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 Understand the differences in solids, liquids, gases, and plasmas.
 Understand the fundamental properties of hydrostatics and hydrodynamics.
 Understand and apply the various laws of fluid behavior.
 State Pascal’s Principle and solve problems using the principle.

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 Describe Archimede’s Principle and the forces involved in that
principle.
 Be able to calculate surface tension and understand the forces
responsible.
 State Bernoulli’s Equation and solve related problems.
 Describe Poiseullie’s Law.

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 Recognize Stoke’s Law and its relation to viscous drag.
 Be familiar with the equation for computing the drag coefficient.

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 Has definite volume
 Has definite shape
 Molecules are held in specific locations
 vibrate about equilibrium positions
 Can be modeled as springs connecting molecules

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 Atoms have an ordered structure
 This example is salt
 Red spheres represent Na^{+} ions
 Blue spheres represent Cl^{} ions

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 Atoms are arranged randomly
 Examples include glass

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 Has a definite volume
 No definite shape
 Exist at a higher temperature than solids
 The molecules “wander” through the liquid in a random fashion
 The intermolecular forces are not strong enough to keep the molecules
in a fixed position

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 Has no definite volume
 Has no definite shape
 Molecules are in constant random motion
 The molecules exert only weak forces on each other
 Average distance between molecules is large compared to the size of the
molecules

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 Matter heated to a very high temperature
 Many of the electrons are freed from the nucleus
 Result is a collection of free, electrically charged ions
 Plasmas exist inside stars

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 All objects are deformable
 It is possible to change the shape or size (or both) of an object through the
application of external forces
 when the forces are removed, the object tends to its original shape
 This is a deformation that exhibits elastic behavior

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 Stress is related to the force causing the deformation
 Strain is a measure of the degree of deformation
 The elastic modulus is the constant of proportionality between stress
and strain
 For sufficiently small stresses, the stress is directly proportional to
the strain
 The constant of proportionality depends on the material being deformed
and the nature of the deformation

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 The elastic modulus can be thought of as the stiffness of the material
 A material with a large elastic modulus is very stiff and difficult to
deform

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 Tensile stress is the ratio of the external force to the crosssectional
area
 Tensile is because the bar is under tension
 The elastic modulus is called Young’s modulus

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 SI units of stress are Pascals, Pa
 The tensile strain is the ratio of the change in length to the original
length

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 Young’s modulus applies to a stress of either tension or compression
 It is possible to exceed the elastic limit of the material
 No longer directly proportional
 Ordinarily does not return to its original length
 If stress continues, the object may break

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 Forces may be parallel to one of the objects faces
 The stress is called a shear stress
 The shear strain is the ratio of the horizontal displacement and the
height of the object
 The shear modulus is S

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 A material having a large shear modulus is difficult to bend

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 Bulk modulus characterizes the response of an object to uniform
squeezing
 Suppose the forces are perpendicular to, and acts on, all the surfaces
 Example: when an object is immersed in a fluid
 The object undergoes a change in volume without a change in shape

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 Volume stress, ΔP, is the ratio of the force to the surface area
 This is also the Pressure
 The volume strain is equal to the ratio of the change in volume to the
original volume

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 A material with a large bulk modulus is difficult to compress
 The negative sign is included since an increase in pressure will produce
a decrease in volume
 The compressibility is the reciprocal of the bulk modulus

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 Solids have Young’s, Bulk, and Shear moduli
 Liquids have only bulk moduli, they will not undergo a shearing or
tensile stress
 The liquid would flow instead

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 The ultimate strength of a material is the maximum force per unit area
the material can withstand before it breaks or factures
 Some materials are stronger in compression than in tension

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 A horizontal beam is supported by two columns
 Used in Greek temples
 Columns are closely spaced
 Limited length of available stones
 Low ultimate tensile strength of sagging stone beams

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 Developed by the Romans
 Allows a wide roof span on narrow supporting columns
 Stability depends upon the compression of the wedgeshaped stones

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 First used in Europe in the 12^{th} century
 Extremely high
 The flying buttresses are needed
to prevent the spreading of the arch supported by the tall, narrow
columns

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 The density of a substance of uniform composition is defined as its mass
per unit volume:
 Units are kg/m^{3} (SI) or g/cm^{3} (cgs)
 1 g/cm^{3} = 1000 kg/m^{3}

