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Thermal ExpansionCreated by: Marlon Flores Sacedon Physics section, DMPS

June 2010

Thermal Expansion

Two Kinds of Thermal Expansion Linear Expansion Volume Expansion

Most materials expand when their temperature increase. The decks of bridges need special joints and supports to allow for

expansion. A completely filled and tightly capped bottle of water cracks when

it is heated You can loosen a metal jar lid by running hot

water over it. .

These are examples of Thermal Expansion

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Linear Expansion

Lo

T2

T2>T1

T1

L

Volume Expansion

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Linear Expansion

Lo

T2

T2>T1

T1

L

Volume Expansion

Vo @ T1

@ T2VT2>T1

TLL o TLL o

12oo

o TTLLL

TLL

Where: = coef. of linear expansion (1/K) L = change in length (m) L & Lo = final & initial length (m) T2&T1= final & initial temperature (oC)

TVV o TVV o

12oo

o TTVVV

TVV

Where: = coef. of volume expansion (1/K) V = change in volume (m3) V & Vo = final & initial volume (m3) T2&T1= final & initial temperature (oC)

L

Material [K-1 or (Co)-1]

Aluminum 2.4x10-5

Brass 2.0x10-5

Copper 1.7x10-5

Glass 0.4-0.9x10-5

Invar 0.09x10-5

Quartz 0.04x10-5

Steel 1.2x10-5

Coefficients of Linear Expansion

Material [K-1 or (Co)-1]

Aluminum 7.2x10-5

Brass 6.0x10-5

Copper 5.110-5

Glass 1.2-2.7x10-5

Invar 0.27x10-5

Quartz 0.12x10-5

Steel 3.6x10-5

Ethanol 75x10-5

Carbon Disulfide 115x10-5

Glycerine 49x10-5

Mercury 18x10-5

Coefficients of Volume Expansion

3Relationship between coefficient of volume expansion & coefficient of linear expansion MFS

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Example 1: A surveyor uses a steel measuring tape that is exactly 50.000 m long at a temperature of 20 oC. What is its length on a hot summer day when the temperature is 35 oC? Answer: 50.009 m

50 mTemperature of tape @ 20oC

Transforming

From the formula:

Solution Lo

L = ?Temperature of tape @ 35oC

From table, the coefficient of linear expansion 1o5 )C(10x2.1

12oo

o TTLLL

TLL

12o12oo TT1LTTLLL = 50.0009 m Ans

Example 2: The surveyor uses the measuring tape (in Example 1) to measure a distance when the temperature is 35 oC; the value that she reads off the tape is 35.794m. What is the actual distance?

Answer: 35.800 m

SolutionLo = 50 m

Temperature of tape @ 20oC

L = 50.009 mTemperature of tape @ 35oC

Let: x = actual distance @ T=35 oC

m50m009.50

794.35x

m8.35x Ans

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Example 3: A glass flask with volume 200 cm3 is filled to the brim with mercury at 20 oC. How much mercury overflows when the temperature of the system is raised to 100 oC? The coefficient of linear expansion of the glass is 0.40x10-5 K-1. Answer: 2.7 cm3

Solution

Glass flask filled w/ mercury

@ T1=20oC @ T2=100oC

Mercury column

Glass flask

glassHgover VVV 12oglass12oHgover TTVTTVV

Mercury, expanded

volume( VHg)

Mercury overflows ( Vover)

Glass flask, expanded volume ( Vglass)

From table: 1o5Hg C10x18 glassHgoover TTVV 12

1o51o5glass C10x2.1C10x4.03 151040.0 oglass Cx

55 102.1101820100200 xxVover

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Seat Work1: The Humber Bridge in England has the worlds longest single span, 1410m in length. Calculate the change in length of steel deck of the span when the temperature increases from -5.0oC to 18.0oC.

Answer: 0.39 m

Seat Work2: A metal rod is 40.125 cm long at 20.0oC and 40.148 cm long at 45.0oC. Calculate the average coefficient of linear expansion of the rod for this temperature range. Answer: 2.3x10-5 (Co)-1

Seat Work3: A glass flask whose volume is 1000.00 cm3 at 0.0oC is completely filled with mercury at this temperature. When flask and mercury are warmed to 55.0oC, 8.95 cm3 of mercury overflow. If the coefficient of volume expansion of mercury is 18.0x10-5 K-1, compute the coefficient of volume expansion of the glass. Answer: 1.7x10-5 (Co)-1

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Assignment1) A Pendulum shaft of a clock is made of brass. What is the fractional change in length of the shaft when it is cooled from 19.50oC to 5.00oC?

