Measurement of Coefficient of Linier Expansions

8/6/2019 Measurement of Coefficient of Linier Expansions

http://slidepdf.com/reader/full/measurement-of-coefficient-of-linier-expansions 1/16

Measurement of Coefficient of Linier Expansions

I. The purpose of the experiment

To determine the coefficient of linier expansions of metal

II. Base theory

Most materials expand when heated through a temperature change that does not

produce a change in phase. The added heat increases the average amplitude of vibration of

the atoms in the material which increases the average separation between the atoms. Suppose

an object of length L undergoes a temperature change of magnitude ΔT. We find

experimentally that if ΔT is reasonably small, the change in length ΔL, is generally

proportional to L and ΔT. Therefore we can write:

ΔL=α L ΔT

Where α is the coefficient of linear expansion and has different values for different

materials. For materials that are not isotropic, such as an asymmetric crystal, α can have a

different value depending on the axis along which the expansion is measured. α can also vary

with temperature so that the degree of expansion depends not only on the magnitude of the

temperature change, but also on the absolute temperature as well.

However, typically these variations are negligible compared to the accuracy withwhich engineering measurements need to be made. We can often safely take the coefficient

of linear expansion as a constant for a given material. Shown below are some values of α for

some common solids.

No SubstancesCoefficient of Linier Expansions

(α) ( 10-6/ o C)

1 Lead 29

2 Aluminum 23

3 Brass 17

4 Copper 195 Steel 11

6 Glass 9

The order of magnitude of the expansion is about 1mm per meter length per 100 oC.



III. Apparatus of experiment

- Steam Generator

- Copper, and steel tube

- Expansion base

- Ruler( sms 1cm)

- Thermometer

- Multi-meter digital

- Bucket

IV. Experimental Procedure

1. Measure the length L of the Steel tube at room temperature.

2. Mount the copper tube in the expansion base as shown in the figure. The stainless

steel pin on the tube fits into the slot on the slotted mounting block and the bracket on the

tube presses against the spring arm of the dial gauge.

3. Use one of the provided thumbscrews to attach the thermistor lug to the threaded

hole in the middle of the copper tube. The lug should be aligned with the axis of the tube,

so there is maximum contact between the plug and the tube.

4. Plug the leads of your ohmmeter into the banana plug connectors labeled

THERMISTOR in the center of the expansion base.

5. Measure Room, the resistance of the thermistor at room temperature.

6. Use tubing to attach the steam generator to the end of the copper tube. Attach it to

the end farthest from the dial gauge.

7. Turn on the steam generator. As steam begins to flow, watch the dial gauge and

the ohmmeter. When the thermistor resistance stabilizes, record the resistance R hot



8. Turn of the heater and Periodically measure the change of length at each interval

of decreasing the resistant.

9. Look the conversion at expansion base and record the temperature.

10. Repeat step 1-9 for measuring coefficient linier expansion of copper

V. An arrangement of data

No Temperature(oC)Scale in Digital gauge

spring(mm)

1

2

3

..

VI. Technique of data analysis (Calculation)

First of all by doing this experiment we find the change of the length by using equation:

nnl l l −=∆

+1

After that to determine the uncertainly of the length we use equation:

( ))1(

2

−

∆−∆=∑

N N

l l y

i

Next calculation is finding the change of temperature, to find the change of temperature we

can following equation:

1+−=∆ nn T T T

To determine the uncertainly of the temperature can be following equation:

5.012

1

2

1=== SMS x



For base theory we know that the formula the change of length, the equation can be derived

to find the coefficient of linier expansion, so the formula of linier expansion is:

T l

l

o∆

∆=α

To find the uncertainly of coefficient linier expansion we can following equation:

T l

l

o∆

∆=α

xT

T l l yl

T l l xT

yl )(

)..(

)(

)..(

)()(

11

011

0

∆∂

∆∆∂+∆∂

∆∆∂=∆∂

∂+∆∂

∂=∆

−−−−

α α α

xT l yT l 11

0

11

0 ..−−−−

∆+∆=∆α

xT

yl ∆

+∆

=∆α α

α

To find relative error in this experiment we use equation:

T

x

l

y RE

∆+

∆=

∆=

α

α

VII. Result Of Experiment

A. Result of Experiment using

iron( steel)

No Temperature(oC)

Scale in Digital

gauge spring(mm)

1 80 0

2 75 0.08

3 70 0.12

4 65 0.17

5 60 0.22

6 55 0.27

7 50 0.31

8 45 0.35

9 40 0.3910 35 0.42

11 30 0.44

B. Result of Experiment using

Copper

No Temperature(oC) Scale in Digital



gauge spring(mm)

1 77 0

2 72 0.12

3 67 0.19

4 62 0.27

5 57 0.36

6 52 0.44

7 47 0.51

8 42 0.57

9 37 0.64

10 32 0.70

11 27 0.74



VIII. Data Analysis

A. For Steel

No Temperature(oC)Scale in Digital

gauge spring(mm)ΔT (0C) ΔL (mm)

