Chapter 9 solids and fluids
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Chapter 9: Solids and Fluids. Three states of matter. Normally matter is classified into one of three (four) states: solid, liquid, gas (, plasma). solid : crystalline solid (salt etc.) amorphous solid (glass etc.). ordered structure. atoms arranged at almost at random.

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Chapter 9: Solids and Fluids

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Chapter 9: Solids and Fluids

  • Three states of matter

  • Normally matter is classified into one of three (four) states:

  • solid, liquid, gas (, plasma).

solid : crystalline solid (salt etc.) amorphous solid (glass etc.)

ordered structure

atoms arranged at almost at random

States of Matter

  • Three (four) states of matter (cont’d)

  • Normally matter is classified into one of three (four) states:

  • solid, liquid, gas (, plasma).

liquid : A molecule in a liquid does random-walk through a series

of interactions with other molecules.

  • - For any given substance, the liquid

  • state exists at a higher temperature

  • than the solid state.

  • The inter-molecular forces in a liquid

  • are not strong enough to hold mole-

  • cules together in fixed position.

  • The molecules wander around in

  • random fashion.

States of Matter

  • Three (four) states of matter (cont’d)

  • Normally matter is classified into one of three (four) states:

  • solid, liquid, gas (, plasma).

gas : In gaseous state, molecules are in constant random motion

and exert only weak forces on each other.

  • The average distance between the molecules of a gas is quite

  • large compared with the size of molecules.

  • Occasionally the molecules collide with each other, but most of

  • them move freely.

  • Unlike solids and liquids, gases can be easily compressed.

States of Matter

plasma : At high temperature, electrons of atoms are free from

nucleus. Such a collection of ionized atoms with equal amounts of

positive (nucleus) and negative charges (electrons) forms a state

called plasma.

  • Stress, strain and elastic modulus

  • Until external force becomes strong enough to deform permanently

  • or break a solid object, the effect of deformation by the external

  • force goes back to zero when the force is removed – Elastic behavior.

  • Stress : the force per unit area causing a deformation

  • Strain : a measure of the amount of the deformation

  • Elastic modulus : proportionality constant, similar to a spring constant

stress = elastic modulus x strain

Deformation of Solids

  • Young’s modulus: elasticity in length

  • Consider a long bar of cross-sectional area A and length L0,

  • clamped at one end. When an external force F is applied along

  • the bar, perpendicular to the cross section, internal forces in the

  • bar resist the distortion that F tends to produce.

  • Eventually the bar attains an

  • equilibrium in which:

  • (1) its length is greater than L0

  • (2) the external force is balanced

  • by internal forces.

Deformation of Solids

The bar is said to be stressed.

Young’s modulus

SI unit: dimensionless

tensile stress

tensile strain

SI unit: Pa = 1 N/m2

  • Young’s modulus: elasticity in length (cont’d)

  • Typical values

Deformation of Solids

  • Stress vs. strain

  • Shear modulus: Elasticity of shape

  • Another type of deformation occurs when an object is subjected to

  • a force F parallel to one of its faces while the opposite face is held

  • fixed by a second force.

  • The stress in this

  • situation is called

  • a shear stress.

Deformation of Solids

Shear modulus

SI unit: dimensionless

shear stress

shear strain

SI unit: Pa = 1 N/m2

  • Bulk modulus: Volume elasticity

  • Suppose that the external forces acting on an object are all

  • perpendicular to the surface on which the force acts and are

  • distributed uniformly.

  • This situation occurs when a

  • object is immersed in a fluid.

Deformation of Solids

bulk modulus

SI unit: dimensionless

volume stress

volume strain

SI unit: Pa = 1 N/m2

  • An example

  • Example 9.3 : Stressing a lead ball

A solid lead sphere of volume 0.50 m3, dropped in the ocean, sinks

to a depth of 2.0x103 m, where the pressure increases by 2.0x107 Pa.

Lead has a bulk modulus of 4.2x1010 Pa. What is the change in

volume of the sphere?

Deformation of Solids

  • Density

  • The density r of an object is defined as:

M: mass, V: volume

SI unit: kg/m3 (cgs unit: g/cm3 )

  • The specific gravity of a

  • substance is the ratio of

  • its density to the density

  • of water at 4oC, which is

  • 1.0x103 kg/m3, and it is

  • dimensionless.

Density and Pressure

  • Pressure

  • Fluids do not sustain shearing stresses, so

  • the only stress that a fluid can exert on a

  • submerged object is one that tends to

  • compress it, which is a bulk stress.

