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DEPARTMENT OF PHYSICS AND ASTRONOMY APPALACHIAN STATE UNIVERSITY In-Lab Assignment: Thermal Expansion of Solids PHY 1104, Section 203 Date: 2/25/21 Name: Kristina Hicks ● Make sure that you read the Introduction to the lab and complete the Pre-lab Quiz on Asulearn before starting the lab activities. ● Note: You need to make a copy of this document and save to your own google drive using “ File>Make a Copy ” in order to have editing permission. Do not email asking for permission to edit this document. Use the question forum if you have questions on the Quiz or the lab. Someone else may have the answer or the same question! ● After completing the lab, make sure that you download your document as a pdf and contribute to the Post-lab Discussion forum ! This lab will be conducted on-line and each student will submit an individual lab report . You are allowed to work on the lab content with another student, but you MUST SUBMIT YOUR OWN UNIQUE images, data, calculations, graphs, explanations, conclusions, etc. Copying items on Lab Assignments constitutes an academic integrity code violation and will be brought before the Office of Student Conduct. I worked on this lab with: All images, data, calculations, graphs, explanations, conclusions in this lab assignment are my own and are not copied from another student or a previous semester. Type your name below to accept the honor statement. Kristina Hicks Purpose Investigate how the dimensions of solid materials change with temperature by taking measurements and plotting results. By the end of this lab students will: ● Use a spec sheet to determine the precision of a digital thermometer in order to determine the uncertainty in temperature that may be less than what the digital display indicates. ● Read a change in length from a micrometer dial. ● Plot change in length as temperature changes to determine the coefficient of thermal expansion of a copper tube. Necessary Equipment Vernier Graphical Analysis App : There is a help document on Graphical Analysis if you need help downloaded and using it. PHY 1104: Thermal Expansion of Solids, Page | 1

DEPARTMENT OF PHYSICS AND ASTRONOMY APPALACHIAN STATE UNIVERSITY Introduction: Thermal Expansion of Solids The materials that we touch and come into contact with every day are made of atoms and molecules that are bonded together and separated by very small distances. These bonds act like springs such that they keep materials together and that they can stretch. In general, when these materials are cold, the bonds between atoms and molecules have the lowest energy level, IE they contain the smallest amount of every. When materials are warmed, the atoms and molecules are given energy and this has the effect of stretching these bonds or increasing the distance between the particles. We can measure this thermally induced expansion, called thermal expansion, and define it to be the tendency of matter to change in size in response to a change in temperature. We know from the study of thermodynamics that an object's temperature using the Kelvin [K] scale is proportional to the object's internal kinetic energy. When energy in the form of heat, which is measured in Joules [J], is transferred to an object, its constituent particles maintain a greater average separation. Since all the distances between particles increases, the material’s volume will expand. While not impossible, materials which contract with increasing temperature are uncommon. Water, over a very narrow range of temperatures, is such a material. Linear Thermal Expansion Consider an object that is being thermally excited. We will look at just one dimension of the material: for example, if we have a long metal rod we will look at its length. The specifics of exactly how the material changes shape with temperature depends on many material properties and many physical laws, such as those of quantum mechanics. Luckily, many materials behave in a very reproducible and linear way over a rather wide range of temperatures around room temperature. Specifically, many materials will expand linearly with the change in temperature. Thus, if we double the temperature change, we double the change in length. This relationship is called linear thermal expansion . Since materials differ greatly in terms of chemical and physical construct, each material has its own coefficient of linear thermal expansion coefficient α . This coefficient indicates the fractional change of its original length per change in temperature °C. The coefficient is found by dividing the percent change in length by the change in temperature , Thus the change in Δ𝑇 = (𝑇 ? − 𝑇 𝑖 ) length ΔL of an object will be equal to the linear thermal expansion coefficient α times original length L o times change in temperature ΔT: (1) ∆𝐿 = 𝐿 ? α ∆𝑇 ( ) where ΔT is the change in temperature or T f – T i . We can rewrite equation 1 as: (2) ∆𝐿 = 𝐿 ? α(𝑇 ? − 𝑇 𝑖 ) This week, you will set up an experiment to measure how the length of an object changes with temperature. If this relationship is linear, the linear thermal expansion coefficient α can be determined and compared to an accepted value, such as that given by table 13-1 in your textbook. PHY 1104: Thermal Expansion of Solids, Page | 2

