As 22 Materials

8/12/2019 As 22 Materials

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2.2 Materials

MaterialsBreithaupt pages 162 to 171

April 11th, 2010


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AQA AS Specification

Lessons Topics

1 to 4 Bulk properties of solids

Density

= m / V

Hookes law, elastic limit, experimental investigations.

F = k L

Tensile strain and tensile stress.Elastic strain energy, breaking stress.

Derivation of energy stored = FL

Description of plastic behaviour, fracture and brittleness; interpretation of simple

stress-strain curves.

5 & 6 The Young modulus

The Young modulus = tensile stress = FLtensile strain AL

One simple method of measurement.

Use of stress-strain graphs to find the Young modulus.


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Density ()

density = massvolume

= m / V

unit = kg m-3

Note: 1 g cm-3is the same

as 1000 kg m-3


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Density examplesdensity

/ kg m-3

density

/ kg m-3Interstellar medium iron

hydrogen lead

helium mercuryair uranium

wood (average) gold

lithium

water Suns core

plastics neutron star

aluminium black hole

0.0989

0.1791.29

0.534

19 100

850to1400

10-25to 10-15

13 500

150 000

700

1000

2 700

7 900

11 300

22 610

19 300

1017

> 4 x 1017

osmium


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Question

Calculate the weight of a gold ingot of dimensions(20 x 10 x 4) cm

volume of gold = 800 cm3

= 0.0008 m3

mass = volume x density

= 0.0008 x 19 300 = 15.4 kg

weight = 15.4 x 9.81

weight of gold ingot = 152 N


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Answers

density mass volume

240 g 40 cm3

3000 kg m-3 4500 kg

0.80 g cm-3 80 cm3

9 kg 0.003 m3

6 g cm-3

3 g cm-3

1.5 m3

64 g

Complete:


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Hookes lawThe force(F ) needed to stretch a spring is directlyproportional to the extension(L ) of a spring from itsnatural length.

F L

Adding a constant of proportionality:

F = k Lkis called the spring constant

The spring constant is the force required to produce anextension of one metre.

unit = Nm-1


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Elastic limit

Up to a certain extension if the force isremoved the spring will return to its originallength. The spring is said to be behavingelastically.

If this critical extension is exceeded, known asthe elastic limit, the spring will be permanentlystretched.

Plasticbehaviour then occurs and Hookes lawis no longer obeyed by the spring.


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Question

A spring of natural length15cm is extended by 3cmby a force of 6N. Calculate(a) the spring constant and(b) the length of the spring

if a force of 18N is applied.

(a)F = k L k = F / L= 6N / 0.03mspr ing constant , k

= 200 Nm-1

(b)F = k L L = F / k

= 18N / 200 Nm-1

L= 0.09 m= 9 cm

And so the

springs length

= 24 cm


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Tensile stress ()

A stretching force is also called a tensileforce.

Tensile stress = tensile force

cross-section area

= F/ A

unitPa (pascal) or Nm-2

Note: 1 Pa = 1 Nm-2


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Breaking stress

This is the stress required to cause a

material to break.


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Tensile strain ()

Tensile strain = extension

original length

=L / L

unitnone (its a ratio like pi)


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Question

A wire of natural length 2.5 m and diameter 0.5mm is extended by 5 cm by a force of 40 N.Calculate:

(a) the tensile strain

(b) the tensile stress(c) the force required to break the wire if itsbreaking stress is 1.5 x 109Pa.

(a) =L / L= 0.05m / 2.5mtensile strain, = 0.02


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Question(b) = F / A

A = Area

= D2/ 4= x 0.0005m2 / 4

= 1.96 x 10-7m2

= 40N / 1.96 x 10-7m2

stress, = 2.04 x 108

Pa

(c) = F / A

F = A = 1.5 x 109 Pa x 1.96 x 10-7m2

Breaking Force, F = 294 N


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The Young Modulus (E )

This is a measure of the stiffness of a material.

Young modu lus = tens i le st ress

tensi le strain

E = /

unitpascal (same as stress)


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Also: tensile stress = F / A

and tensile strain =L / L

Therefore: E = (F / A )

(L / L)

which is the same as:

E = F L

A L


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Examples of Young Modulus

Material E/ x 109Pa

diamond 1200

titanium carbide 345

steel 210copper 130

brass 100

glass 80oak 12

rubber band 0.02


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Question 1

Calculate the tensile strain caused to a steel wirewhen put under 4.0 x 10 7Pa of stress.

E = /

= / E= (4.0 x 10 7Pa) / (210 x 10 9Pa)

= 0.01904

tensile strain = 0.00019


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Question 2

A metal wire of originallength 1.6m, cross sectionalarea 0.8 mm2extends by4mm when stretched by atensile force of 200N.

Calculate the wires

(a) strain,

(b) stress

(c) Young Modulus.