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 The densities of most liquids and solids vary slightly with changes in
temperature and pressure
 Densities of gases vary greatly with changes in temperature and pressure

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 The specific gravity of a substance is the ratio of its density to the
density of water at 4° C
 The density of water at 4° C is 1000 kg/m^{3}
 Specific gravity is a unitless ratio

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 The force exerted by a fluid on a submerged object at any point if
perpendicular to the surface of the object

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 The spring is calibrated by a known force
 The force the fluid exerts on the piston is then measured

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 If a fluid is at rest in a container, all portions of the fluid must be
in static equilibrium
 All points at the same depth must be at the same pressure
 Otherwise, the fluid would not be in equilibrium

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 Hydrostatic pressure is the pressure at the bottom of a column of fluid
caused by the weight of the fluid. Hydrostatic pressure exists at all
points below the surface, but it is not constant at all points. The
hydrostatic pressure at any point depends on both the fluid density and
the depth below the fluid surface.

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 Examine the darker region, assumed to be a fluid
 It has a crosssectional area A
 Extends to a depth h below the surface
 Three external forces act on the region

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 P_{o} is normal atmospheric pressure
 1.013 x 10^{5} Pa = 14.7 lb/in^{2}
 The pressure does not depend upon the shape of the container

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 A change in pressure applied to an enclosed fluid is transmitted
undimished to every point of the fluid and to the walls of the
container.

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 The hydraulic press is an important application of Pascal’s Principle
 Also used in hydraulic brakes, forklifts, car lifts, etc.

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 One end of the Ushaped tube is open to the atmosphere
 The other end is connected to the pressure to be measured
 Pressure at B is P_{o}+ρgh

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 Invented by Torricelli
 A long closed tube is filled with mercury and inverted in a dish of
mercury
 Measures atmospheric pressure as ρgh

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 One atmosphere (1 atm) =
 76.0 cm of mercury
 1.013 x 10^{5} Pa
 14.7 lb/in^{2}

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 Any object completely or partially submerged in a fluid is buoyed up by
a force whose magnitude is equal to the weight of the fluid displaced by
the object.

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 The upward force is called the buoyant force
 The physical cause of the buoyant force is the pressure difference
between the top and the bottom of the object

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 The magnitude of the buoyant force always equals the weight of the
displaced fluid
 The buoyant force is the same for a totally submerged object of any
size, shape, or density

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 The buoyant force is exerted by the fluid
 Whether an object sinks or floats depends on the relationship between
the buoyant force and the weight

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 The upward buoyant force is B=ρ_{fluid}gV_{obj}
 The downward gravitational force is w=mg=ρ_{obj}gV_{obj}
 The net force is Bw=(ρ_{fluid}ρ_{obj})gV_{obj}

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 The object is less dense than the fluid
 The object experiences a net upward force

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 The object is more dense than the fluid
 The net force is downward
 The object accelerates downward

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 The object is in static equilibrium
 The upward buoyant force is balanced by the downward force of gravity
 Volume of the fluid displaced corresponds to the volume of the object
beneath the fluid level

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 Streamline flow
 every particle that passes a particular point moves exactly along the
smooth path followed by particles that passed the point earlier
 also called laminar flow
 Streamline is the path
 different streamlines cannot cross each other
 the streamline at any point coincides with the direction of fluid
velocity at that point

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 The flow becomes irregular
 exceeds a certain velocity
 any condition that causes abrupt changes in velocity
 Eddy currents are a characteristic of turbulent flow

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 Viscosity is the degree of internal friction in the fluid
 The internal friction is associated with the resistance between two
adjacent layers of the fluid moving relative to each other

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 The fluid is nonviscous
 There is no internal friction between adjacent layers
 The fluid is incompressible
 The fluid is steady
 Its velocity, density and pressure do not change in time
 The fluid moves without turbulence
 No eddy currents are present

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 A_{1}v_{1} = A_{2}v_{2}
 The product of the crosssectional area of a pipe and the fluid speed is
a constant
 Speed is high where the pipe is narrow and speed is low where the pipe
has a large diameter
 Av is called the flow rate