DEPARTMENT OF PHYSICS AND ASTRONOMY APPALACHIAN STATE UNIVERSITY The Micrometer Metric Dial Examine the micrometer metric dial in figure 2, a similar metric dial is attached to the apparatus we will use in this experiment to measure the change in distance of two different types of metal rods as they expand and contract due to temperature changes. The micrometer metric dial is used to measure very small changes in distance. The least count, or distance between the smallest tick marks, of the big needle is 0.01 mm. There are 100 increments in a full circle of the needle. The small needle counts the number of times the large needle has made a complete circle; we will not use this needle since the dial will not make a full rotation in today’s experiment. There is a set screw on the side of the dial that will allow one to zero the markers to the needle, then retighten. You can estimate the needle position to the nearest half of a division, which will give you an estimated error of 0.005 mm. For example, consider the micrometer dial in the picture. This micrometer is reading 0.628 mm +/- 0.005 mm = 0.000628 m +/- 0.0000005 m If you are uncertain about how to use this device, there are many articles and videos on the web, use the search term “How to read a metric dial indicator”. We will not be using the smaller dial at all (this measures mm), just the large outer dial which measures hundredths of a mm and has a least count of 0.001 mm. The Digital Thermometer To measure temperature, we will use an Extech EasyView digital thermometer (or a similar device) placed into the end of the metal sample to measure temperature. The digital thermometer is an example of digital reading where the device is less accurate than the display implies. PHY 1104: Thermal Expansion of Solids, Page | 4

DEPARTMENT OF PHYSICS AND ASTRONOMY APPALACHIAN STATE UNIVERSITY An excerpt of the user manual is shown in figure 3. Examine the section that describes the accuracy of the digital thermometer. This accuracy describes the confidence or uncertainty of the reading in a particular temperature range. Notice that the accuracy of the measurement is a percentage of the temperature itself plus an additional amount of error. The steps you must take to determine the measurement uncertainty are a little bit different than for other digital devices that are more accurate than the display shows. For the thermometer the accuracy depends on what temperature you are measuring, and is reported as a percentage of the measurement with an additional uncertainty. Steps to take to find uncertainty in a digital thermometer Determine the uncertainty in your measurement (with units): 1. Consider the temperature range that you will be using to find the correct uncertainty formula in the manual. 2. Take your measurement. 3. Calculate the uncertainty for that measurement using the correct formula for the range you are in. Express your measurement with uncertainty in the correct format: 4. Round your uncertainty to one DIGIT. Keep in mind that you will be rounding the value up, except in extreme cases. 5. Determine the place (tens, ones, tenths, hundreds) where the uncertainty digit is. Round your measurement to that place, making sure that both the measurement and uncertainty have the same units. You are rounding your measurement to the least significant digit, since any digit in a place less than the least significant one is INSIGNIFICANT. 6. Report your measurement and uncertainty with the same number of decimal places and units. PHY 1104: Thermal Expansion of Solids, Page | 5

DEPARTMENT OF PHYSICS AND ASTRONOMY APPALACHIAN STATE UNIVERSITY Lab Activities Activity One – Finding the linear thermal expansion coefficient for a metal rod through direct calculation We will heat up a metal rod and measure the expansion due to an increase in thermal energy (as measured by the temperature) in the rod. The rod will expand as the temperature goes up, and contract in the same manner as the temperature cools down following the linear equation (equation 2 in the Introduction). ∆𝐿 = 𝐿 ? α(𝑇 ? − 𝑇 𝑖 ) The amount that the rod will expand and contract is directly proportional to the linear expansion coefficient, α . Once you have determined the linear expansion coefficient for a particular metal, you can then find the expansion of that metal between any two temperature extremes. This activity will focus on direct calculation of the linear expansion coefficient with two data points acquired at room temperature and after the tube is heated by steam. From these two data points we can directly solve for α . Experiment Steps and Video Using the thermal expansion apparatus, we heated up a metal rod by filling the rod with steam. We then used a precise measurement tool called a micrometer metric dial to measure the expansion of the metal rod after the steam and the rod achieved thermal equilibrium. Here are the steps we followed in the video below: 1. We measured the initial length of the rod from clip to clip. Although the entire metal pipe is elongating, we only measure the expansion that takes place between the two ring clips, as shown in the figure above. One clip is anchored in the mounting bracket while the plunger of the metric digital indicator rests on the other clip to read any change in length. Thus, when determining L 0 , we are not concerned with the total length of the rod, just the distance between the clips. We measured L 0 for the metal rod to be L 0 = 0.348 m ± 0.005 m . 2. We placed the metal rod in the apparatus between the micrometer dial and the holder, making sure that the rod was held from clip to clip in the apparatus and the micrometer was resting against one of the clips. 3. We zeroed the micrometer metric dial turning the face of the dial to zero the micrometer. 4. We turned on the steam generator while the rubber tube remained disconnected in order to heat up the water, in preparation for the experiment. PHY 1104: Thermal Expansion of Solids, Page | 7