1 80 02 75 0.08 5 0.08

3 70 0.12 5 0.04

4 65 0.17 5 0.05

5 60 0.22 5 0.05

6 55 0.27 5 0.05

7 50 0.31 5 0.04

8 45 0.35 5 0.04

9 40 0.39 5 0.04

10 35 0.42 5 0.03

11 30 0.44 5 0.02

Total 50 0.44

Average 5 0.044

The result of calculation:

No ΔL (mm) l l ∆−∆2

l l ∆−∆

1 0.08 0.036 0.001296

2 0.04 0.004 0.000016

3 0.05 0.006 0.000036

4 0.05 0.006 0.000036

5 0.05 0.006 0.000036

6 0.04 0.004 0.000016

7 0.04 0.004 0.000016

8 0.04 0.004 0.000016

9 0.03 0.014 0.000196

10 0.02 0.024 0.000576

Total 0.44 0.00224

Average 0.044

After that we find the uncertainly of length using equation:

( )

)1(

2

−

∆−∆=∑

N N

l l y

i

51048.290

00224.0

)110(10

00224.0 −==

−= x y



005.0= y

So the value of ΔL

yl l ±=∆

mml )005.004.0( ±=∆

While the value of ΔT is

xT T ±=∆

C T o

)5.05( ±=∆

Finally we determine the value of coefficient of linier expansion using equation:

T l

l

o∆

∆=α

51008.15740

04.0 −== x x

α

Thus to determine the value uncertainly of coefficient of linier expansion of copper can be

following equation:

xT

yl ∆

+∆

=∆α α

α

5.051008.1005.0

04.01008.1

55 −−

+=∆ x xα

65 104.2)1.0125.0(108.1 −−=+=∆ x xα

So

α α α ∆+=

65 104.21008.1 −−+= x xα


%22%1001008.1

104.2%100

5

6

==∆

=−

−

x x

x x RE

α

α

B. For Copper



No Temperature(oC)Scale in Digital

gauge spring(mm)ΔT (oC) ΔL (mm)

1 77 0

2 72 0.12 5 0.12

3 67 0.19 5 0.07

4 62 0.27 5 0.08

5 57 0.36 5 0.09

6 52 0.44 5 0.08

7 47 0.51 5 0.07

8 42 0.57 5 0.06

9 37 0.64 5 0.07

10 32 0.70 5 0.06

11 27 0.74 5 0.04

Total 50 0.62

Average 5 0.068

The result of calculation:

No ΔL (mm) l l ∆−∆2

l l ∆−∆

10.12 0.052

0.00270

4

20.07 0.002

0.00000

4

30.08 0.012

0.00014

4

4 0.09 0.022

0.00048

4

50.08 0.012

0.00014

4

60.07 0.002

0.00000

4

70.06 0.008

0.00006

4

80.07 0.002

0.00000

4

90.06 0.008

0.00006

4

100.04 0.028

0.000784

Total 0.62 0.0044

Average 0.068

After that we find the uncertainly of length using equation:



( )

)1(

2

−

∆−∆=∑

N N

l l x

i

6108.490

00440.0

)110(10

00440.0 −==

−= x x

0022.0= y

So the value of ΔL

yl l ±=∆

mml )0022.0068.0( ±=∆

While the value of ΔT is

xT T ±=∆

C T o)5.05( ±=∆

Finally we determine the value of coefficient of linier expansion using equation:

T l

l

o∆

∆=α

5108.15.740

068.0 −== xα

Thus to determine the value uncertainly of coefficient of linier expansion of copper can be

following equation:

xT

yl ∆

+∆

=∆α α

α

5.05

108.10022.0

068.0

108.1 55 −−

+=∆x x

α

65 104.2)1.0032.0(108.1 −−=+=∆ x xα

So

α α α ∆+=

65 104.2108.1 −−+= x xα




%13%100108.1

104.2%100

5

6

==∆

=−

−

x x

x x RE

α

α

IX. Interpretations

1. For steel

From the table on base theory we know that the value of coefficient linier expansion of steel

that accepted is 1.1x10

-5

o

C

-1

, so to calculate the %error in experiment that we done using steel

tube is following equation:

%100% xValueTheory

ValueTheoryValue Experiment error

−=

%100101.1

101.11008.1%

5

55

x x

x xerror

−

−− −=

%8.1%100018.0%100101.1

1002.0%

5

5

===−

−

x x x

xerror

The value of coefficient linier expansion of steel that we that from experiment is 1.08oC-1.

Meanwhile, the value of coefficient linier expansion of steel from the theory is 1.1x10-5 oC-1.

Error in experiment using steel is just 1.8 %. Because the error less than 10%, so this experiment

can be accepted.

In this experiment that we done has relative error 22%, so this result not to accurately this

can happen probably caused when we calculate the data there are more rounding of data.

2.

For copper



From the table on base theory we know that the value coefficient of linier expansion of

copper that accepted is 1.7x10

-5

o

C

-1

, so to calculate %error in experiment that we done using

steel tube is following equation:

%100% xValueTheory


−=

%100107.1

107.1108.1%

5

55

x x

x xerror

−

−−−

=

%9.5%100059.0%100107.1

101.0%

5

5

===−

−

x x x

xerror

The value of coefficient linier expansion of copper that we get from experiment is 1.8 oC-1.

