Chapter 9

Solids and Fluids

States of matter

• Solid

• Liquid

• Gas

• Plasma

Solids

• Have definite volume and shape

• Molecules:1) are held in specific locations by electrical forces2) vibrate about equilibrium positions3) can be modeled as springs connecting molecules

Solids

• Crystalline solid: atoms have an ordered structure (e.g., diamond, salt)

• Amorphous solid: atoms are arranged almost randomly (e.g., glass)

Fluids

• Fluids – substances that can flow (gases, liquids)

• Fluids conform with the boundaries of any container in which they are placed

• Fluids lack orderly long-range arrangement of atoms and molecules they consist of

• Fluids can be compressible and incompressible

Liquids

• Have a definite volume, but 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

Gases

• Have neither definite volume nor 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

Plasmas

• 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

Indeterminate structures

• Indeterminate systems cannot be solved by a simple application of the equilibrium conditions

• In reality, physical objects are not absolutely rigid bodies

• Concept of elasticity is employed

Elasticity

• All real “rigid” bodies can change their dimensions as a result of pulling, pushing, twisting, or compression

• This is due to the behavior of a microscopic structure of the materials they are made of

• Atomic lattices can be approximated as sphere/spring repetitive arrangements

Stress and strain

• All deformations result from a stress – deforming force per unit area

• Deformations are described by a strain – unit deformation

• Coefficient of proportionality between stress and strain is called a modulus of elasticity

stress = modulus * strain

Tension and compression

• Strain is a dimensionless ratio – fractional change in length of the specimen ΔL/Li

• The modulus for tensile and compressive strength is called the Young’s modulus

iLLY

AF

Thomas Young (1773 – 1829)

Tension and compression

• Strain is a dimensionless ratio – fractional change in length of the specimen ΔL/Li

• The modulus for tensile and compressive strength is called the Young’s modulus

Shearing

• For the stress, force vector lies in the plane of the area

• Strain is a dimensionless ratio Δx/h

• The modulus for this case is called the shear modulus

hxS

AF

Hydraulic stress

• The stress is fluid pressure P = F/A

• Strain is a dimensionless ratio ΔV/V

• The modulus is called the bulk modulus

iVVB

AF

Blaise Pascal(1623 - 1662)

Density and pressure

• Density

• SI unit of density: kg/m3

• Pressure

• SI unit of pressure: N/m2 = Pa (pascal)

• Pressure is a scalar – at a given point in a fluid the measured force is the same in all directions

• For a uniform force on a flat area

Vm

V

0lim

AF

PA

0lim

AFP

Atmospheric pressure

• Atmospheric pressure:

• P0 = 1.00 atm = 1.013 x 105 Pa

• Specific gravity of a substance is the ratio of its density to the density of water at 4° C (1000 kg/m3)

• Specific gravity is a unitless ratio

Fluids at rest

• For a fluid at rest (static equilibrium) the pressure is called hydrostatic

• For a horizontal-base cylindrical water sample in a container

mgFF 12 gyyAAPAP )( 2112 gyyPP )( 2112

hgPP 0

Fluids at rest

• The hydrostatic pressure at a point in a fluid depends on the depth of that point but not on any horizontal dimension of the fluid or its container

• Difference between an absolute pressure and an atmospheric pressure is called the gauge pressure

hgPPPg 0

hgPP 0

Measuring pressure

• Mercury barometer

• Open-tube manometer

hgP 0

0;;0

22

011

PhyPPy

gyyPP )( 2112

PPhyPPy

22

011

;;0

gyyPP )( 2112

hgPPPg 0

Chapter 9Problem 19

A collapsible plastic bag contains a glucose solution. If the average gauge pressure in the vein is 1.33 × 103 Pa, what must be the minimum height h of the bag in order to infuse glucose into the vein? Assume that the specific gravity of the solution is 1.02.