Answer: -2.9x10-4

2) An underground tank with a capacity of 1700L (1.70m3) is filled with ethanol that has an initial temperature of 19.0oC. After the ethanol has cooled off to the temperature of the tank and ground, which is 10.0oC, how much air space will there be above the ethanol in the tank? (Assume that the volume of the tank doesnt change.)

Answer: 2.3x10-5 (Co)-1

3) A metal rod that is 30.0 cm long expands by 0.0650 cm when its temperature is raised from 0oC to 100oC. A rod of a different metal and of the same length expands by 0.0350 cm for the same rise in temperature. A third rod, also 30.0 cm long is made up of pieces of each of the above metals placed end-to-end and expands 0.0580 cm between 0oC and 100oC. Find the length of each portion of the composite bar. Answer: 23.0cm, 7.0cm

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F

TLL

thermalo

Thermal Stress soAYF

LL

tensiono

0AYFT

LL

LL

tensionothermalo

TYAF (thermal stress)

FA

Tensile Stress)A(Area)F(Force

Tensile Strain)L(LengthInitial)L(lengthinChange

o

LoL

oLLA

F

StrainStress)Y(Moduluss'Young

oLLA

FY

Where: F = Tensile force, (N) A = cross-section area, (m2) Y = Youngs Modulus, (Pa or N/m2) = coef. of linear expansion, (K-1) = change in tempersture, (K) T

Approximate Youngs Modulus

Substance Youngs Modulus, Y

(Pa)Aluminum 7.0x1010

Brass 9.0x1010

Copper 11x1010

Crown glass 6.0x1010

Iron 21x1010

Lead 1.6x1010

Nickel 21x1010

Steel 20x1010

MFSLo=10cm

Lo=10cm

Example 1: An aluminum cylinder 10 cm long, with a cross-section area of 20 cm3, is to be used as a spacer between two steel walls. At 17.2 oC it just slips in between the walls. When it warms to 22.3oC, calculate the stress in the cylinder and the total force it exerts on each wall, assuming that the walls are perfectly rigid and a constant distance apart. Answer: -8.6x106 and -1.7x104N

@ T1=17.2 oC

F = 0

Area A=20cm3

@ T1=22.3 oC

F > 0

TYAF

KKxPaxAFStress )2.173.22)(104.2(1070.0 1511

)1200(106.8 26 inlborPax

AF

Stress is - 8.6x106 PaNegative sign indicates compression

PaxinF 6

2 106.820

F = -1.7x104 N (compression)

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Assignment: a) A wire that is 1.50 m long at 20oC is found to increase in length by 1.9 cm when warmed to 420oC. Compute its average coefficient of linear expansion for this temperature range. b) The wire is stretched just taut (zero tension) at 420oC. Find the stress in the wire if it is cooled to 20oC without being allowed to contact. Youngs modulus for the wire is 20.x1011 Pa. Answer: a) 3.2x10-5 (Co)-1; b)2.5x109 Pa

1.50 m@ 20oC

0.019 m

@ 420oC

151017.3)20420(50.1

019.0

KxTL

L

o

1.519 m

@ 420oC

1.519 m

@ 20oCF>0 F>0

TYAF

)(105.2

)42020)(1017.3(1020

9

511

stressPax

xxAF

Ans

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Assignment1) A brass rod is 185 cm long and 1.60 cm in diameter. What force must be applied to each end of the rod to prevent it from contracting when it is cooled from 120oC to 10oC? Answer: 4.0x104 N

2) Steel train rails are laid in 12.0 m-long segments placed end-to-end. The rails are laid on a winter day when their temperature is -2.0oC. a) How much space must be left between adjacent rails if they are to just touch on a summer day when their temperature is 33.0oC? b) If the rails are originally laid in contact, what is the stress in them on a summer day when their temperature is 33.0oC?

Answer: a) 5.0x10-3 m; b) 8.4x107 Pa

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