Lab Manual 101L S24 (4)

1 PHYSICS 101L LABORATORY MANUAL K.W. Gentle J.A. Yeazell Department of Physics University of Texas at Austin January 2024 Copyright 2023, The University of Texas at Austin Note : The laboratory sessions do not begin immediately, you should consult the syllabus on Canvas for the schedule. You must prepare for each Lab. For Lab 0, you must read the Lab before joining your session of the class. You should always arrive prepared to begin work without further instruction. The course has no lectures; reading is required. If you do not arrive prepared, you will be unable to complete the Lab in the time available.

2 Getting Started The focus of this course is on the measurement and modeling of the motion of objects (Mechanics). You will design experiments and develop computational models that explore the key concepts in this area, e.g. conservation of momentum. The laboratory activities will start with a warmup exercise and then you will extend this to investigate a more complex phenomenon. This course has a few goals: 1) Foster a physics/science community. Find fellow physics/science majors to work with. Science and physics is often a product of teamwork. The development of colleagues and hopefully friends can make it easier, more interesting (and more fun) to learn and do physics. 2) Introduce you to attacking physics problems experimentally and numerically. It asks you to design experiments and introduces you to the computer modeling of physics problems. 3) Help you to communicate scientific results in a written format and introduce you to oral presentations of scientific topics. 4) Introduce you to finding scientific information. How to search databases for peer-reviewed articles on a topic. You are expected to read through the scheduled activity before the start of the lab session so that you are familiar with the topics and start thinking about the design of an experiment or of a computational model. With this preparation, you should be able to complete all the work for this course within the lab session (including any required scientific writing describing your experiment and results). The laboratory has Apple computers that will be used for data acquisition and can be used for computer modeling (you are welcome to use your own laptop for modeling). These Apple computers can be accessed in the usual way with your UTEID and associated password. Please save any data files you generate to UTBox or your favorite Cloud system (not to the local drives on the computers in the lab).

4 Lab 0. Measurement, Uncertainties, Accuracy, and Using Excel This activity addresses a range of topics essential to experimental science. It covers: • What is a measurement? • What is the precision of a number or measurement? • What is the uncertainty of a measurement? • Error propagation: If you need to make more than one measurement to determine a more complicated quantity, how do you combine the uncertainties of each measurement to find the uncertainty of this complicated quantity? • What is accuracy and how can you judge experimental results? • How to use Excel to analyze data and to display experimental results graphically (including error bars). Accuracy and Uncertainty in Experimental Measurements Measurement is a central aspect of all science. Our ability to model the phenomena we observe is tied to the accuracy and confidence we have in those measurements. Here we describe how to determine the limits of experimental measurements. This topic is commonly known as “error analysis”. It is important to note that real errors or mistakes in a measurement are not included in this analysis. Real errors (also known as systematic errors) are things like using the measurement device improperly, forgetting to zero the device, the device is out of calibration, etc. If known these real errors should be eliminated, and the measurement repeated. The errors studied here typically stem from the basic accuracy of the measurement device and the random fluctuations in measurement due to noise. Precision and accuracy have two different meanings in error analysis. Precision is tied to number of digits to which a number is known or the significant digits in that number. Accuracy is how well a measurement compares with an accepted value. For example, you use a digital scale to measure a 1 kg standard mass. Your result is 1.0000252 kg. The difference from 1 kg is the accuracy (0.0000252). Repeated measurements reveal that the last digit fluctuates and this fluctuation leads to the precision of the measurement. The measurement has 3 significant digits and this is the precision. Precision is best defined by examples: for a measurement in decimal notation, the precision is the number of significant digits, e.g. 7.235 has 4 significant digits; 𝜋 or √2 has infinite precision, while the approximation 3.14 has only 3 significant digits; an integer 1, 2, … has infinite precision, but 2.0 only has 2 significant digits; in scientific notation the number 3.287 × 10 −5 has 4 significant digits. Note zeros to the right of the decimal point may act only as place holders and have no significance, e.g. 0.03 has only one significant digit while 0.0308 has 3 significant digits. The number of significant digits of a quantity that results from a multiplication or division is determined by the number with the least significant digits, e.g. ( 5.00 7.145 ) ( √3 2 )