Meanwhile, the value of coefficient linier expansion of steel from the theory is 1.7x10-5 oC-1.

Error in experiment using steel is just 5. %. Because the error less than 10%, so this experiment

can be accepted.

In this experiment that we done has relative error 13%, so this result not to accurately this

can happen probably caused when we calculate the data there are more rounding of data. And

there are some error happen when we done the experiment. The error will be explained in

comment in above.

X. Comment

From the experiment that we have done, the result of the experiment is not accurately, it’s

caused by some error that we have when this experiment did. The error is classified to three,

there are:

1. Gross Error (error that caused by human):

among them is missreading when read the scale of the digital gauge spring because our scale

not perpendicular to the scale on digital gauge spring, and error in rounding the number when

we calculate the data in analysis the data because the data is in decimal.

2. Systematic errors (error that caused by instrument and environment):



a) Instrumental error: the error that occur’s because of tool’s has bigger smallest

scale such as thermometer that we use. that can make our experiment has high

relative error and our experiment not accurately.

b)Enviromental error: the error that occur because of the disturbance of enviromentsuch as the temperature not constant.

3. Random error : due to unknown causes and occur even when all systematic error have

been accounted for

Suggestions:

for the next experiments should use the temperature in interval between 40 0Celsius and

70 0Celsius, because if you use temperature higher than 70 0Celsius then the data obtained will

has high deviation level this is due because in the high temperature, the change length of the

material is not constant (changes too fast for the same interval temperature(ΔT)), while if the

experiment using a lower temperature than 40 0Celsius then data obtained will be deviated,

because at low temperatures the change of the length of a material is very slowly (the change

relatively small for the same ΔT)

XI. Conclusion

From the experiment that we done we get that The value of coefficient linier expansion of

steel that we that from experiment is 1.08oC-1, with accuracy 1.8%, and relative error 22%.While

The value of coefficient linier expansion of copper that we that from experiment is 1.8 oC-1, with

accuracy 5.9%, and relative error 13%. From both experiment we can conclude that the value of

coefficient linier expansion of copper bigger then the value of coefficient linier expansion of

steel.

Questions and Solution:



1. What can you conclude about measuring coefficient of linier expansion?

Solution:

From the experiment that we done we get that The value of coefficient linier expansion of

steel that we that from experiment is 1.08oC-1, with accuracy 1.8%, and relative error 22%.While

The value of coefficient linier expansion of copper that we that from experiment is 1.8 oC-1, with

accuracy 5.9%, and relative error 13%. From both experiment we can conclude that the value of

coefficient linier expansion of copper bigger then the value of coefficient linier expansion of

steel.

2. Explain the errors of this experiment!

Solution:

From the experiment that we have done, the result of the experiment is not accurately, it’s

caused by some error that we have when this experiment did. The error is classified to three,

there are:

1. Gross Error (error that caused by human):

Among them is missreading when read the scale of the digital gauge spring because

our scale not perpendicular to the scale on digital gauge spring, and error in rounding the

number when we calculate the data in analysis the data because the data is in decimal.

2. Systematic errors (error that caused by instrument and environment):

a) Instrumental error: the error that occur’s because of tool’s has bigger smallest scale

such as thermometer that we use. that can make our experiment has high relative error

and our experiment not accurately.

b) Enviromental error: the error that occur because of the disturbance of enviroment

such as the temperature not constant.

3. Random error : due to unknown causes and occur even when all systematic error

have been accounted for

3. Calculate the percentage errors (the order of accuracy) of this experiment!

To find %error for Steel is following equation:



%100% xValueTheory


−=

%100101.1

101.11008.1%

5

55

x x

x xerror

−

−−−

=

%8.1%100018.0%100101.1

1002.0%5

5

===−

−

x x x

xerror

To find %error for Copper is following equation:

%100% xValueTheory


−=

%100107.1

107.1108.1%

5

55

x x

x xerror

−

−−−

=

%9.5%100059.0%100107.1

101.0%5

5

===−

−

x x x

xerror



Reference

Djonoputro, B.D. 1977. Teori Ketidakpastian. Bandung: Universitas ITB.

Halliday, D., Resnick, R., and Walker, J. (1993), Fundamentals of Physics, 4th edn (extended),

John Wiley & Sons, New York.

Unname. Coefficient of Linier Expansion. (online)www.wikipedia.com. be access on April

14th2011.

http://www.wikipedia.com/

http://www.wikipedia.com/



Measurement Coefficient of Linier Expansion(Physics Laboratory II)

Lab Report

WRITTEN BY,

KOMANG GEDE YUDI ARSANA (NIM. 1013021018)

PHYSICS DEPARTMENT OF EDUCATION

FACULTY OF MATHEMATIC AND SCIENCE

GANESHA UNIVERSITY OF EDUCATION

April 2011

Measurement of Coefficient of Linier Expansions

Documents