Pharos University in
Alexandria
Faculty of Allied Medical Science
Biomedical Physics (GRBP-101)
Prof. Dr. Mostafa. M. Mohamed
Vice Dean
Dr. Mervat Mostafa
Department of Medical Biophysics
Pharos University
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Chapter (3)
Part (1)
Fluids
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The differences in the physical properties of solids,
liquids, and gases are explained in terms of the
forces that bind the molecules. In a solid, the
molecules are rigidly bound; a solid therefore has a
definite shape and volume. The molecules
constituting a liquid are not bound together with
sufficient force to maintain a definite shape, but the
binding is sufficiently strong to maintain a definite
volume.
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Force and Pressure in a Fluid
Solids and fluids transmit forces differently. When a force is applied
to one section of a solid, this force is transmitted to the other parts
of the solid with its direction unchanged. Because of a fluid’s ability
to flow, it transmits a force uniformly in all directions. Therefore, the
pressure at any point in a fluid at rest is the same in all directions.
The pressure in a fluid increases with depth because of the weight
of the fluid above. In a fluid of constant density ρ, the difference in
pressure, P2 - P1, between two points separated by a vertical
distance h is
P2 - P1 = ρ gh
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Fluid pressure is often measured in millimeters of mercury, or torr [after
Evangelista Torricelli (1608-1674), the first person to understand the
nature of atmospheric pressure]. One torr is the pressure exerted by a
column of mercury that is 1 mm high. Pascal, abbreviated as Pa is
another commonly used unit of pressure. The relationship between the
torr and several of the other units used to measure pressure follows:
1 torr = 1mm Hg
= 13.5 mm water
= 1.33 x 103 dyn/cm2
= 1.32 x 10-3 atm
= 1.93x10-2 psi
= 1.33 x 102 Pa (N/m2 )
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Pascal’s Principle
When a force F1 is applied on a surface of a liquid that has an area A1, the
pressure in the liquid increases by an amount P, given by
The ratio A2/ A1 is analogous to the mechanical advantage of a lever.
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An illustration of Pascal’s principle.
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Hydrostatic Skeleton
Let us now calculate the hydrostatic forces inside a moving worm. Consider a
worm that has a radius r. Assume that the circular muscles running around its
circumference are uniformly distributed along the length of the worm and that
the effective area of the muscle per unit length of the worm is AM. As the circular
muscles contract, they generate a force fM, which, along each centimeter of the
worm’s length, is
fM = SAM
The hydrostatic skeleton.
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Here S is the force produced per unit area of the muscle. (Note that
fM is in units of force per unit length.) This force produces a
pressure inside the worm. The magnitude of the pressure can be
calculated with the aid, which shows a section of the worm. The
length of the section is L. If we were to cut this section in half
lengthwise,, the force due to the pressure inside the cylinder would
tend to push the two halves apart. This force is calculated as
follows. The surface area A along the cut midsection is
A = L x 2r
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Calculating pressure inside a worm.
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and
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Archimedes’ Principle
Archimedes’ principle states that a body partially or wholly
submerged in a fluid is buoyed upward by a force that is equal in
magnitude to the weight of the displaced fluid. The derivation of
this principle is found in basic physics texts. We will now use
Archimedes’ principle to calculate the power required to remain
afloat in water and to study the buoyancy of fish.
Power Required to Remain Afloat
Whether an animal sinks or floats in water depends on its density. If its density
is greater than that of water, the animal must perform work in order not to
sink. We will calculate the power P required for an animal of volume V and
density ρ to float with a fraction f of its volume submerged. but our approach
to the problem will be different.
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Because a fraction f of the animal is submerged, the animal is
buoyed up by a force FB given by
FB
=
gf Vρw
where ρw is the density of water. The force FB is simply the weight
of the displaced water. The net downward force FB on the animal is
the difference between its weight gVρ and the buoyant force; that
is,
FD = gV ρ — gVf ρ w = gV(ρ — f ρ w)
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Surface Tension
The molecules constituting a liquid exert attractive forces on each
other. A molecule in the interior of the liquid is surrounded by an
equal number of neighboring molecules in all directions.
