Chapter 9: Solids and Fluids
State of matters: Solid, Liquid, Gas and Plasma.
Solids
• Has definite volume and shape• Can be crystalline or amorphous• Molecules are held in specific
locations by electrical forces• vibrate about equilibrium positions• Can be modeled as springs
connecting molecules• External forces can be applied to
the solid and compress the material– In the model, the springs would be
compressed• When the force is removed, the
solid returns to its original shape and size
– This property is called elasticity
Liquid
• Has a definite volume• No definite shape• Exists 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
Gas
• 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
Plasma
• 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
Elastic Deformation: Young’s Modulus
⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ=
oLLY
AF
Young’s modulus
unit of modulus: N/m2 (= pascal, Pa)
Young’s modulus
• Stress is the force per unit area causing the deformation• Strain is a measure of the amount of deformation• The elastic modulus is the constant of proportionality
between stress and strain
tensile stress uniaxial strain
Elastic Deformation: Shear Modulus
shear strain
shear modulus
⎟⎠⎞
⎜⎝⎛ Δ
=hXS
AF
shear modulus
shear stress
• The elastic modulus can be thought of as the stiffness of the material• It is possible to exceed the elastic limit of the material
-- No longer directly proportional-- May not return to its original length-- Break
Notes on Young’s and Shear Modulus
Pressure and Bulk Modulus
AFP =
The pressure P is the magnitude F of the force acting perpendicular to a surface divided by the area A over which the force acts
Unit of Pressure: N/m2 = pascal (Pa)
⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ−=Δ
oVVBP
bulk modulus
Bulk modulus
• 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
Elastic Modulus of Some Materials
Mass Density:The mass density ρ is the mass m of a substance divided by its volume V
ρ = m / V
Unit of Mass Density: kg/m3
Specific Gravity: (for any substance)Specific gravity =
Unit of specific gravity: unit-less, number
Density
Density of substanceDensity of water at 4oC
• 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
Pressure
The pressure P has a general definition of (normal) force per unit area
SI unit: pascal (N/m2)
AFP =
In a static liquid (or gas), the pressure is isotropic (non-directional). For a gas (low mass density) the pressure in a small volume can be regarded as homogeneous (constant throughout space). For liquids (high mass density), the pressure depends on the depth of the liquid.
Atmospheric pressure at sea level is
atmPa 110013.1 5 =×
Exam 2
Average: 51.6
98 64 3695 60 3695 58 3493 58 2893 58 2885 55 2584 53 2583 52 2483 52 2283 52 2082 51 2081 50 1978 46 1877 44 1676 42 1575 41 1470 41 1469 41 1467 38 966 3864 38
Pressure and Depth in a Static Fluid
PA = P0A + Mg
P = P0 + ρgh
Hoover Dam. Can we use a less massive structure to hold the water if the size (volume) of the reservoir (with same depth) is much narrower?
h: depth below the face of fluid
Absolute and Relative (Gauge) Pressure
P2 = P1 + ρgh
absolute pressure
relative pressure
Barometer, invented by Torricelli in 1643.
Pressure in different units:One atmosphere (1 atm)
= 76.0 cm of mercury = 760 Torr= 1.013 x 105 Pa= 14.7 lb/in2 (psi)
Pascal’s Principle
Pascal’s PrincipleAny change in the pressure applied to a completely enclosed fluid is transmitted undiminished to every point of the fluid and the enclosing walls.
F2 = F1A2
A1
⎛
⎝ ⎜
⎞
⎠ ⎟
1
1
2
2
AF
AF
=
Can we get something out of nothing?
Of course not!
Example: r1=5.00 cm, r2=30.0 cm. mcar=4000 lb. Can you lift the car?
Buoyant Forces and Archimedes’ Principle
Magnitude of buoyant force = Weight of displaced fluid
Archimedes’ Principle
The physical cause of the buoyant force is the pressure difference between the top and the bottom of the object
• The magnitude of the buoyant force always equals the weight of the displaced fluid
• The buoyant force is exerted by the fluid. The buoyant force is the same for a totally submerged object of any size, shape, or density
fluid fluid fluidB V g wρ= =
Totally Submerged Object
The net force isB-mg=(ρfluid-ρobj)gVobj
Will it float?If ρobj < ρfluid, float,If ρobj > ρfluid, sink.
Floating Object
fluid
obj
total
fluiddisplaced
total
submerged
VVV V
ρρ
== _
• The object is in static equilibrium• The upward buoyant force is balanced by the downward force of gravity• Volume of the fluid displaced is equal to the volume of the object beneath the fluid level
How much volume will be submerged?
Tip of the iceberg!ρice=0.9167 g/cm3, ρwater=0.9998 g/cm3 at 0 °
Problems On Static Fluid
A light spring of constant k = 160 N/m rests vertically on the bottom of a large beaker of water. A 5.00-kg block of wood (density – 650 kg/m3) is connected to the spring, and the block-spring system is allowed to come to static equilibrium (Fig. P9.34b). What is the elongation ΔL of the spring?
A bargain hunter purchases a gold crown at a flea market. After she gets home, she hangs it from a scale and finds its weight to be 7.84N. She then weights the crown while it is immersed in water of density 1000 kg/m3, and now the scale reads 6.86N. Is the crown made of pure gold (density of gold is 19.3 x 103 kg/m3).
Fluids in Motion: Streamline Flow
• Streamline flow – Every particle that passes a
particular point moves exactly along the smooth path followed by particles that passed the point earlier
• 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
Fluids in Motion: Turbulent Flow
• The flow becomes irregular– exceeds a certain velocity– any condition that causes
abrupt changes in velocity
• Eddy currents are a characteristic of turbulent flow
Characteristics of an Ideal Fluid
• The fluid is nonviscous– There is no internal friction between adjacent layers
• The fluid is incompressible– Its density is constant
• The fluid motion is steady– Its velocity, density, and pressure at a certain spatial
point do not change in time• The fluid moves without turbulence
– No eddy currents are present– The elements have zero angular velocity about its center
Equation of Continuity
• A1v1 = A2v2• A consequence of
Conservation of Mass• The product of the cross-
sectional 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
Bernoulli’s Equation
In the steady flow of a nonviscous, incompressible fluid of density ρ, the pressure P, the fluid speed v, and the elevation y at any two points (1 and 2) are related by
22221
211 2
121 gyvPgyvP ρρρρ ++=++
Bernoulli’s equation is a consequence of Conservation of Energy applied to an ideal fluid
Application of Bernoulli’s Equation
P1 +12
ρv12 = P2 +
12
ρv22
At the same height:
Example Problems
The figure at right shows a water tank with a valve at the bottom. If this valve is opened, what is the maximum height attained by the water stream coming out of the right side of the tank? Assume that h = 10.0 m, L = 2.00 m, and θ = 30.0° and that the cross‐sectional area at A is very large compared with that at B.
A siphon is a device that allows a fluid to seemingly defy gravity. The flow must be initiated by a partial vacuum in the tube, as in a drinking straw. (a) Show that the speed at which the water emerges from the siphon is given by . (b) For what values of y will the siphon work?
ghv 2=
Surface Tension
• 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
– The net effect of this pull on all the surface molecules is to make the surface of the liquid contract
When liquid meets solid
• 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
Capillary Action
• 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• The liquid drops in the tube when cohesive forces are greater than
adhesive forces
Chapter 9 Summary
Mass DensitySpecific GravityElastic modulus (bulk modulus, Young’s modulus, shear modulus)Fluid pressure and depthPascal’s principleArchimedes’ principle on buoyant forceEquation of continuityBernoulli’s equation