(a) =L / L= 0.004m / 1.6m

= 0.0025

strain = 0.0025


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(b) = F / A= 200N / 0.8 x 10 - 6m2

(1m2= 1 000 000 mm2)

stress = 2.5 x 108Pa

(c)E = / = 2.5 x 10 8/ 0.0025

Young m odu lus= 1.0 x 1011Pa


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Measurement of E

With equal control and test weightsof 10N adjust the micrometer

attached to the test wire so that the

spirit level between the two wires is

horizontal.

Note the reading on the micrometer

and also the length of the test, L

wire using a metre ruler.

Use another micrometer to measurethe diameter of the test wire at

various places along the wire and

calculate an average value, D.

hinge

test

weights

rigid support

spiritlevel

long

wires

control

weight

micrometer


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Measurement of E

Calculate the average cross-section area of the wire, A fromA = D2/4

Add an additional load, Fof 5N

to the test wire.

Readjust the micrometer tobring the spirit level again andnote the new reading

hinge

test

weights

rigid support

spiritlevel

long

wires

control

weight

micrometer


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Measurement of E

Stop before the strain reaches0.01 in order to prevent the wireexceeding its limit ofproportionality (just before theelastic limit).

Draw a graph of stress againststrain. This should be a straightline through the origin.

Measure the gradient of thisgraph which will be equal to theYoung Modulus, Eof the testwire.

0

Stress, / Pa

Strain,

Gradient= / = E


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Stressstrain curves(a) Metal wire (e.g. steel)

P = Limit of

proportionality

Up to this point

the stress is

proportional to

the strain.

stress

strain

P


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E = Elastic limit

This is close to P

Beyond this point the

wire will become

permanently

stretched and sufferplastic deformation.

stress

P

E

strain


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Y1= Yield point

This is where the

wire weakens

temporarily.

BeyondY2, a small

increase in stress

causes a large

increase in strain asthe wire undergoes

plastic flow.

Y1stress

P

E

strain

Y2


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UTS = Ultimatetensile stress

Beyond the

maximum stress,(UTS), the wire losesits strength, extendsand becomesnarrower at itsweakest point whereit fractures at B

Y1stress

P

E

UTSbreaking

point B

strain

Y2


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Stressstrain curves(b) Brittle material (e.g. glass)

A brittle material does

not undergo plastic

deformation and willfracture at its

elastic limit.

stress

P

E

breaking

point B

strain


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Stressstrain curves(c) Ductile material (e.g. copper)

A ductile material can bedrawn into a wire.

Both steel and copper areboth ductile but copper ismore ductile because itcan withstand a greaterstrain than steel beforebreaking although it is notas strong or as stiff assteel.

stress

copper

steel

strain


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Elastic strain energy

When a spring or wire is stretched potentialenergy is stored.

This form of potential energy is called elasticstrain energy.

Consider a spring of original length L

undergoing an extensionLdue to a tensileforce F.


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The graph opposite showshow the force varies as thespring extends.

The work done in extendingthe spring is given by:

work = force x distance0

force

extension


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= average tensile force x extension

= F L= area under the cur ve

= energy s tored in the spr ing

and so:

elastic strain energy = F L

0

F

area = F L

L

force

extension


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Stretching rubberThe work done in stretching rubberup to extensionLis equal to thearea under the loading curve.

The unloading curve for rubber isdifferent from its loading curve.

When the rubber is unloaded onlythe energy equal to the area underthe unloading curve is returned.

The area between the two curves is

the energy transferred to internalenergy, due to which the rubberband becomes warmer.

0

un load ing

L

force

extension

loading

energy lost

to heating

the rubber


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Answers

tensile force extension strain energy

120 N 2 m

40 N 15 cm

3 kN 50mm 150 J

2MN 6 m 12 J

Complete:

120 J

3 J

100

4


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QuestionA spring of original length

20cm extends to 25cm

when a weight of 4N is

hung from it. Calculate:

(a) the elastic strain energystored in the spring,

(b) the spring constant

(c) the length of the springwhen it is storing 0.5 J of

energy.

(a)strain energy

= F L

= x 4N x 0.05m

strain energy

= 0.10 J


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(b)F = k L k = F / L= 4N / 0.05m

spring constant, k = 80 Nm-1

(c)strain energy = F Land F = k Lwhen combinedgive: strain energy = k (L)2

L = (2 x strain energy / k)= (2 x 0.5 / 80)= (0.0125)= 0.112m

Therefore spring length = 20cm + 11.2cm

= 31.2 cm


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Internet Links Balloons & Bouyancy- PhET - Experiment with a helium

balloon, a hot air balloon, or a rigid sphere filled with differentgases. Discover what makes some balloons float and others

sink.

