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THERMAL EXPANSION WORKSHEET
PS 253 – Physics Lab for Engineers
/ 100pts
Name:
Section # :
DERIVE UNCERTAINTY IN THERMAL EXPANSION COEFFICIENT
You will be determining thermal expansion coefficients (α) for various metal rods based on measurements of rod
length, change in length, and change in temperature. Each of these measured variables has experimental
uncertainty. Given the equation =
∆
( − )
−1
where each variable x has measured uncertainty δx derive an
equation for the thermal expansion coefficient uncertainty δα.
EXPLAIN YOUR METHOD FOR DETERMINING TEMPERATURE UNCERTAINTY
In the space below, describe how you determined the uncertainty of your temperature measurements using the
digital temperature probe.
USEFUL MATHEMATICAL EQUATIONS
Percent Relative
Uncertainty, %δx
% =
∙ %
Percent Difference, %Diff
% =
| − |
∙ %
Discrepancy Factor, t
=
| − |
Page 1
UNKNOWN METAL ROD #1
Variable, x
Value [units]
Uncertainty, δx
Rod Length, Lo
60
±
Initial Temperature, Ti
23.77
±
Final Temperature, Tf
63.3008
±
Change in Length, ΔL
±
Change in Temp, ΔT
±
% Relative
Uncertainty
0.5mm
Thermal Expansion Coefficient (α) ± Uncertainty (δα)
Secondary Identifying Characteristics
Include any other measurements or observations you wish to use to help identify the type of material.
Silver
Mass 53.2g
Material Identity and Reference Expansion Coefficient
Using an external reference source of material thermal expansion coefficients, indicate what material you believe
the rod is made of. Give the material name, its reference thermal expansion coeff, and the reference source used.
Comparison of Measured Result to Reference Material
Percent Difference
Discrepancy Factor & Significance1
1
A discrepancy greater than 1.96δx has only a 5% chance of occurring randomly. This is unlikely and
considered a significant discrepancy. Given your data and assumptions, your result is unlikely to agree with
the reference. The lower the discrepancy the more acceptable the result. Results can also be inconclusive.
Page 2
UNKNOWN METAL ROD #2
Variable, x
Value [units]
Uncertainty, δx
Rod Length, Lo
59.9cm
±
Initial Temperature, Ti
26.556
±
Final Temperature, Tf
99.7553
±
Change in Length, ΔL
±
Change in Temp, ΔT
±
% Relative
Uncertainty
0.5
Thermal Expansion Coefficient (α) ± Uncertainty (δα)
Secondary Identifying Characteristics
Include any other measurements or observations you wish to use to help identify the type of material.
Brown
Mass:145grams. +- 0.5
Material Identity and Reference Expansion Coefficient
Using an external reference source of material thermal expansion coefficients, indicate what material you believe
the rod is made of. Give the material name, its reference thermal expansion coeff, and the reference source used.
Comparison of Measured Result to Reference Material
Percent Difference
Discrepancy Factor & Significance2
2
A discrepancy greater than 1.96δx has only a 5% chance of occurring randomly. This is unlikely and
considered a significant discrepancy. Given your data and assumptions, your result is unlikely to agree with
the reference. The lower the discrepancy the more acceptable the result. Results can also be inconclusive.
Page 3
UNKNOWN METAL ROD #3
Variable, x
Value [units]
Uncertainty, δx
Rod Length, Lo
60cm
±
Initial Temperature, Ti
26.97
±
Final Temperature, Tf
99.8847
±
Change in Length, ΔL
±
Change in Temp, ΔT
±
% Relative
Uncertainty
Thermal Expansion Coefficient (α) ± Uncertainty (δα)
Secondary Identifying Characteristics
Include any other measurements or observations you wish to use to help identify the type of material.
Dark brass color
160.2g
Material Identity and Reference Expansion Coefficient
Using an external reference source of material thermal expansion coefficients, indicate what material you believe
the rod is made of. Give the material name, its reference thermal expansion coeff, and the reference source used.
Comparison of Measured Result to Reference Material
Percent Difference
Discrepancy Factor & Significance3
3
A discrepancy greater than 1.96δx has only a 5% chance of occurring randomly. This is unlikely and
considered a significant discrepancy. Given your data and assumptions, your result is unlikely to agree with
the reference. The lower the discrepancy the more acceptable the result. Results can also be inconclusive.
Page 4
DISCUSSION OF RESULTS
Looking at your results and how they compare to the reference materials you identified them as, discuss if you
believe your results are successful, inconclusive, or unsuccessful. Do you feel you accurately identified all three
rods? Was more information needed (like your secondary characteristics) to justify your identifications, or could
you fully support your conclusions based on the thermal expansion test alone?