  • The force F exerted by the fluid on the

  • object is always perpendicular to the

  • surfaces of the object.

Density and Pressure

  • If F is the magnitude of a force exerted

  • perpendicular to a given surface of area A,

  • then the pressure P is defined as:

F: force, A: area

SI unit: Pa = N/m2

  • Variation of pressure with depth

  • When a fluid is at rest in a container, all portions of the fluid must

  • be in static equilibrium – at rest with respect to the observer.

  • All points at the same depth must be at the same pressure. If this

  • were not the case, fluid would flow from the higher pressure region

  • to the lower pressure region.

  • Consider an object at rest

  • with area A and height h in

  • a fluid.

Density and Pressure

  • Effect of atmospheric pressure:

P0 : atmospheric pressure, P: pressure at depth h

  • Examples

  • Example 9.5 : Oil and water

r=0.700 g/cm3

h1=8.00 m

r=1025 kg/m3

h2=5.00 m

Density and Pressure

  • Pascal’s principle

  • A change in pressure applied to an enclosed fluid is transmitted

  • undiminished to every point of the fluid and to the walls of the

  • container.

  • Hydraulic press

Density and Pressure

F2 > F1 if A2 > A1

  • Car lift

  • Example 9.7 : Car lift

(a) Find necessary force by compressed air at piston 1.

weight=13,300 N

(b) Find air pressure.

Density and Pressure

circular x-sec

(c) Show the work done

by pistons is the same.

r1=5.00 cm

r2=15.0 cm

  • Absolute and gaugepressure

  • An open tube manometer (Fig.(a))


measures the gauge pressure P-P0

P : absolute pressure

P0 : atmospheric pressure

  • A mercury barometer (Fig.(b))

Pressure Measurements

measures the atmospheric pressure


  • One atmospheric pressure

defined as the pressure equivalent of a

column of mercury that is exactly 0.76 m

in height.

  • Blood pressure measurement

  • A specialized manometer

  • (sphygmomanometer)

  • A rubber bulb forces air into a cuff wrap.

  • A manometer is attached under cuff and

  • is under pressure.

  • -The pressure in the cuff is increased until

  • the flow of blood through brachial artery

  • is stopped.

  • -Then a valve on the bulb is opened, and

  • measurer listens with a stethoscope to

  • the artery at a point just below the cuff.

  • -When the pressure at the cuff and the

  • artery is just below the max. value

  • produced by heart (the systolic pressure),

  • the artery opens momentarily on each beat.

  • -At this point, the velocity of the blood is high,

  • and the flow is noisy and can be heard…

Pressure Measurements

  • Archimedes’s principle

Any object completely or partially submerged in a fluid is buoyed

up by a force with magnitude equal to the weight of the fluid

displaced by the object.

Upward force (buoyant force) :

Buoyant Forces and

Archimedes’s Principle

Downward force:

  • Archimedes’s principle and a floating object

Upward force (buoyant force) :


Downward force:


Buoyant Forces and

Archimedes’s Principle

  • Examples

  • Example 9.8 : A fake or pure gold crown?

Is the crown made of pure


Tair =7.84 N

Twater =6.86 N

Buoyant Forces and

Archimedes’s Principle

rgold=19.3x103 kg/m3

  • Examples

  • Example 9.9 : Floating down the river

What depth h is the bottom of

the raft submerged?

A=5.70 m2

rwood=6.00x102 kg/m3

Buoyant Forces and

Archimedes’s Principle

  • Some terminology

  • When a fluid is in motion:

  • (1) if every particle that passes a particular point moves along exactly

  • the same smooth path followed by previous particles passing the

  • point, this path is called streamline. If this happens, this flow is said

  • to be streamline or laminar.

  • (2) the flow of a fluid becomes irregular, or turbulent, above a certain

  • velocity or under any conditions that can cause abrupt changes in

  • velocity.

  • Ideal fluid :

Fluid in Motion

  • The fluid is non-viscous :

  • There is no internal friction force between adjacent layers.

  • The fluid is incompressible :

  • Its density is constant.

  • The fluid motion is steady :

  • The velocity, density, and pressure at each point in the fluid do not

  • change with time.

  • The fluid moves without turbulence :

  • Each element of the fluid has zero angular velocity about its center.

  • Equation of continuity

  • Consider a fluid flowing through a pipe of non-uniform size. The

  • particles in the fluid move along the streamlines in steady-state flow.

