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Unit SevenSolids and Fluids

GE253 Physics

- John Elberfeld
- JElberfeld_at_itt-tech.edu
- 518 872 2082

Schedule

- Unit 1 Measurements and Problem Solving
- Unit 2 Kinematics
- Unit 3 Motion in Two Dimensions
- Unit 4 Force and Motion
- Unit 5 Work and Energy
- Unit 6 Linear Momentum and Collisions
- Unit 7 Solids and Fluids
- Unit 8 Temperature and Kinetic Theory
- Unit 9 Sound
- Unit 10 Reflection and Refraction of Light
- Unit 11 Final

Chapter 7 Objectives

- Distinguish between stress and strain, and use

elastic moduli to compute dimensional changes. - Explain the pressure-depth relationship and state

Pascal's principle and describe how it is used in

practical applications. - Relate the buoyant force and Archimedes'

principle and tell whether an object will float

in a fluid, on the basis of relative densities. - Identify the simplifications used in describing

ideal fluid flow and use the continuity equation

and Bernoulli's equation to explain common

effects of ideal fluid flow. - Describe the source of surface tension and its

effect and discuss fluid viscosity.

Reading Assignment

- Read and study College Physics, by Wilson and

Buffa, Chapter 7, pages 219 to 250 - Be prepared for a quiz on this material

Written Assignments

- Do the homework on the handout.
- You must show all your work, and carry through

the units in all calculations - Use the proper number of significant figures and,

when reasonable, scientific notation

Introduction

- Until now, we have beendealing with ideal

objects - Now we will look at objects that can stretch,

bend, break, flow, and compress when forces are

applied

Introduction

- In this week, we will study real, macroscopic

objects, such as wires, blocks, and containers of

fluids. - We will keep the concept of force and see what

happens to macroscopic objects when forces act on

them and see what forces macroscopic objects

exert.

Stress and Strain

- The stress of taking physics may cause a strain

on your brain - But in physics, these terms have different and

precise meanings. - We will learn to use these terms as we study the

effects of forces on solids and fluids in this

unit, not to accelerate objects, but to deform

them.

Stress and Strain

- Stress (force/area) causes strain ( deformation

of some type)

Stress and Strain

- In physics, stress is related to the force

applied to an object - Strain is related to how the object is deformed

as a result. - Both also depend on the size of the object.

Stress

- The diagram shows a wire or rod of length Lo and

area A. - When you apply a force F on the wire, it

stretches by a distance . - When you apply a force F in the opposite

direction on a rod, the rod is compressed by a

distance . - The greater the force F, the greater is the

amount of stretching or compressing.

A

F

L0

Stress

- It is clear that a thick wire is harder to

stretch than a thin one. - For this reason, you can define the concept of

stress as - Stress F / A
- The units of stress are N/m2.

Strain

- The longer the wire or rod, the greater is the

amount of stretching. For this reason, you can

define strain as - Strain ?L / L
- The vertical lines stand for absolute value,

which means the same strain results from positive

?L due to stretching or negative ? L due to

compressing.

Strain

- A strain of 0.05 on a wire means that the wire

has been stretched to a length 5 greater than

its original length.

Youngs Modulus

- NOTE This graph shows Youngs Modulus how much

a wire will stretch if you apply different forces

on it.

YoungsModulusRegion

Graph

- Engineers need to know the force need to stretch

material a certain distance - What force cause the bridge to sag into the

waves? - Graphs show strain on the X axis and stress on

the Y axis

Youngs Modulus

- Youngs Modulus works in the straight line part

of the graph only - The ratio of stress to strain in a material is

known as Youngs modulus - Y (F/A) / ( ?L / L)
- Where, F is force A is cross sectional area

perpendicular to the applied force and L is

length of the object being stressed. - NOTE Youngs Modulus is an ELASTIC Modulus

because materials bounce back to their original

shape in that part of the graph.

Youngs Modulus

- The bigger the modulus, the stronger the

material, and the more force it takes to cause a

specific deformation. - Because material bounce back to their original

shapes, moduli (plural) are elastic - The units of Youngs modulus are the same as

those of stress, N/m2. - Youngs modulus for steel is 20 x 1010 N/m2 and

for bone is 1.5 x 1010 N/m2 for example.

Practice

- The femur (upper leg bone) is the longest and

strongest bone in the body. - Let us assume that a typical femur is circular

and has a radius of 2.0 cm. - How much force is required to extend the bone by

0.010?