Therefore, the net resultant intermolecular force on an interior
molecule is zero. The situation is different, however, near the
surface of the liquid. Because there are no molecules above the
surface, a molecule here is pulled predominantly in one direction,
toward the interior of the surface. This causes the surface of a
liquid to contract and behave somewhat like a stretched
membrane. This contracting tendency results in a surface tension
that resists an increase in the free surface of the liquid.
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Surface tension.
total force FT produced by surface tension tangential to a liquid surface of
boundary length L is
FT = TL
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When a liquid is contained in a vessel, the surface molecules near the wall are
attracted to the wall. This attractive force is called adhesion. At the same time,
however, these molecules are also subject to the attractive cohesive force
exerted by the liquid, which pulls the molecules in the opposite direction. If the
adhesive force is greater than the cohesive force, the liquid wets the container
wall, and the liquid surface near the wall is curved upward.
If the adhesion is greater than the cohesion, a liquid in a narrow tube
will rise to a specific height h, which can be calculated from the following
considerations. The weight W of the column of the supported liquid is
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Fm = 2πRT
The upward component of this force supports the weight of the
column of liquid that is,
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Angle of contact when (a) liquid wets the wall and (b) liquid does not wet the wall.
(a) Capillary rise. (b) Capillary depression.
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Therefore, the height of the column is
If the adhesion is smaller than the cohesion, the angle Ө is greater
than 90◦. In this case, the height of the fluid in the tube is
depressed. Equation still applies, yielding a negative number for h.
These effects are called capillary action.
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Soil Water
Most soil is porous with narrow spaces between the smallm
particles. These spaces act as capillaries and in part govern the
motion of water through the soil. When water enters soil, it
penetrates the spaces between the small particles and adheres to
them. If the water did not adhere to the particles, it would run
rapidly through the soil until it reached solid rock. Plant life would
then be severely restricted. Because of adhesion and the resulting
capillary action, a significant fraction of the water that enters the
soil is retained by it. For a plant to withdraw this water, the roots
must apply a negative pressure, or suction, to the moist soil.
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Fine-grained soil (a) holds water more tightly than coarse-grained soil (b).
saturated with water. As the amount of water in the soil decreases, the SMT
increases. In loam, for example, with a moisture content of 20% the SMT is
about 0.19 atm. When the moisture content drops to 12%, the SMT increases
to 0.76 atm.
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Contraction of Muscles
An examination of skeletal muscles shows that they consist of smaller muscle
fibers, which in turn are composed of yet smaller units called myofibrils. Further,
examination with an electron microscope reveals that the myofibril is composed
of two types of threads, one made of myosin, which is about 160 A˚ (1 A˚ = 10-8
cm) in diameter, and the other made of actin, which has a diameter of about 50
A˚. Each myosin-actin unit is about 1 mm long. The threads are aligned in a
regular pattern with spaces between threads so that the threads can slide past
one another
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Contraction of muscles.
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If the average diameter of the threads is D, the number of threads N per
square centimeter of muscle is approximately
The maximum pulling force Ff produced by the surface tension on each fiber
The total maximum force Fm due to all the fibers in a 1-cm2 area of muscle is
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The average diameter D of the muscle fibers is about 100 A˚ (10-6 cm).
Therefore, the maximum contracting force that can be produced by
surface tension per square centimeter of muscle area is
Fm = T x 4 x 106 dyn/cm2
A surface tension of 1.75 dyn/cm can account for the 7 x 106 dyn/cm2
measured force capability of muscles. Because this is well below surface
tensions commonly encountered, we can conclude that surface tension
could be the source of muscle contraction.
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Surfactants
Surfactants are molecules that lower surface tension of liquids. (The word is
an abbreviation of surface active agent.) The most common surfactant
molecules have one end that is water-soluble (hydrophilic) and the other
end water insoluble (hydrophobic)
Surface layer of surfactant molecules.
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Exercises
(Problems For Part (1))
• In Eq. 7.14, it is assumed that the density of the animal is greater than the density
of the fluid in which it is submerged. If the situation is reversed, the immersed
animal tends to rise to the surface, and it must expend energy to keep itself below
the surface. How is Eq. 7.14 modified for this case?
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• Calculate the perimeter of a platform required to support a 70 kg person
solely by surface tension.
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