DISCUSSION OF UNCERTAINTIES
Identify at least 3 likely sources of uncertainty that you believe affected your results in a non-trivial way. Be specific
in the source, what was affected, and how it was affected (+bias, -bias, or +-random, etc.). Discuss how significant
you think each source of uncertainty is (does one have a greater effect than others). % Relative Uncertainty is a
decent measure for comparing the overall relative effects of uncertainties. However, know that the values found
only take into account direct measurement uncertainties (they do not account for any incorrect assumptions or
systematic effects).
THOUGHTS FOR IMPROVEMENT
Think back on how you conducted the experiment and analysis. If repeated, would you perform it the same or
would you do something different? Try to come up with at least 1 practical, non-trivial improvement you would
make. Describe why you think this would improve the experiment and better meet its objectives.
Page 5
Thermal Expansion
PS253 – Physics Laboratory for Engineers
Embry-Riddle Department of Physical Sciences
All materials expand and contract as their temperatures vary, some more so than
others. In this lab you will investigate the material property of thermal expansion and
analyze it quantitatively using a simple linear model. You will be given a selection of
metal rods made of different materials. Based on your experimental data and a
literature search you must successfully identify three different metals from their
thermal expansion coefficients to within a 5.0% statistical difference of the literature
values.
Introduction
!WARNING! Do not turn on the hot plate until you are actually told it is time to use it.
Having it sit on while unused is a significant burn hazard.
We will measure the expansion of metal rods, as temperature increases from room temperature
to approximately the boiling point of water. The change in length is measured using a very
accurate machinist’s dial indicator, and the temperature will be measured using a thermistor or
possibly a thermocouple electronic sensor. The physics of the thermistor is quite different
from the thermocouple. Thermistors are made of semiconductors that have a relatively large
change in resistance over a small temperature variation. Thermocouples directly produce a
voltage that varies by fractions of a millivolt over small temperature ranges. As a lesson on the
functions and processes of electronic instrumentation used in research and industry, students
will also perform tests to determine the accuracy of the temperature sensor and the electronics
that process the sensor output, including effects of analog to digital conversion.
The process of thermal expansion, as you may well imagine, is one that we need to minimize
for designs that require very accurate position control. For example, the curve on a quality
lens or mirror surface has to be accurate within one ten thousandth of a millimeter for visible
light. Therefore, a small deviation due to thermal expansion can ruin an optical instrument.
Thermal properties must also be well determined when various materials are attached together
in panels, trusses, and other structures. This becomes incredibly complicated for say a satellite
or space telescope, which might change from a fixed attitude with respect to the Sun to a
varying, rotating one and likely has some components casting shadows on others.
One important point that is overlooked in a general physics text is that the thermal expansion
property is not a fixed constant over all temperature ranges. It varies by small amounts for
some materials over certain temperature ranges, and in professional design analyses one must
carefully research the materials that may be used and compute the effects of the change in the
thermal expansion “constant”.
Metals that show very small expansion coefficients include “Invar” and “Super-Invar.” They
are used to make chassis for devices that must maintain positions within a millionth of a
millimeter or so. However, even though the expansion coefficient may be tiny for a very
limited temperature range, it can grow very fast outside the design temperature range.
Thermal Expansion
PS253 – Physics Lab for Engineers
On the other hand, there are some interesting materials that contract upon heating. A
Titanium-Nickel alloy has been discovered that shows very large contraction when heated. It
is available under the product name “Muscle-Wire” and, as the name suggests, hair thin strands
of the alloy can be made to contract by simply sending an electrical current through the wire,
and the resistive heating provides a relatively easy means for controlling the contraction by
changing the applied current.
Linear Thermal Expansion Model
We assume that over a sufficiently small range of temperatures, that the thermal expansion,
L, follows a simple linear model with a fixed proportionality constant. It is directly
proportional to the temperature change and to the initial length, Lo. The constant of
proportionality is defined as . Its value depends on the composition of the object under
study. The formula for thermal expansion is then:
∆ = ( − )
(1)
In our experiment:
L is measured with a delicate machinist’s dial indicator with length precision
±0.005mm. Check the dial so you know how to read it properly.
Lo is measured with a meter stick which has ±0.5mm position precision.
Tf and Ti are measured using electronic instrumentation and the precision will be
determined by statistical analysis of “steady state” temperature measurements.
will be calculated from the measured quantities using Eq. 1. Notice the units:
length/[length degree]. Although the lengths cancel, you need to include them as a scale
factor, i.e. mm/[mm K]. will be determined experimentally for three of four different
metals, and compared against values that you will determine from literature research.