5 The precisions of √3 and the integer 2 are infinite so the result of the calculation is determined the 3 significant digits in 5.00 . While doing the calculations you should keep more digits than are significant in the intermediate steps and only round to the correct number of significant digits at the end. The correct way to report the result of the above calculation is 0.606 . There are different rules for the significant figures resulting from addition and subtraction. The number with the least number of decimal places to the right of the decimal is the determining factor. If one has 5 places to the right of the decimal point while the other only has 3, then the sum or the difference is rounded to the 3 rd decimal place. For example, correctly rounded the following sum of 1.51475 + 0.322 = 1.836. Precision is closely tied to topic of uncertainty. The uncertainty in a measured value describes how well you know that value. Let’s say you me asure the length of a table with a ruler or meter stick and you find it be 2.154 m. Why did you only report 4 significant digits? On most rulers or meter sticks, the smallest division corresponds to a millimeter. The uncertainty in this measurement is typically defined as ±0.5 of this smallest division. All experimental measurements are reported as two numbers: the first number is the best estimate of the quantity being measured (usually the average of several measurements) and the second gives the bounds of the uncertainty. Exercise 1 : You and your partner should measure the length of a table twice with a meter stick and calculate the average. Then report your measurement below, ________________ ± ________________ The reported length should have a form like 2.154m ±0.0005m. Note that the uncertainty has the same units as the best estimate. This is known as absolute uncertainty. The uncertainty can also be given in terms of the ratio of uncertainty to the best estimate. This is known as the relative uncertainty. In the above example, the measurement of the table could also be written, 2.154m ± [0.0005/2.154] = 2.154m ± [0.0002] or 2.154m ± 0.02% The square brackets indicate it is a relative uncertainty.

7 1) Sums and Differences : If the desired quantity, ? , is the found by summing two (or more) measurements ? and ? (e.g. total length = sum of the two different lengths) then the absolute uncertainty ∆? in terms of the absolute uncertainties ∆? and ∆? is given by, (∆?) 2 = (∆?) 2 + (∆?) 2 + ⋯ (Note: the quantities MUST have the same units.) The uncertainty for quantities derived from differences also add using the exact same rule. Examples: (6.1cm ± 0.2cm) + (4.2cm ± 0.2cm) = 10.3cm ± √(0.2) 2 + (0.2) 2 cm = 10.3cm ± 0.3 cm (6.1cm ± 0.2cm) - (4.2cm ± 0.2cm) = 1.9cm ± √(0.2) 2 + (0.2) 2 cm = 1.9cm ± 0.3 cm The subtraction of quantities leads to an increased relative uncertainty in the result. In the second example, the relative uncertainty of the result is larger than either of quantities being subtracted. In the extreme case, where two nearly identical quantities are being subtracted, essentially all certainty can be lost (the relative uncertainty nears infinity)! Exercise 2 : Think back to exercise 1. Did you have to add together several measurements with the meter stick to get the length of the table? How would you report the measurement now? ________________ ± ________________ 2) Products and Quotients : If the desired quantity, ? , is the found by a product of two (or more) measurements ? and ? (e.g. area = product of length and width) then the relative uncertainty of ? is ∆?/?, where ? is the best estimate and ∆? is the absolute uncertainty and is given by, (∆?/?) 2 = (∆?/?) 2 + (∆?/?) 2 + ⋯ The exact same formula applies for quotients of measurements. Note ? and ? are best estimates and their absolute uncertainties are ∆? and ∆? , so the ratios are relative uncertainties. Example: (8.0N ± 1.0 N) / (2.0 m ± 0.2 m) = = 4.0 N/m ± [ √(1.0/8.0) 2 + (0.2/2.0) 2 ] = 4.0 N/m ± [0.16] 3) Quantities raised to Powers (not exponentials): If the desired quantity ? is given by raising the measurement ? to the power ? gives, ? = ? ? , then the relative uncertainty of the desired quantity is given by, (∆?/?) 2 = (|?|∆?/?) 2 Examples: (5.0cm ± 0.3cm ) 3 = (5.0cm ) 3 ± [ √3 2 (0.3/5.0) 2 ] =125 cm 3 ± [0.18] = 125 cm 3 ± 22.5cm 3 (9.0m 2 /s 2 ± 1.0m 2 /s 2 ) 1/2 = 3.0m/s ± [ √(0.5) 2 (1.0/9.0) 2 ] = 3.0m/s ± [0.055] = 3.0m/s ± 0.17m/s Exact numbers, like 𝝅 or numbers not from measurements, have no uncertainty.