In a small time interval Dt, the

fluid entering the bottom end of

the pipe moves a distance:

The mass contained in the bottom

blue region :

Fluid in Motion

From a similar argument :

Since DM1=DM2(flow is steady):

Equation of continuity

  • An example

  • Example 9.12 : Water garden

Fluid in Motion

  • Bernoulli’s equation

  • Consider an ideal fluid flowing through a pipe of non-uniform size.

Work done to the fluid at Point 1

during the time interval Dt:

Work done to the fluid at Point 2

during the time interval Dt:

Fluid in Motion

Work done to the fluid :

  • Bernoulli’s equation (cont’d)

If m is the mass of the fluid passing

through the pipe in Dt , the change

in kinetic energy is:

The change in gravitational

potential energy in Dt is:

Fluid in Motion

From conservation of energy:

  • Bernoulli’s equation (cont’d)

From conservation of energy:

Fluid in Motion

Bernoulli’s equation

  • Venturi tube

Consider a water flow through

a horizontal constricted pipe.

Fluid in Motion

  • Examples

h =0.500 m

y1 =3.00 m

  • Example 9.13 : A water tank

  • Consider a water tank with a hole.

  • Find the speed of the water

  • leaving through the hole.



Fluid in Motion

(b) Find where the stream hits the ground.

  • Examples

  • Example 9.14 : Fluid flow in a pipe

A2=1.00 m2

A1=0.500 m2

h =5.00 m

Find the speed at Point 1.

Fluid in Motion

  • Surface tension

  • The net force on a molecule at A is zero

  • because such a molecule is completely

  • surrounded by other molecules.

  • The net force on a molecule at B is downward

  • because it is not completely surrounded by

  • other molecules. There are no molecules

  • above it to exert upward force. this asymmetry

  • makes the surface of the liquid contract and

  • the surface area as small as possible.

Surface Tension, Capillary Action, and Viscous Fluid Flow

  • The surface tension is defined as :

where the surface tension force F

is divided by the length L along

which the force acts.

SI unit : N/m=(N m)/m2=J/m2

  • Surface tension (cont’d)

  • The surface tension of liquids

  • decreases with increasing

  • temperature, because the faster

  • moving molecules of a hot liquid

  • are not bound together as strongly

  • as are those in a cooler liquid.

Surface Tension, Capillary Action, and Viscous Fluid Flow

  • Some ingredient called surfactants

  • such as detergents and soaps decrease surface tension.

  • The surface tissue of the air sacs in the lungs contain a fluid that has

  • a surface tension of about 0.050 N/m. As the lungs expand during

  • inhalation, the body secretes into the tissue a substance to reduce

  • the surface tension and it drops down to 0.005 N/m.

  • Surface of liquid

Surface Tension, Capillary Action, and Viscous Fluid Flow

  • Forces between like-molecules such as between water molecules are

  • called cohesive forces.

  • Forces between unlike-molecules such as those exerted by glass on

  • water are called adhesive forces.

  • Difference in strength between cohesive and adhesive forces creates

  • the shape of a liquid at boundary with other materials.

  • Viscous fluid flow

  • Viscosity refers to the internal friction of a fluid. It is very difficult for

  • layers of a viscous fluid to slide past one another.

  • When an ideal non-viscous fluid flows

  • through a pipe, the fluid layers slide

  • past one another with no resistance.

  • If the pipe has uniform cross-section

  • each layer has the same velocity.

Surface Tension, Capillary Action, and Viscous Fluid Flow

ideal fluid, non-viscous

  • The layers of a viscous fluid have

  • different velocities. The fluid has the

  • greatest velocity at the center of the

  • pipe, whereas the layer next to the wall

  • does not move because of adhesive

  • forces between them.

viscous fluid

  • Viscous fluid flow

  • Consider a layer of liquid between two solid surfaces. The lower surface

  • is fixed in position, and the top surface moves to the right with a velocity

  • v under the action of an external force F.

  • A portion of the liquid is distorted from

  • its original shape, ABCD, at one instance

  • to the shape AEFD a moment later. The

  • force required F to move the upper plate

  • at a fixed speed v is :

  • where h is the coefficient of viscosity of

  • the fluid, and A is the area in contact with fluid.

Surface Tension, Capillary Action, and Viscous Fluid Flow

1 poise=10-1 N s/m2

1 cp (centipoise) = 10-2 poise

SI unit : N s/m2

cgs unit: dyne s/cm2= poise


  • Poiseuille’s law

  • Consider a section of tube of

  • length L and radius R

  • containing a fluid under

  • pressure P1 at the left end and

  • a pressure P2 at the right.