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 Relates pressure to fluid speed and elevation
 Bernoulli’s equation is a consequence of Conservation of Energy applied
to an ideal fluid
 Assumes the fluid is incompressible and nonviscous, and flows in a
nonturbulent, steadystate manner

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 States that the sum of the pressure, kinetic energy per unit volume, and
the potential energy per unit volume has the same value at all points
along a streamline

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 Shows fluid flowing through a horizontal constricted pipe
 Speed changes as diameter changes
 Can be used to measure the speed of the fluid flow
 Swiftly moving fluids exert less pressure than do slowly moving fluids

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 Net force on molecule A is zero
 Pulled equally in all directions
 Net force on B is not zero
 No molecules above to act on it
 Pulled toward the center of the fluid

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 The net effect of this pull on all the surface molecules is to make the
surface of the liquid contract
 Makes the surface area of the liquid as small as possible
 Example: Water droplets take on
a spherical shape since a sphere has the smallest surface area for a
given volume

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 Surface tension allows the needle to float, even though the density of
the steel in the needle is much higher than the density of the water
 The needle actually rests in a small depression in the liquid surface
 The vertical components of the force balance the weight

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 The surface tension is defined as the ratio of the magnitude of the
surface tension force to the length along which the force acts:
 SI units are N/m
 In terms of energy, any equilibrium configuration of an object is one in
which the energy is a minimum

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 The surface tension of liquids decreases with increasing temperature
 Surface tension can be decreased by adding ingredients called surfactants
to a liquid

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 Cohesive forces are forces
between like molecules
 Adhesive forces are forces between unlike molecules
 The shape of the surface depends upon the relative size of the cohesive
and adhesive forces

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 The adhesive forces are greater than the cohesive forces
 The liquid clings to the walls of the container
 The liquid “wets” the surface

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 Cohesive forces are greater than the adhesive forces
 The liquid curves downward
 The liquid does not “wet” the surface

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 In a, Φ > 90° and cohesive forces are greater than adhesive
forces
 In b, Φ < 90° and adhesive forces are greater than cohesive
forces

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 Capillary action is the result of surface tension and adhesive forces
 The liquid rises in the tube when adhesive forces are greater than
cohesive forces
 At the point of contact between the liquid and the solid, the upward
forces are as shown in the diagram

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 Here, the cohesive forces are greater than the adhesive forces
 The level of the fluid in the tube will be below the surface of the
surrounding fluid

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 The height at which the fluid is drawn above or depressed below the
surface of the surrounding liquid is given by:

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 Viscosity refers to friction between the layers
 Layers in a viscous fluid have different velocities
 The velocity is greatest at the center
 Cohesive forces between the fluid and the walls slow down the fluid on
the outside

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 Assume a fluid between two solid surfaces
 A force is required to move the upper surface
 η is the coefficient
 SI units are Ns/m^{2}
 cgs units are Poise

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 Gives the rate of flow of a fluid in a tube with pressure differences

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 At sufficiently high velocity, a fluid flow can change from streamline
to turbulent flow
 The onset of turbulence can be found by a factor called the Reynold’s
Number, RN
 If RN = 2000 or below, flow is streamline
 If 2000 <RN<3000, the flow
is unstable
 If RN = 3000 or above, the flow
is turbulent

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 Movement of a fluid may be due to differences in concentration
 The fluid will flow from an area of high concentration to an area of low
concentration
 The processes are called diffusion and osmosis

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 Molecules move from a region of high concentration to a region of low
concentration
 Basic equation for diffusion is given by Fick’s Law
 D is the diffusion coefficient

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 Concentration on the left is higher than on the right of the imaginary
barrier
 Many of the molecules on the left can pass to the right, but few can
pass from right to left
 There is a net movement from the higher concentration to the lower
concentration

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 Osmosis is the movement of water from a region where its concentration
is high, across a selectively permeable membrane, into a region where
its concentration is lower
 A selectively permeable membrane is one that allows passage of some
molecules, but not others

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 When an object falls through a fluid, a viscous drag acts on it
 The resistive force on a small, spherical object of radius r falling
through a viscous fluid is given by Stoke’s Law:

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 As the object falls, three forces act on the object
 As its speed increases, so does the resistive force
 At a particular speed, called the terminal speed, the net force is zero

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