Calculations

- Y (F/A) / ( ?L / L)
- From the chart Ybone 1.5 x 1010N/m2
- R 2cm(1m/100cm) .02m
- A ?R2 ?(.02m)2 1.26 x 10-3m2
- ?L / L .01 .01/100 1 x 10-4
- Y ( ?L / L) (F/A)
- F A Y ( ?L / L)
- F 1.26 x 10-3m2 1.5 x 1010N/m2 1 x 10-4
- F 1.89 x 103 N (about 425 pounds)

Practice

- A mass of 16 kg is suspended from a steel wire of

0.10-cm-diameter. - By what percentage does the length of the wire

increase? - Youngs modulus for steel is 20 x 1010 N/m2.

Calculation

- Y (F/A) / ( ?L / L)
- F W mg 16kg 9.8m/s2 157 N
- R .1cm(1m/100cm)/2 .0005m
- A ? R2 ? (.0005m)2 7.85 x 10-7m2
- ( ?L / L) (F/A) / Y (F/ ? R2 ) / Y
- (?L/L) (157N/ 7.85 x 10-7m2) / 20x1010N/m2
- (?L/L) .001 0.1
- NOTE The change in length is inversely

proportional to the SQUARE of the radius!

Shear Modulus

- To distort a rectangular solid, apply a force on

one of its faces in a direction parallel to the

face. - Simultaneously, you must also apply a force on

the opposite face in the opposite direction. - The diagram on the screen shows how you can

distort an object by applying force.

Shear

- The diagram on the screen shows that the amount

of distortion can be measured by the new angle Ø

, which the faces make. - Shearing stress is defined as F/A, where F is the

tangential force and A is the area of the surface

that the force acts on. - Shearing strain is the angle Ø .
- Similar to linear distortions, shearing strain

is directly proportional to shearing stress. - The constant of proportionality is called the

shear modulus (S) - S (F/A) / F

Shear

- Shear modulus has the same units as Youngs

modulus, N/m2. - For many substances, shear modulus is

approximately one-third of the Youngs modulus.

Bulk Modulus

- You can also distort a rectangular solid by

applying forces perpendicular to its surfaces. - This stress causes the solid to become smaller

and is known as volume stress or pressure. - Volume stress is defined as F/A, where F is the

force perpendicular to the surfaces, and A is the

area of the surface.

Bulk Modulus

- Volume strain is the change in volume divided by

the original volume. - Similar to shearing stress and stress, strain is

directly proportional to volume stress. - The constant of proportionality is called bulk

modulus (B) and is defined as - B (F/A) / ( ?V / V)

Bulk Modulus

- Solids are usually surrounded by air that exerts

a compressing force on their surface. - Pressure Force/Area
- Therefore, when you apply a force F, you increase

this force, or more precisely, the volume stress. - Bulk modulus is usually written as
- B (F/A) / ( ?V / V)
- B ?p / ( ?V / V)
- ?p is the increase in pressure above normal air

pressure

Table

1.0 x 10 9 4.5 x 109 26 x 109 2.2 x 109

Compare Bulk Modulus

- NOTE
- B ?p / ( ?V / V)
- For a gas, it takes the smallest pressure to

create a change in volume, so gasses have the

smallest bulk modulus - Solids require a big pressure to have a change in

volume, so they have the biggest bulk modulus - Bigger modulus implies a stronger material

Practice

- By how much must you change the pressure on a

liter of water to compress it by 0.10?

Calculation

- B ?p / ( ?V / V)
- B 2.2 x 109N/m2
- ( ?V / V) .1 (1/100) .001
- ?p B ( ?V / V)
- ?p 2.2 x 109 N/m2 .001 2.2 x 106 N/m2
- Because there is always air pressure, this is the

INCREASE in pressure to cause the change in

volume

Pressure in a Fluid

- The molecules in a solid are tightly bound.
- In a liquid or a gas, however, the molecules are

in motion and free to move. - The diagram shows a cubical container filled with

gas. - The molecules of the gas are represented by the

red dots. - The blue arrows show the direction they are

moving in.

Pressure

- When a gas molecule with momentum p hits the wall

and bounces back, it exerts an impulse equal to

2p on the wall. - To calculate the force exerted by a single

molecule, you need to know how long the collision

lasted. - The effect of a single molecule is very small,

but the effect of a room full, like in a tornado,

can be huge.

Measuring Pressure

- The diagram shows a device that measures the

pressure of a gas. - Because gas molecules can move freely, the

pressure of a gas will be the same no matter

where you place the gauge. - Furthermore, any changes in pressure applied to

the container will be transmitted throughout the

gas volume by the moving molecules.