Pascal’s principle

• Pascal’s principle: A change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of its container

• Hydraulic lever

• With a hydraulic lever, a given force applied over a given distance can be transformed to a greater force applied over a smaller distance

2

2

1

1

AF

AFP

2

121 A

AFF

2211 xAxAV

1

221 A

Axx 11 xFW 22 xF

Archimedes’ principle

• Buoyant force: For imaginary void in a fluidp at the bottom > p at the top

• Archimedes’ principle: when a body is submerged in a fluid, a buoyant force from the surrounding fluid acts on the body. The force is directed upward and has a magnitude equal to the weight of the fluid that has been displaced by the body

gmB f

Archimedes of Syracuse

(287-212 BCE)

Archimedes’ principle

• Sinking:

• Floating:

• Apparent weight:

• If the object is floating at the surface of a fluid, the magnitude of the buoyant force (equal to the weight of the fluid displaced by the body) is equal to the magnitude of the gravitational force on the body

Bmg

Bmg

Bmgweightapparent

Chapter 9Problem 34

A light spring of force 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. What is the elongation ΔL of the spring?

Motion of ideal fluids

Flow of an ideal fluid:

• Steady (laminar) – the velocity of the moving fluid at any fixed point does not change with time (either in magnitude or direction)

• Incompressible – density is constant and uniform

• Nonviscous – the fluid experiences no drag force

• Irrotational – in this flow the test body will not rotate about its center of mass

Equation of continuity

• For a steady flow of an ideal fluid through a tube with varying cross-section

xAV tAv tvAtvA 2211

2211 vAvA

constAv

Equation of continuity

Bernoulli’s equation

• For a steady flow of an ideal fluid:

• Kinetic energy

• Gravitational potential energy

• Internal (“pressure”) energy

Daniel Bernoulli(1700 - 1782)

intEUKE gtot

2

2mvK

mgyU g

VPE int

2

2vV

gyV

Bernoulli’s equation

• Total energy

intEUKE gtot

VPgyVvV

2

2

PgyvVEtot

2

2

const

22

22

11

21

22PgyvPgyv

Chapter 9Problem 43

A hypodermic syringe contains a medicine having the density of water. The barrel of the syringe has a cross-sectional area A = 2.50 × 10-5 m2, and the needle has a cross-sectional area a = 1.00 × 10-8 m2. In the absence of a force on the plunger, the pressure everywhere is 1 atm. A force F of magnitude 2.00 N acts on the plunger, making medicine squirt horizontally from the needle. Determine the speed of the medicine as it leaves the needle’s tip.

Surface tension

• Net force on molecule A is zero, because it is pulled equally in all directions

• Net force on B is not zero, because no molecules above to act on it

• It is pulled toward the center of the fluid

Surface tension

• 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 – surface tension

Surface tension

• Surface tension: the ratio of the magnitude of the surface tension force to the length along which the force acts:

• SI units are N/m

• Surface tension of liquids decreases with increasing temperature

• Surface tension can be decreased by adding ingredients called surfactants to a liquid (e.g., a detergent)

LF

Liquid surfaces

• Cohesive forces are forces between like molecules, adhesive forces are forces between unlike molecules

• The shape of the surface depends upon the relative strength of the cohesive and adhesive forces

• If adhesive forces are greater than cohesive forces, the liquid clings to the walls of the container and “wets” the surface

Liquid surfaces

• Cohesive forces are forces between like molecules, adhesive forces are forces between unlike molecules

• The shape of the surface depends upon the relative strength of the cohesive and adhesive forces

• If cohesive forces are greater than adhesive forces, the liquid curves downward and does not “wet” the surface

Contact angle

• If cohesive forces are greater than adhesive forces, Φ > 90°

• If adhesive forces are greater than cohesive forces, Φ < 90°

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

Capillary action

• Capillary action is the result of surface tension and adhesive forces

• The level of the fluid in the tube is below the surface of the surrounding fluid if cohesive forces are greater than adhesive forces

Viscous fluid flow

• Viscosity: friction between the layers of a fluid

• 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

Questions?

Answers to the even-numbered problems

Chapter 9

Problem 14:

1.9 × 104 N

Answers to the even-numbered problems

Chapter 9

Problem 22:

10.5 m;no, some alcohol and water evaporate

Answers to the even-numbered problems

Chapter 9

Problem 26:

0.611 kg