8 Exercise 3 : You and your partner should each measure the length and diameter of a cylinder. From these measurements, calculate the volume and the uncertainty in the volume using error propagation. Make sure you use the rules on the previous page. Show your work and report your measurement of the volume, ________________ ± ________________ Checkpoint: Raise your hand and show the results of these three exercises to your TA. Make sure you can explain the steps of your calculation and justify any assumptions. If you have to wait for the TA, you may read on. How do you find the uncertainty for a given measurement? Often the measurement device has a given resolution or uncertainty. For example, an analytical balance may have an uncertainty of 1 𝜇g . There may be other times that the uncertainty of a measurement may not depend solely on the uncertainty of the instrument. Consider using a stopwatch to time an event. The time resolution of the stopwatch is typically at least 0.01 seconds. Unfortunately, a person’s reaction time is less certain. One way to estimate the uncertainty in the timing of the event is to repeatedly time the event and use either the standard deviation or the maximum deviation for the uncertainty. Before discussing accuracy, you will review some of the features of Excel which will simplify your calculations and statistical tests of your results. You will also use Excel to calculate standard deviations for sets of measurements. These can be used to plot error bars on a graph. Excel Summary (based on Office 365 education version) Open a new workbook in Excel and follow along to learn some of the basic ways we will be using Excel in this course. Entering an equation For example, say you wish to multiply the entries in B1 and C1 and put the product in D1 (see below). You can select B1 and type in a 2 and similarly put a 3 in C1. You can then select the D1 cell and type = B1*C1 and hitting Enter would give the result of 6. You could also select D1 and type = then click on the cell B1, type * , then click on the cell C1 and hit return to accomplish the same.

10 calculate the new result. For example, in the above screenshots, the cell A3 contains the formula =A2+1 and the cell B6 contains the formula =A6^2 . Absolute referencing Suppose you have a certain value (say a constant like ? ) that will be used in many calculations, and you want to type it in once, and then reference it after that. Putting dollar signs in front of the row and the column of the cell with this constant tells Excel to not use relative referencing. To use an absolute reference to the cell C1 in a formula, instead of typing C1 you type $C$1. In the screen shot below, a constant value of 9.8 was typed into C1 and used in the formula in column B which now represents the function ?(?) = 9.8? 2 . Standard deviation Standard deviation describes of how far a set of measurements varies from the mean or average of the measurement. In cells D1 thru D5 type in 5 random numbers that are close to 10, e.g. 9.875 (you might imagine these are 5 measurements of something). In D6, type =average(D1:D5) and hit enter to get the average of these measurements . In D6, type =stdev(D1:D5) and hit enter to get the standard deviation . Change some of the numbers in D1 thru D5 and see how the average and standard deviation changes. It is desirable that an experiment yield a standard deviation that is similar in size to the estimated of the uncertainty in an experiment found via the error propagated from the known uncertainties of the measurement devices. If the standard deviation is much larger than the estimated uncertainty it is likely there is unknown systematic error, i.e. a mistake. If the standard deviation is much smaller than the estimated uncertainty then you should be suspicious of the results. If these numbers represented measurements of a 10kg mass on a scale with a known uncertainty of 0.001 kg, what would you think about this set of measurements (Was it a good experiment? Should you redo it? Should you be suspicious of the results?)