Pascals Principle

- Because liquid molecules are also free to move,

the same principles also apply to liquids and

thus all fluids. - Pascals principle states this effect
- Pressure applied to an enclosed fluid is

transmitted undiminished to every point in the

fluid and to the walls of the container.

Atmospheric Pressure

- When a gas is in a container, it exerts uniform

pressure throughout the container. - In the case of atmosphere, however, the air stays

at the Earths surface because of the force of

gravity. - Air pressure is greatest at sea-level and

decreases as the altitude increases. - The pressure of our atmosphere at sea-level is
- This amount of pressure is defined as 1

atmosphere (atm).

Density

- A liquid is much more dense than a gas.
- As a result, the effect of gravity on a liquid is

much greater than that on a gas in the same sized

container on the earth. - In ordinary sized containers, a gas exerts

uniform pressure throughout the container it is

in, but a liquid does not. - To refresh your memory, density is the mass of an

object or system of particles divided by the

volume it occupies. - ? m / V

Depth and Pressure in a Liquid

- The diagram explains the relationship between

pressure and depth in a liquid. - It shows a container of water with an imaginary

rectangular column of water. - The column of water has a surface area A, and a

weight mg. - Hence, a person holding the column of water

experiences a force mg, and a pressure mg/A,

which is the pressure of the water at the bottom

of the container.

Demonstration

Pressure

- You can derive the equation for pressure using

the relationship between mass and density - ? m/V gt m ? V
- V A h gt m ? Ah
- p F/A mg/A ? Ah g / A
- p ? g h
- The total pressure of the liquid at the bottom of

the container is - p p0 ? g h where p0 normal air pressure

Density

Practice

- What is the total pressure exerted on the back of

a scuba diver in a lake at a depth of 8.00 m?

Total Pressure

- Pressing down on the divers back is all the water

above him, plus all the AIR above the water. - pwater ? g h 1000kg/m3 x 9.8m/s2 x 8m
- pwater 7.84 x 104 N/m2 7.84 x 104 Pa
- pAir 1 Atmosphere 101kPa
- Ptotal 7.84 x 104 Pa 101kPa
- Ptotal 1.79 x 105 Pa

Barometers

- A barometer is a device that measures atmospheric

pressure. - To make a barometer at home, turn a glass upside

down under water. - When you lift it straight up, it will remain

filled with water. - Air pressure holdsthe water up

Barametric Pressure

- At the surface of the Earth, the mercury in a

barometer rises to a height of 760 mm. - This means atmospheric pressure is
- The density of mercury is taken from the density

table shown previously. - Here we have introduced another unit of pressure

called the torr, named after a famous scientist,

and representing the pressure corresponding to 1

mm of mercury.

Pascals Principle

- The pressure of a gas in a container is the same

at every point in the gas and on the walls of the

container. - Pressure in a liquid varies with depth
- The diagram on the screen shows a piston that

exerts a force F over an area A on a body of

water. - This creates a pressure p that can be experienced

throughout the fluid.

Pressure

- Total pressure is the sum of the pressure from

the weight of the fluid AND the added pressure

from the piston

Hydraulic Lift

- Two pistons are connected to a hydraulic lift.
- When you press the input piston, the output

piston rises. - The pressure on the input piston is the same as

that on the output piston. - Keep in mind that the force is given by the

pressure times the area. F P A - The output piston has a much larger area than the

input piston. - Therefore, the output piston exerts a much larger

force than the input piston, and you can easily

lift an automobile. F P A

Lift a Car

- p1 p2
- F1/A1 F0/A0
- F0 F1 (A0 /A1 )

Practice

- The input and lift (output) pistons of a garage

lift have diameters of 10 cm and 30 cm,

respectively. The lift raises a car with a weight

of 1.4 x 104 N. - (a) What is the force on the input piston?
- (b) What is the pressure to the input piston?

Calculations

- p1 p2
- F1/A1 F0/A0
- R1 10cm (1m/100cm)/2 .05m
- R2 30cm (1m/100cm)/2 .15m
- A1 ? R12 ?(.05m)2 7.85x10-3m2
- A2 ? R22 ?(.15m)2 7.07x10-2m2
- F1 F0(A1/A0)
- F1 1.4 x 104 N (7.85x10-3m2/ 7.07x10-2m2)
- F1 1,550N
- p1 F1/A1 1,554N / 7.85x10-3m2 2.0x105N/m2
- p2 F2/A2 1.4 x 104 N / 7.07x10-2m2
- p2 2.0x105N/m2 checks!