DEPARTMENT OF PHYSICS AND ASTRONOMY APPALACHIAN STATE UNIVERSITY 5. We inserted the thermometer at the end of the rod. The rod is hollow, so the steam can travel down the length. By inserting the thermometer at the end, we are actually measuring the mixture of air and steam, but after a short time the rod will be at equilibrium with the air and steam and we can assume they are roughly the same temperature. In Question 1, you will record the initial temperature of the rod before it is connected to the steam using the information provided in the video. 6. Connect the metal rod to the steam and wait for the rod and steam to reach thermal equilibrium. 7. Disconnect the metal rod from the steam and observe the contraction of the rod as it cools down again. Watch the video ( https://youtu.be/KAB4-18FIr4 ) of the experiment before you continue. You will use a video of the experiment to get your measurements. The video includes: 1. An overview of the equipment 2. The micrometer starts at zero, with the initial temperature shown right before connecting to the steam. 3. The micrometer and thermometer at the max temperature/expansion distance. The cool down portion of the experiment will be in a separate video. It is recommended to watch these videos with 720 p resolution. Question 1: We will be using the Extech Digital Thermometer to record our temperature measurements. The digital thermometer is an example of digital reading where the device is less accurate than the display implies. You will need to use the user manual for the thermometer to determine the uncertainty in the digital display. (Help doc: Measurement and Error ) PHY 1104: Thermal Expansion of Solids, Page | 8

DEPARTMENT OF PHYSICS AND ASTRONOMY APPALACHIAN STATE UNIVERSITY Question 3: Method 1, direct calculation of α Method 1 will use equation 2: ∆𝐿 = 𝐿 ? α(𝑇 ? − 𝑇 𝑖 ) and the data obtained during the heat up portion of the experiment and directly solve for α . α = ∆𝐿 𝐿 0 𝑇 ? −𝑇 𝑖 ( ) a) In the video, watch the micrometer dial and temperature as the rod expands with temperature, when it reaches a stable maximum value record the final temperature, T f in °C with an appropriate uncertainty in your measurement. Also, determine the ∆L from the micrometer metric dial , and record it with an appropriate uncertainty in your measurement. The micrometer reads in mm, so make sure that you convert your micrometer reading to meters. T f (with uncertainty) = 99.5 ℃ ℃ ± 0. 1 Δ L (with uncertainty) = 0.378 x 10 0.005 m −3 ?± b) Using the data obtained during the heat up portion of the experiment, where T i is the initial (room) temperature of the rod, and T f is the final maximum temperature of the rod and L 0 is the initial length (measured to be L 0 = 0.348 m ± 0.005 m ). Calculate the thermal expansion coefficient α for the metal. You do not need to propagate your uncertainty in the calculation. This value will be called α 1 , and has units of 1/ ℃ . Show work: α = ∆𝐿 𝐿 0 𝑇 ? −𝑇 𝑖 ( ) = 14.6 x 10 1/ ℃ α 1 = 0.378 ? 10 −3 ? 0.348 ? (99.5 ℃ − 25.1 ℃) −6 = 14.6 x 10 1/ ℃ −6 α 1 = 14.6 x 10 1/ ℃ −6 c) Compare your experimental value of α 1 with the known value (the metal should be copper with 𝛼 =17 x 10 -6 1/ ℃ ) using percent error ( ). If you have more than 20% error, do not % ????? = ?????−?????𝑖?????? ????? ????? | | | | ×100% continue. Look over your calculations and make sure you recorded the data from the video correctly. Show work: % ????? = ?????−?????𝑖?????? ????? ????? | | | | ×100% x 100% = 14.118 % | (17 ? 10 −6 1/℃) − (14.6 ? 10 −6 1/℃) 17 ? 10 −6 1/℃ | % error = 14% PHY 1104: Thermal Expansion of Solids, Page | 10