Analysis of Electronic Instrument Uncertainties
The following description pertains to the Pasco Capstone software, the sensors, and data
acquisition devices that are supported by this software; however, it can be generalized for
nearly any digital sensor or device. For more technical information on the Pasco thermistor
temperature probe you will be using, follow the link- CI-6605A.
We will consider only the uncertainties in the electronic measuring system that can be
determined by computing the standard deviation of many samples recorded under controlled
“constant” conditions. We will also review the data to find the minimum change in
temperature that can be measured with the electronic system- its resolution. Keep in mind
for this analysis we are ignoring any potential systematic errors, such as a poorly calibrated or
manufactured temperature probe. This could of course be checked for by running many
probes in parallel at the same location to see how well they agree on the temperature. However,
the statistical analysis of steady state conditions alone will only determine random uncertainties
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Thermal Expansion
PS253 – Physics Lab for Engineers
whether they be inherent to the sensor and its capabilities or to the steady temperature of the
room itself.
The minimum measurable temperature change, or resolution, is determined by the device that
converts the continuously varying sensor/amplifier signal to a discrete form for computer
processing. This is done by an Analog to Digital Converter (ADC). The resolution of an ADC
is determined by its internal circuit design, specifically:
Upper and lower limit of voltage values that it is designed to accept
Number of binary digits that it is designed to output.
To illustrate the basics of ADC operation, consider an ADC that is designed to measure
voltages from 0 to 2V [Volts], with an output of 4 binary digits, corresponding to a temperature
range of 10-20°C. The assignment of voltages could be:
Input Voltage [V]
Binary Value
Temperature Output [°C]
x≤0
0000
10.00000
0 < x ≤ 0.133
0001
10.66666
0.133 < x ≤ 0.267
0010
11.33333
0.267 < x ≤ 0.400
0011
12.00000
…
…
…
1.867 < x ≤ 2.000
1111
20.00000
Consider the column on the right. Note that the computer output shows seven digits but that
the minimum measurable temperature change is 2/3 of a °C. The device cannot distinguish
between temperatures within these 2/3 °C intervals. Typical ADC’s have 8 to 24 binary digit
outputs so the resolution is better than in this simple example, but the same limitations occur.
There is a tradeoff in any electronic sensor between maximizing the range of physical values
to be covered and maximizing the resolution within that range.
Determining Sensor Resolution and Temperature Precision
Set up Capstone to recognize your temperature probe:
1. Make sure that the temperature sensor is plugged into one of the Analogue Channel
ports on the Pasco Interface box.
2. Using the PC that is attached to the Pasco interface by USB, open the program Capstone.
3. When Capstone opens, in the left-side menu, click on the Hardware Setup button to open
a window where you can add your sensors.
4. Click on the virtual input where you plugged in the temperature sensor and select
Stainless Steel Temperature Sensor from the menu to add the sensor. Then exit the
Hardware Setup window.
5. In the central display area select the option Two Small, One Large Display. Click on each
display area that appears and set the small display panels to a Digits display and a Table
display; set the large display to a Graph (Display options are in the right-side menu).
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Thermal Expansion
PS253 – Physics Lab for Engineers
In the displays, wherever it indicates set the fields so that the
Digits shows temperature, the Table lists temperature and time, and the Graph plots
temperature over time.
6. Click Record to collect 1-2 seconds of data. If no one was holding the steel part of the
sensor and the temperature is vastly different from room temperature, 19-23°C, speak
to your instructor.
7. Select the Digits display and adjust the decimal precision to show temperature to two
decimal places. Select the Table display and adjust the decimal precision on the
temperature column to show four or five decimals for temperature.
8. You can adjust the size of each display by moving the cursor to the inner edges and
click/drag the display edges around. The smaller displays only need to be a column on
the left or right, most of the screen should be used by the graph.
9. In the bottom menu, adjust the sampling rate to 25 samples per second.
10. With the probe in equilibrium, in the ambient air, Record 5-9 seconds of temperature
values. Inspect the data set to determine whether it was nearly in equilibrium or whether
it was actually cooling or heating due to prior handling of the probe. Repeat for another
run until you obtain data at equilibrium.
11. Select the Table display and in its menu options find the Statistics button (Σ). Select it and
use the dropdown settings to only show the Standard Deviation.
12. Inspect the dataset to find the minimum temperature resolution of the ADC (remember
this will be a small step size change between two values that likely repeat often). The
Graph might be helpful in visually finding this smallest step size, but the Table will give
you the values to actually calculate it. This will give you two options for choosing an
estimated temperature uncertainty.
Record both values for your report, but use the larger of the two as the uncertainty estimate for all
temperature measurements taken in this experiment.