Thought Experiment

- If the output piston of a hydraulic lift has a

very large area, a two-year-old can lift the

Empire State Building. - Does this make sense?
- Is energy conserved?

Results

- Yes, it does make sense.
- Simple machines, such as levers, transform small

input forces into large output forces. - Energy is the ability to do work.
- It is the product of force and distance.
- Machines transform small forces applied over

large distances to large forces applied over

small distances.

Practice

- A 60-kg athlete does a single-hand handstand.
- If the area of the hand in contact with the floor

is 100 cm2, what pressure is exerted on the

floor?

Calculation

- p F/A
- A 100 cm2 (1m/100cm)2 1.00x10-2m2
- p mg/A 60kg 9.8m/s2/ 1.00x10-2m2
- p 5.88x104N/m2

Practice

- An oak barrel with a lid of area 0.20 m2 is

filled with water. A long, thin tube of

cross-sectional area 5.0 X 10-5 m2 is inserted

into a hole at the center of the lid, and water

is poured into the tube. - When the water reaches 12 m high, the barrel

bursts. - What was the weight of the water in the tube?
- What was the pressure of the water on the lid of

the barrel? - What was the net force on the lid due to the

water pressure?

Calculations

- W m g ? V g ? Ah g
- A 5.0 X 10-5 m2 , h 12 m
- W1000kg/m3 5.0 X 10-5 m2 12 m 9.8m/s2
- W 5.88 N
- p F/A 5.88 N / 5.0 X 10-5 m2
- p 1.18x105N/m2
- OR
- p ?gh 1000kg/m3 9.8m/s2 12m 1.18x105N/m2
- Checks!

Calculations

- p F / A
- F p A
- F 1.18x105N/m2 0.20 m2
- F 2.36 x 104N (More than 5000 pounds!)

Archimedes

- Have you ever yelled, 'Eureka!' when you figured

out how to solve a challenging problem? - Well the story goes that when Archimedes figured

out the principle named after him, he was so

excited that he not only yelled, 'Eureka,' but

also went running through town from the public

bath naked! - You may not get that excited over Archimedes

principle, but you will learn some useful and

interesting things about fluid buoyancy and fluid

flow in this lesson.

Archimedes' Principle

- The upward force experienced by an object when it

is immersed in a liquid or a gas is called

buoyant force. - Buoyant force is equal in magnitude to the weight

of the volume of fluid displaced. - This rule is called Archimedes' principle.
- BF mg ? V g

Archimedes Principle

- The diagrams on the screen illustrate Archimedes

principle. - Part (a) shows buoyant force as the force that

pushes up when you hold a piece of wood under

water. - Part (b) shows buoyant force as the force that

reduces the weight of an object weighing 10 N

to 8 N.

Practice

- A spherical helium balloon has a radius of 30 cm.

What is the buoyant force acting on it in air?

Calculations

- Find the volume if R 30 cm .3 m
- V (4/3) ? R3 .113 m3
- BF mg ? V g 1.29kg/m3 .113 m3 9.8m/s2
- BF 1.43 N (about 1/3 of a pound)
- If the balloon and the air in the balloon has

weight less then 1.43 N, it will float if more,

it will sink

Practice

- The diagram shows a partially submerged iceberg

at rest. - It is in equilibrium because its buoyant force is

equal to the weight of the iceberg. - Suppose a uniform solid cube of material 10 cm on

each side has a mass of 700 g. Will the cube

float? If it will float, how much of the cube

will be submerged?

Calculations

- Suppose the cube were completely SUBMERGED!
- LSide 10 cm .1m
- V LWH (.1m)(.1m)(.1m) 10-3 m3
- BF ? V g 1x103Kg/m3 10-3 m3 9.8m/s2
- BF 9.8N
- Weight of the object mg
- W 700 g (1kg/1000g) 9.8m/s2 6.86 N
- The buoyant force is greater than the weight!
- The object will be pushed up to the surface and

will rise above the surface until the weight of

the fluid displaced just balances the weight of

the object

Floating Object

- V LWH (.1m)(.1m)(.1m) 10-3 m3
- W 700 g (1kg/1000g) 9.8m/s2 6.86 N
- BF 6.86 N ? V g 1x103Kg/m3 V 9.8m/s2
- V 6.86 N / (1x103Kg/m3 9.8m/s2 )
- V 7 x 10-4 m3
- 7 x 10-4 m3 is the volume of the cube below the