DEPARTMENT OF PHYSICS AND ASTRONOMY APPALACHIAN STATE UNIVERSITY d) Was your value close to the expected value? If your value originally had a high percent error it is likely that there is a problem with your data or units. If this is the case, discuss what you did to correct the problem. The value was about 14% error from the expected value, which could have been due to at first improperly calculating α 1. Activity two – Graphically determine the coefficient of linear expansion The second method we will use to find the coefficient of linear expansion will use multiple data sets obtained during the cool down portion of the experiment and plot ΔL versus ΔT and perform a linear curve fit to the graph. Acquiring multiple data points and performing a curve fit on them should yield much more accurate results than just using two data points as in the previous activity. We see that the rod will expand as the temperature goes up, and contract in the same manner as the temperature cools down following the linear equation (equation 1 in the Introduction): ∆𝐿 = 𝐿 ? α ∆𝑇 ( ) Video of cooling down (https://youtu.be/-6MGRtv2g0M ) shows: 1. Max temperature 2. Disconnection from the steam 3. The metal as it cools down, although not all the way back to room temperature It is recommended to watch these videos with 720 p resolution. Question 4: Watch the video of the cool down period (you don’t have to watch the whole thing at this time, just enough to answer the question.) During this cool down period, PHY 1104: Thermal Expansion of Solids, Page | 11

DEPARTMENT OF PHYSICS AND ASTRONOMY APPALACHIAN STATE UNIVERSITY y = m x + b ∆𝐿 = L ? (α) ∆𝑇 + 0 Question 6: Write out a detailed, step-by-step description of how you would take data using the video provided and use your independent and dependent variables defined above to graphically measure the linear expansion coefficient, α , for a metal rod. Include information about a) how you are going to collect data, b) how many data points you will take, c) what you expect the graph to look like and d) how to determine α from the slope after you have performed the linear curve fit. (Hint: You should take data while the rod is cooling down, not heating up.) The purpose of this question is to describe how you would use the scientific method and the materials shown in the video to find the coefficient of thermal expansion experimentally if you were in-person. This does not need to exactly match the actual method employed in the rest of the video. The data will be collected by watching a video and pausing every few seconds to record multiple points, which will later be plotted onto a graph. The independent variable, , ∆𝑇 will be plotted on the x-axis and this will be calculated by The dependent variable, 𝑇 ? − 𝑇 𝑖 . , will be plotted on the y-axis and will be calculated by . Eight or more data ∆𝐿 ∆𝐿 = 𝐿 ? α ∆𝑇 ( ) points will be collected and I expect the graph to be a significant downward slope since we are taking data from when the rod will be cooling down. The thermal expansion coefficient, ,will be calculated by equating it to α ? 𝐿 ? . Question 7: Watch the video of the experiment again (https://youtu.be/-6MGRtv2g0M ) to acquire data. You should collect 8 or more data points as the metal rod is cooling. You should be plotting ∆T on the x -axis and ∆L on the y -axis. For convenience, use the absolute value of ∆T so your graph is linearly increasing, as the slope will be the same. Your ∆T and ∆ L should be starting at high values and going down as you get closer to the initial room temperature. a) Record your data in the table below, adjust your uncertainties if necessary. You will calculate the change in temperature from the original initial temperature, ( T i =26 ° C) that you recorded in Activity 1. You do not need to have uncertainties in this table. (Hint: the values close to the max temp are hard to read, start collecting data around 80°C for increased accuracy.) Data Point T f ∆T=|T f - T i | ∆L 1 80.4 ℃ 54.4 ℃ 3.8 x 10 −4 ? PHY 1104: Thermal Expansion of Solids, Page | 13

DEPARTMENT OF PHYSICS AND ASTRONOMY APPALACHIAN STATE UNIVERSITY 2 62.5 ℃ 36.5 ℃ 3.0 x10 −4 ? 3 53.7 ℃ 27.7 ℃ 2.5 x 10 −4 ? 4 46.1 ℃ 20.1 ℃ 2.0 x 10 −4 ? 5 41.7 ℃ 15.7 ℃ 1.8 x 10 −4 ? 6 36.6 ℃ 10.6 ℃ 1.5 x 10 −4 ? 7 32.9 ℃ 6.9 ℃ 1.3 x 10 −4 ? 8 31.7 ℃ 5.7 ℃ 1.2 x 10 −4 ? b) Plot your values from your table in Graphical Analysis. Make sure that you add labels to your columns, and a title to your graph. Here is a refresher on how to enter and plot data in Graphical Analysis . Perform a linear curve fit to your data to determine the slope of your data. Your slope should be in units of m/ ℃ . slope, m = 5.41 x 10 m/ ℃ −6 c) Your slope should be in the range of 4.00 x10 -6 m/ ℃ and 7.00x10 -6 m/ ℃ . i) Yes, my slope is in the correct range. (You may continue the lab.) ii) No, my slope is not in this range. (DO NOT CONTINUE. Check your equation and graph or ask your instructor for help.) Question 8: You are using the slope of your linear graph to determine the linear thermal expansion coefficient α from this plot. We will call this α 2 , and has units of 1/ ℃ . a) Show all work for your calculation of α 2 from your slope. The slope will be , ? = 𝐿 0 α 2 where L 0 is the initial length as measured in the first activity ( L 0 = 0.348 m ± 0.005 m ). Show work: ? = 𝐿 0 α 2 5.41 x 10 m/ =(0.348 m)( −6 ℃ α 2 ) Divide m by 0.348 to get α 2 1.55 x 10 1/ ℃ −5 α 2 = 1.55 x 10 1/ ℃ or 15.5 x 10 1/ ℃ −5 −6 PHY 1104: Thermal Expansion of Solids, Page | 14