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Thermal Expansion
PS253 – Physics Lab for Engineers
Determining Thermal Expansion Coefficients of Unknown Metal Rods
13. Obtain your choice of three out of the four different types of metal rods available (not
multiple rods of the same material). Using a meter stick, measure the initial length of
your first trial rod to the nearest 0.5mm.
Note that when measuring length on a ruled scale you are actually measuring the
displacement between two position measurements. This is important for correct
propagation of uncertainties (required for this lab and your report).
14. Make other observations about your rods such as color, weight/mass, luster, magnetic
properties, etc. to assist in any difficult identification later; however, the primary means
of successfully identifying the metal must be the thermal expansion coefficient.
15. Fill the metal boiler tank no more than 1/3 full with hot water from the sink. Connect
as shown in Figure 1. Also, fill the overflow beaker about half full with cool water to
condense steam that passes through the system.
!WARNING! There are many hot objects that can cause contact or steam burns in this
experiment. Only the boiler should be on/near the hotplate, clear the area of other
objects. Steam is likely to escape from the temperature probe inlet so do not place your
face or ungloved hands near it. The plastic tubing might dislodge at some point, if it
does immediately remove the boiler from the hotplate and with gloved hands reattach
the tubing as quickly as possible. All metals in contact with the steam will be hot long
after the steam is gone.
Test Sample
Aluminum
Rod
Temperature
Probe
Dial Indicator,
L
Pull to
protect
indicator
Figure 1: Thermal Expansion Apparatus. The metal steam boiler on the upper right
supplies steam to the pipe running from the lower right to the upper left where the
steam then vents and condenses in a cool bath of water.
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Thermal Expansion
PS253 – Physics Lab for Engineers
To insert a metal sample rod:
16. Obtain a hollow insulated pipe; insert a metal sample rod into the pipe center.
17. Pull back the dial indicator shaft and hold it pulled back. Gently place the pipe in the
apparatus so that one end of the metal sample rod rests against the end stop screw.
18. Place the pipe flat, and slowly release the dial indicator shaft so it rests against the metal
rod.
19. Connect the inlet and outlet plastic tubing to the pipe. Insert the temperature probe
Recording data during a trial:
20. Set the sampling rate to 2 Hz and begin recording temperature data continuously. When
you are sure the temperature is steady at equilibrium, record the initial temperature Ti.
Continue recording temperature continuously for the rest of the trial run.
21. By twisting, adjust the outer machinist dial scale so zero is aligned with the pointer. The
large outer dial scale is in increments of 0.01mm while the smaller inner dial counts
revolutions of the outer dial [1mm].
22. Record the initial position and uncertainty of the dial indicator, both scales summed
together, for the room temperature rod.
!CAUTION! Make sure the cord for the temperature probe is not in contact with the hot
plate and does not touch any surface that will get hot from the steam. The cord will melt
and damage the wires, making the probe unusable.
23. Turn on the hotplate, place the boiler centered on the hotplate, and wait for the water
to start boiling. It will take 5-10 minutes for the steam to reach all the way to the end of
the pipe and fully heat the rod, watch the temperature readings.
24. Watch the temperature and dial indicator: after the temperature has spiked and held
steady and the dial reaches its maximum value and holds steady for 15 seconds or so,
record the final temperature and the final dial indicators, both dial scales summed
together.
25. Turn off the heater. Let it cool down a little before removing the sample rod.
To remove a metal sample rod:
26. With the hotplate off, put on gloves since the pipe will be hot, and remove the
temperature probe.
27. At the pipe, disconnect the plastic tube from the boiler to the pipe, and raise it above
the boiler to empty any trapped water back into the boiler.
28. Pull back the dial indicator shaft and hold it pulled back. Slowly tip the pipe upward at
the indicator end to drain any trapped hot water into the cool bath beaker.
!WARNING! The hot metal rod and some hot water will remain in the pipe.
29. At the pipe, remove the plastic tube connecting the pipe to the cool bath and carefully
walk the pipe over to the sink.
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Thermal Expansion
PS253 – Physics Lab for Engineers
30. At the sink remove the metal rod from the pipe and leave the rod in the sink. Run the
pipe under cold water at the sink flushing out the pipe through the tubing connectors
until the pipe is near room temperature again.
31. Repeat Steps 16-28 to test the other metal rod(s) until you have successfully tested three
different material rods.
32. When finished make sure the hotplate is turned off, place the metal boiler on the table,
and leave your hollow pipe in the sink to cool off.
Data Analysis
Complete the remainder of your work and analysis on the lab worksheet. The lab worksheet
mostly replicates the type of analysis, thought processes, and points of discussion that your
typical lab report or lab summary would have you do.
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