surface - The remainder
- 10-3 m3 - 7 x 10-4 m3 3 x 10-4 m3 is above the

surface - It is 70 submerged

Ideal Fluid Flow

- To mathematically describe fluid flow like we

have projectile motion, for example can be very

difficult in real cases. - Consider the upper portion of the rising smoke

it looks almost random! - In order to obtain a basic description of fluid

flow, we will make some simplifications and

consider ideal fluid flow, so the flow looks more

like the lower part of the rising smoke stream. - The conditions for an ideal fluid are that its

flow is

Ideal Fluid Flow

- Steady The flow is not turbulent particles in

the fluid do not collide, and their paths do not

cross. The flow looks smooth. - Irrotational There is no rotational motion in

small volumes of the fluid, so there are no

whirlpools. A paddle wheel completely

embedded in the stream does not rotate. - Nonviscous Viscosity is negligible. This means

the fluid flows easily, without significant

friction or resistance to the flow. - Incompressible The density is constant

throughout the fluid. - When a fluid's flow has these four

characteristics, we say it is an ideal fluid.

Unless specified otherwise, we will consider

these four conditions to be true in the fluids we

consider.

Equation of Continuity

- When a liquid flows through a tube, the amount of

liquid entering the tube must equal the amount of

liquid coming out of it. - If the cross-sectional area of the tube varies

across its length, then the speed of the liquid

must vary too. - The diagram on the screen shows what happens to

the speed of water when the nozzle has a

smaller cross-sectional area than the tube.

Equation of continuity

- If the liquid is incompressible, the density of

the liquid remains the same throughout the tube.

If the liquid is incompressible, the density of

the liquid remains the same throughout the tube. - Hence, as the above equation simplifies to
- Av constant
- Area x velocity constant
- Volume/time constant
- Equation of continuity is an equation that

describes the fact that the amount of fluid

entering a tube is equal to the amount of fluid

leaving the tube.

Equation of Continuity

- High cholesterol in the blood can cause fatty

deposits called plaques to form on the walls of

blood vessels. - Suppose plaque reduces the effective radius of an

artery by 25. - Because the area is smaller, what must happen to

the velocity?

Bernoullis Equation

- Bernoulli's equation states that when a liquid is

moving fast, its pressure is reduced. - The diagram below shows a tube of varying

cross-sectional areas. - The pressure indicators show that the pressure is

lower in the middle region, where the smaller

cross sectional area results in the flow rate

being greater.

Bernoulli's Equation

- To observe Bernoulli's equation, conduct the

following experiment. - Hold two corners of a sheet of paper with both

hands. - Rotate your hands so that the paper takes the

shape of the wing of an airplane. - Blow over the top of the paper.
- You will observe that the sheet of paper rises.
- It lifts up because air flows on one side of the

paper, but is stationary on the other. - This is the principle behind the lifting of air

planes the orientation and shape of the wing of

an airplane causes the air flowing over it to

flow faster than the air under it.

Wing Design

Surface Tension

- When there is a boundary between a liquid and a

gas, some remarkable phenomena occur. - The figures on the screen show two water/air

phenomena and one soap solution/air boundary

phenomena.

Surface Tension

- 1) An insect is able to walk on water. 2) Water

in air forms circular droplets.3) Soap solutions

form bubbles. - In the phenomenon where bubbles are made, there

are two surfaces the inside and the outside of

the bubble. - Although the bubble is thin compared to a piece

of paper, it is very thick compared to the size

of a molecule. - The surface in an air-liquid boundary is only two

or three molecules thick.

Surface Tension

- The forces of attraction between the molecules of

a liquid are lower than those of a solid. - All the same, they do exist because otherwise the

molecules of a liquid would escape into the

atmosphere. - In a liquid, a molecule, which is at a distance

from the surface, is surrounded by other

molecules. - A molecule on the surface, however, is not

entirely surrounded by molecules. - The molecules on the surface are held together by

a horizontal force between them, which is called

surface tension.

Viscosity

- You must have noticed that in January, molasses

flows slower than water or kerosene. - This happens because of viscosity.
- Viscosity is a fluids internal resistance to

flow. - It can be measured with a device called a

viscosimeter. - There are various types of viscosimeters.

Flow Rate

- When fluid is flowing in a pipe, as a result of

viscosity, the speed of the fluid near the

surface of the pipe is less than that at the

center. - The flow rate is the average volume of a fluid

that flows beyond a given point during a time

interval - The diagram on the screen shows that is not easy

to calculate the average flow rate. - Flow rate has the unit of m3/s.

Summary

- What are the elastic modulie?
- What is the relationship between pressure and

depth in a fluid? - What is Archimedes principle?
- What are the essential principles of fluid flow?
- What causes surface tension and what is

viscosity?