  • Poiseuille’s law describes the

  • flow rate of a viscous fluid

  • under pressure difference:

Surface Tension, Capillary Action, and Viscous Fluid Flow

  • Reynolds number

  • At sufficiently high velocities, fluid flow changes from simple streamline

  • flow to turbulent flow, characterized by a highly irregular motion of the

  • fluid. Experimentally the onset of the turbulence in a tube is determined

  • by a dimensionless factor called Reynolds number, RN, given by:

r : density of fluid

v : average speed of the fluid along the direction of flow

Surface Tension, Capillary Action, and Viscous Fluid Flow

d : diameter of tube

h : viscosity of fluid

  • If RN is below about 2000, the flow of fluid through a tube is streamline.

  • If RN is above about 3000, the flow of fluid through a tube is turbulent.

  • If RN is between 2000 and 3000, the flow is unstable.

  • Examples

  • Example 9.18 : A blood transfusion

A patient receives a blood transfusion through a needle of radius 0.20

mm and length 2.0 cm. The density of blood is 1,050 kg/m3. The bottle

supplying the blood is 0.50 m above the patient’s arm. What is the rate

of the flow through the needle?

Surface Tension, Capillary Action, and Viscous Fluid Flow

  • Examples

  • Example 9.19: Turbulent flow of blood

Determine the speed at which blood flowing through an artery of

diameter 0.20 cm will become turbulent.

Surface Tension, Capillary Action, and Viscous Fluid Flow

  • A fluid can move place to place as a result of difference in

  • concentration between two points in the fluid. There are two

  • processes in this category : diffusion and osmosis.

  • Diffusion

  • In a diffusion process, molecules move from a region where their

  • concentration is high to a region where their concentration is lower.

  • Consider a container in which a

  • high concentration of molecules

  • has been introduced into the left

  • side (the dashed line is an

  • imaginary barrier).

Transport Phenomena

All the molecules move in random

direction. Since there are more

molecules on the left side, more

molecules migrate into the right

side than otherwise. Once a

concentration equilibrium is reached, there will be no net movement.

  • Diffusion (cont’d)

  • Fick’s law

where D is a constant of proportion called

the diffusion coefficient (unit : m2/s), A is

the cross-sectional area, (…) is the change

in concentration per unit distance

(concentration gradient), and DM/Dt is the

mass transported per unit time. The

concentrations, C1 and C2 are measured

in unit of kg/m3.

Transport Phenomena

  • Size of cells and osmosis

  • Diffusion through cell membranes is vital in supplying oxygen to the

  • cells of the body and in removing carbon dioxide and other waste

  • products from them.

  • A fresh supply of oxygen diffuses from the blood, where its

    concentration is high, into the cell, where its concentration is low.

  • Likewise, carbon dioxide diffuses from the cell into the blood where

    its concentration is lower.

Transport Phenomena

  • A membrane that allows passage of some molecules but not others

    is called a selectively permeable membrane.

  • Osmosis is the diffusion of water across a selectively permeable

  • membrane from a high water concentration to a low water

  • concentration.

  • Motion through a viscous medium

  • The magnitude of the resistive force on a very small spherical

  • object of radius r moving slowly through a fluid of viscosity h with

  • speed v is given by:




Stokes’s law

  • Consider a small sphere of radius r falls

  • through a viscous medium.

Transport Phenomena

force of




  • Motion through a viscous medium (cont’d)

  • At the instance the sphere begins to fall, the force of friction is

  • zero because the velocity of the sphere is zero.

  • As the sphere accelerates, its speed increases

  • and so does Fr.




  • When the net force goes to zero, the speed

  • of the sphere reaches the so-called terminal

  • speed vt.

Transport Phenomena

force of




  • Sedimentation and centrifugation

  • If an object is not spherical, the previous argument can still be

  • applied except for the use of Stokes’s law. In this case, we assume

  • that the relation Fr=kv holds where k is a coefficient.

Terminal speed condition

Transport Phenomena

  • Sedimentation and centrifugation (cont’d)

  • The terminal speed for particles in biological samples is usually

  • quite small; the terminal speed for blood cells falling through plasma

  • is about 5 cm/h in the gravitational field of Earth.

  • The speed at which materials fall through a fluid is called

  • sedimentation rate. The sedimentation rate in a fluid can be

  • increased by increasing the effective acceleration g: for example

  • by using radial acceleration due to rotation

  • (centrifuge).

Transport Phenomena

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