DEPARTMENT OF PHYSICS AND ASTRONOMY APPALACHIAN STATE UNIVERSITY Question 10: Compare your experimental results α 1 and α 2 . Use percent difference ( ), which is the difference between two experimental % ?𝑖???????? = ?𝑖???????? ??????? | | | | ×100% values found using different methods. If you have more than 10% error, look over your calculations. α 1 (in 1/ ℃ )= 14.6 x 10 1/ ℃ −6 α 2 (in 1/ ℃ )= 15.5 x 10 1/ ℃ −6 % difference = % ?𝑖???????? = ?𝑖???????? ??????? | | | | ×100% x 100% = (15.5 ? 10 −6 )−(14.6 ? 10 −6 ) 0.00001505 5.98% = 6% difference Experimental Error and Conclusion PHY 1104: Thermal Expansion of Solids, Page | 16

DEPARTMENT OF PHYSICS AND ASTRONOMY APPALACHIAN STATE UNIVERSITY Question 11: Discuss at least two main sources of error that affected both the heating up and cooling down process and how they impacted the experiment. Would the source of error make your coefficient of linear expansion larger than you expected or smaller than you expected? How does this compare to your actual results, were the coefficients of linear expansion larger than the known values or smaller? Do NOT use the phrase ‘human error’, be more specific in order to identify how the error would impact the experiment! (Note: Miscalculations, numerical rounding, or misreading instructions are NOT errors. Focus on errors with the process of collecting and analyzing the data.) Two main sources of error that affected both the heating up and cooling down process include acquiring the 8 data points in the cooling down video. Due to stopping and starting the video, an error could have occurred in achieving data but not stopping the video every certain number of seconds. In addition, the length, converting to meters may have been improperly calculated, which could have thrown off my percent error. All of the percent errors were less than 20%, which is what was aimed for. THe coefficients of linear expansion were smaller than the accepted values. Question 12: Summarize your experiment and interpret the major results. It should include: 1) a short description of what you did and WHY it was done (what was the key physics explored); 2) your MAJOR results with values and units (NOT all values measured or calculated); 3) the accuracy of your measured values as shown by your % error or % difference values; 4) identify the improvements that would have the most impact on your accuracy and/or precision and how would you implement them in a practical manner (you do not need to reiterate what the errors were, just propose a solution for them); 5) were the goals/purposes of the lab achieved? The conclusion should be concise (not wordy) and written in paragraph form. In this experiment, thermal expansion was found using the equation: . A metal ∆𝐿 = 𝐿 ? α(𝑇 ? − 𝑇 𝑖 ) rod was watched as it heated up and then cooled down and compared to this equation. The initial temperature of the rod was 25.1 ℃ with an accuracy setting of . In the expanding video of watching the rod, the ± 0. 3% ????𝑖?? + 1℃ ( )@ 0℃ ?? 1000℃ temperature reached 99.5 ℃ with a change of length of 0.378 x 10 . ± 0. 1 ℃ −3 ?± 0. 005 ? The thermal expansion coefficient, , was determined to be 14.6 x 10 1/ ℃ , using the equation α −6 . This thermal expansion coefficient, also known as was compared to copper α = ∆𝐿 𝐿 0 𝑇 ? −𝑇 𝑖 ( ) α 1 with 𝛼 =17 x 10 -6 1/ ℃ and gave me a percent error of 14%. This error may have been due to the improper calculation of the thermal expansion coefficient. Then, a video of the rod cooling down was watched and 8 points were recorded to compare it to the linear equation: y= mx +b and compared to +0 to the linear equation. The was used in the x-axis and was ∆𝐿 =𝐿 ? (α)∆𝑇 ∆𝑇 ∆𝐿 used in the y-axis and the points from the video were imputed in the vernier graphing analysis. A slope was formed to be 5.41 x 10 m/ ℃ and was used to calculated using the equation −6 α 2 , which was found to be 1.55 x 10 1/ ℃ or 15.5 x 10 1/ ℃ . This was compared to ? = 𝐿 0 α 2 −5 −6 PHY 1104: Thermal Expansion of Solids, Page | 17