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THERMAL EXPANSION WORKSHEET PS 253 – Physics Lab for Engineers / 100pts Name: Nicholas Donatelli 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 𝛼 = ∆𝐿 𝐿𝑜 (𝑇𝑓 −𝑇𝑖 ) −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. Repeating Temps: 24.140918 – 24.129958 Min step: 0.01096 Std Dev 0.012307 USEFUL MATHEMATICAL EQUATIONS Percent Relative Uncertainty, %δx %𝜹𝒙 = 𝜹𝒙 ∙ 𝟏𝟎𝟎% 𝒙 Percent Difference, %Diff %𝑫𝒊𝒇𝒇 = |𝒙 − 𝒙𝒓𝒆𝒇 | ∙ 𝟏𝟎𝟎% 𝒙𝒓𝒆𝒇 Discrepancy Factor, t 𝑡= |𝒙 − 𝒙𝒓𝒆𝒇 | 𝛿𝑥 Page 1 UNKNOWN METAL ROD #1 Variable, x Value [units] Rod Length, Lo 599.5 mm ± 0.5 mm Initial Temperature, Ti 23.82 deg ± 0.05 deg Final Temperature, Tf 102.06 deg ± 0.05 deg Change in Length, ΔL 0.52 mm ± 0.01 mm Change in Temp, ΔT Uncertainty, δx % 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. Magnetic, maroon/rust color 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] Rod Length, Lo 599.5 mm ± Initial Temperature, Ti 23.82 ± Final Temperature, Tf 102.03 ± Change in Length, ΔL 0.93 mm ± Change in Temp, ΔT Uncertainty, δx % 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. Non-magnetic, brass/bronze color, 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] Rod Length, Lo 600.5mm ± Initial Temperature, Ti 23.82 deg ± Final Temperature, Tf 102.1 ± Change in Length, ΔL 0.49 ± Change in Temp, ΔT Uncertainty, δx % 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. 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: ∆𝐿 = 𝛼𝐿𝑜 (𝑇𝑓 − 𝑇𝑖 ) (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 – 2 of 8- 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] 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 … … 1111 20.00000 x≤0 … 1.867 < x ≤ 2.000 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). – 3 of 8- 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. – 4 of 8- 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. – 5 of 8- 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 indicat…

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