Density Lab- Explore Science

Floating Log- Explore Science Stretching Springs- PhET - A realistic mass and spring

laboratory. Hang masses from springs and adjust the spring

stiffness and damping. You can even slow time. Transport the

lab to different planets. A chart shows the kinetic, potential,and thermal energy for each spring.
http://phet.colorado.edu/new/simulations/sims.php?sim=Balloons_and_Buoyancyhttp://subscription.echalk.co.uk/Science/chemistry/atomicStructure/atomicStructure.htmlhttp://www.ionaphysics.org/lab/Explore/dswmedia/density.htmhttp://www.ionaphysics.org/lab/Explore/dswmedia/floatlog.htmhttp://subscription.echalk.co.uk/Science/chemistry/atomicStructure/atomicStructure.htmlhttp://phet.colorado.edu/new/simulations/sims.php?sim=Masses_and_Springshttp://phet.colorado.edu/new/simulations/sims.php?sim=Masses_and_Springshttp://subscription.echalk.co.uk/Science/chemistry/atomicStructure/atomicStructure.htmlhttp://www.ionaphysics.org/lab/Explore/dswmedia/floatlog.htmhttp://www.ionaphysics.org/lab/Explore/dswmedia/floatlog.htmhttp://www.ionaphysics.org/lab/Explore/dswmedia/density.htmhttp://subscription.echalk.co.uk/Science/chemistry/atomicStructure/atomicStructure.htmlhttp://phet.colorado.edu/new/simulations/sims.php?sim=Balloons_and_Buoyancy


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Core Notes from Breithaupt pages 162 to 171

1. Define what is meant by density, include the equation.

2. Define Hookes law. Quote the equation for Hookes law.3. What is meant by (a) the spring constant and (b) the elastic

limit.

4. Define (a) tensile stress; (b) breaking stress; (c) tensile strain &(d) Young modulus.

5. Explain how the Young Modulus of a wire can be foundexperimentally.

6. Copy Figure 3 on page 168 and explain the significance of thelabelled points.

7. Copy Figure 4 on page 169 and use it to explain the meaning of

the terms: (a) strength; (b) brittle & (c) ductile.8. What is meant by strain energy?

9. Copy Figure 4 on page 166 and use it to show that the strainenergy stored by a spring is given by: strain energy = F L.


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Notes on Densityfrom Breithaupt pages 162 & 163

1. Define what is meant by density, include theequation.

2. Calcu late (a) the vo lume of copper that has a massof 178 kg ; (b) the mass o f 14.4m3o f air; (c) thedensi ty of a so l id of mass 2000kg and vo lume 3m3.

3. State the dens i ty of (a) a metal li c so lid ; (b) water &(c) air

4. (a) Exp lain why a dens it y o f 1000 kgm-3is the sameas one of 1 g cm-3. (b) What is the density of waterin g mm-3?

5. Exp lain how to measure the dens ity o f (a) a regu larso l id; (b) a liqu id and (c) an irr egular so l id.

6. Try the Summary Ques tions on page 163


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Notes on Hookes law and Springsfrom Breithaupt pages 164 to 166

1. Define Hookes law. Quote the equation for Hookeslaw.

2. What is meant by (a) the spring constant and (b)the elastic limit.

3. A sp r ing o f natural leng th 40 cm is ex tended to 50cm by a force o f 2N. Calcu late (a) the sp ringconstant in Nm-1(b) the expected length of th esp ring if i t were to be extended by a forc e of 5N.

4. Show that the overal l spr ing cons tan t , k fo r (a)

sp r ings in ser ies is given by k = k1+ k2; (b) spr ingsin paral lel is given by 1 / k = 1 / k1+ 1 / k2where k1and k2are the spr ing con stants of the ind iv idualspr ings.

5. Try Summary Ques tions 1, 2 & 3 on page 166


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Notes on Stress, Strain & Young Modulus

from Breithaupt pages 167 to 169

1. Define (a) tensile stress; (b) breaking stress; (c) tensile strain& (d) Young modulus.

2. Explain how the Young Modulus of a wire can be foundexperimentally.

3. Copy Figure 3 on page 168 and explain the significance ofthe labelled points.

4. Copy Figure 4 on page 169 and use it to explain the meaningof the terms: (a) strength; (b) brittle & (c) ductile.

5. Calcu late the (a) s t ress ; (b ) s t rain & (c ) Young Modu lus fo r aw ire of or ig inal length 2.5m and cros s-sect ional diameter0.4mm that stretches by 2cm when a tensio n of 50N isapplied.

6. Show th at Young Modu lu s is equal to (T x L) / (A x L)wherethese symbo ls have the meaning shown on page 168.

7. Try the Summary Ques tio ns on page 169

N t St i E


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Notes on Strain Energy

from Breithaupt pages 170 & 171

1. What is meant by strain energy?2. Copy Figure 4 on page 166 and use it to show thatthe strain energy stored by a spring is given by:strain energy = F L.

3. A sp r ing o f natural leng th 30 cm is ex tended to36cm by a force of 5N. Calcu late the energy s toredin the sp r ing.

4. Copy Figu re 2 on page 171 and exp lain why arubber band becom es warmer when i t is

cont inual ly stretched and unstretched.5. Show that the s train energy s to red by a sp ring is

g iven by: strain energy = k L2.6. Try Summary Quest ion 4 on page 166 and al l o f the

questions on page 171.

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As 22 Materials