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