
hyaline membrane disease and is a leading cause of death for infants, particularly in premature births. Some success has been achieved in treating
hyaline membrane disease by spraying a surfactant into the infant’s breathing passages. Emphysema produces the opposite problem with alveoli.
Alveolar walls of emphysema victims deteriorate, and the sacs combine to form larger sacs. Because pressure produced by surface tension
decreases with increasing radius, these larger sacs produce smaller pressure, reducing the ability of emphysema victims to exhale. A common test
for emphysema is to measure the pressure and volume of air that can be exhaled.
Making Connections: Take-Home Investigation
(1) Try floating a sewing needle on water. In order for this activity to work, the needle needs to be very clean as even the oil from your fingers can
be sufficient to affect the surface properties of the needle. (2) Place the bristles of a paint brush into water. Pull the brush out and notice that for a
short while, the bristles will stick together. The surface tension of the water surrounding the bristles is sufficient to hold the bristles together. As
the bristles dry out, the surface tension effect dissipates. (3) Place a loop of thread on the surface of still water in such a way that all of the thread
is in contact with the water. Note the shape of the loop. Now place a drop of detergent into the middle of the loop. What happens to the shape of
the loop? Why? (4) Sprinkle pepper onto the surface of water. Add a drop of detergent. What happens? Why? (5) Float two matches parallel to
each other and add a drop of detergent between them. What happens? Note: For each new experiment, the water needs to be replaced and the
bowl washed to free it of any residual detergent.
382 CHAPTER 11 | FLUID STATICS
Adhesion and Capillary Action
Why is it that water beads up on a waxed car but does not on bare paint? The answer is that the adhesive forces between water and wax are much
smaller than those between water and paint. Competition between the forces of adhesion and cohesion are important in the macroscopic behavior of
liquids. An important factor in studying the roles of these two forces is the angle θ between the tangent to the liquid surface and the surface. (See
Figure 11.33.) The contact angle θ is directly related to the relative strength of the cohesive and adhesive forces. The larger the strength of the cohesive force relative to the adhesive force, the larger θ is, and the more the liquid tends to form a droplet. The smaller θ is, the smaller the
relative strength, so that the adhesive force is able to flatten the drop. Table 11.4 lists contact angles for several combinations of liquids and solids.
Contact Angle
The angle θ between the tangent to the liquid surface and the surface is called the contact angle.
Figure 11.33 In the photograph, water beads on the waxed car paint and flattens on the unwaxed paint. (a) Water forms beads on the waxed surface because the cohesive
forces responsible for surface tension are larger than the adhesive forces, which tend to flatten the drop. (b) Water beads on bare paint are flattened considerably because the
adhesive forces between water and paint are strong, overcoming surface tension. The contact angle θ is directly related to the relative strengths of the cohesive and
adhesive forces. The larger θ is, the larger the ratio of cohesive to adhesive forces. (credit: P. P. Urone)
One important phenomenon related to the relative strength of cohesive and adhesive forces is capillary action—the tendency of a fluid to be raised
or suppressed in a narrow tube, or capillary tube. This action causes blood to be drawn into a small-diameter tube when the tube touches a drop.
Capillary Action
The tendency of a fluid to be raised or suppressed in a narrow tube, or capillary tube, is called capillary action.
If a capillary tube is placed vertically into a liquid, as shown in Figure 11.34, capillary action will raise or suppress the liquid inside the tube depending
on the combination of substances. The actual effect depends on the relative strength of the cohesive and adhesive forces and, thus, the contact
angle θ given in the table. If θ is less than 90º , then the fluid will be raised; if θ is greater than 90º , it will be suppressed. Mercury, for example,
has a very large surface tension and a large contact angle with glass. When placed in a tube, the surface of a column of mercury curves downward,
somewhat like a drop. The curved surface of a fluid in a tube is called a meniscus. The tendency of surface tension is always to reduce the surface
area. Surface tension thus flattens the curved liquid surface in a capillary tube. This results in a downward force in mercury and an upward force in
water, as seen in Figure 11.34.
CHAPTER 11 | FLUID STATICS 383
Figure 11.34 (a) Mercury is suppressed in a glass tube because its contact angle is greater than 90º . Surface tension exerts a downward force as it flattens the mercury,
suppressing it in the tube. The dashed line shows the shape the mercury surface would have without the flattening effect of surface tension. (b) Water is raised in a glass tube
because its contact angle is nearly 0º . Surface tension therefore exerts an upward force when it flattens the surface to reduce its area.
Table 11.4 Contact Angles of Some Substances
Interface
Contact angle Θ
Mercury–glass
140º
Water–glass
0º
Water–paraffin
107º
Water–silver
90º
Organic liquids (most)–glass 0º
Ethyl alcohol–glass
0º
Kerosene–glass
26º
Capillary action can move liquids horizontally over very large distances, but the height to which it can raise or suppress a liquid in a tube is limited by
its weight. It can be shown that this height h is given by
(11.51)
h = 2γ cos θ
ρgr .
If we look at the different factors in this expression, we might see how it makes good sense. The height is directly proportional to the surface tension
γ , which is its direct cause. Furthermore, the height is inversely proportional to tube radius—the smaller the radius r , the higher the fluid can be
raised, since a smaller tube holds less mass. The height is also inversely proportional to fluid density ρ , since a larger density means a greater mass
in the same volume. (See Figure 11.35.)
384 CHAPTER 11 | FLUID STATICS
Figure 11.35 (a) Capillary action depends on the radius of a tube. The smaller the tube, the greater the height reached. The height is negligible for large-radius tubes. (b) A
denser fluid in the same tube rises to a smaller height, all other factors being the same.
Example 11.12 Calculating Radius of a Capillary Tube: Capillary Action: Tree Sap
Can capillary action be solely responsible for sap rising in trees? To answer this question, calculate the radius of a capillary tube that wo