11 Accuracy and Judging Scientific Results A peer-reviewed scientific article has been reviewed by other scientists who do research in the same area. They use their experience and knowledge of the topic to judge its suitability to be published. They also expect to see statistical arguments about the experimental results in the paper. One common test of the validity of the results is known as a chi-squared ( 𝜒 2 ) test. It is used to determine how the observed experimental results differ from the expected results. The assumption that the expected results are true is known as the “null hypothesis”. In equation form, it calculates the square of the difference between a particular observed result ( ? 𝑖 ) and the expected result ( ? 𝑖 ) divided by the expected result. It sums this over all the different possible results, 𝜒 2 = ∑ (? 𝑖 − ? 𝑖 ) 2 ? 𝑖 ? 𝑖=1 (Eq. 1) Large values of 𝜒 2 indicate a large deviation from the expected result or null hypothesis. The likelihood of finding a particular 𝜒 2 in a set of measurements can be found in tables or calculated in Excel. This likelihood is known as the ? -value. A ? -value close to 1 indicates that the measured results support the expected results, conversely a small ? -value suggests the expected result or null hypothesis may need to be rejected. In this exercise, you are going to look at the likelihood that a die is fair. You are checking a six- sided die (so ? = 6 ) and you assume it is fair (the null hypothesis) so that each face is just as likely to appear. The die is rolled ? times you would expect each face to appear ? 𝑖 = ?/6 times. Of course, if ? was a relatively small number of rolls the observed number of times a face ( ? 𝑖 ) will appear will deviate from this expected outcome. The ? -values (or likelihood that the observed or measured results confirm the expected results) are a function of both the value of 𝜒 2 and the number of degrees of freedom of the system. In this example, the number of degrees of freedom is 5 since if you know what happens with 5 of the faces of the die the last face is determined. Similarly, for a flipped coin, the number of degrees of freedom is 1. Below is a table of ? -values for different degrees of freedom and 𝜒 2 values. Table of 𝒑 -values as a function of degrees of freedom (rows) and 𝝌 𝟐 values (columns). d.f. / 𝜒 2 0.1 0.5 1 2 4 8 16 1 0.7518 0.4795 0.3173 0.1573 0.0455 0.0047 0.0001 2 0.9512 0.7788 0.6065 0.3679 0.1353 0.0183 0.0003 3 0.9918 0.9189 0.8013 0.5724 0.2615 0.0460 0.0011 4 0.9988 0.9735 0.9098 0.7358 0.4060 0.0916 0.0030 5 0.9998 0.9921 0.9626 0.8491 0.5494 0.1562 0.0068 For example, a 𝜒 2 = 1 for a die experiment would support that the die was fair at the level of 96.26%. However, the same 𝜒 2 = 1 for a flipped coin experiment, would support that the coin was fair only at the level of 31.73% (you should be suspicious). Exercise 4 : Your TA will give you a data set where the die you are testing for fairness was rolled 1500 times. The expected number of times each face should appear is 250 or ? 𝑖 = 250 . The

13 Fitting your data to a line or other function If you want to add a regression line, the “+ box” next to the graph contains a check box for a “Trendline” click that and a best fit line will be drawn on the graph. The arrow next to the Trendline box offer s you More Options. Click “More Options" and near the bottom of the resulting menu, click on the “display the equation” for the fitted line and “display R - squared value” (a measure of how good the fit is; closer to one is better). Often you may wish to plot experimental data vs. a theoretical prediction. The expected graph would be a line with the slope of one (assuming data and theory agree). The linear fit to this line and the correlation are a measure of how well the theoretical prediction fits the data. Note there are also other functions available to fit your data in Excel. y = 68.6x - 91.467 R² = 0.9583 -100 0 100 200 300 400 0 2 4 6 8 f(x) x

14 Uncertainty (Error) Bars To add error bars to the graph, the “+ box” next to the graph contains a check box for “Error Bars”. Check the box and some rather arbitrary error bars appear on each data point. Click on one of the vertical error bars on the graph. An Error Bars menu opens. If all the uncertainties are the same for each data point, choose fixed value and enter the uncertainty in the adjacent box. If the uncertainty varies with each point you can choose Custom. Click on the box “Specify Values” and a small dialog box opens (see below). The example below adds error bars to the ? data points (in column B) using the uncertainties given in column D. Click on the box below “Positive Error Value” and highlight D1 through D6. Then do the same for the box below “Negative Error Value”. The ? error bars are done in a similar fashion using the data in Column C. Exercise 5: Generate a set of data for an object moving in a way described by the formula ?(𝑡) = 𝑡 2 ? −3? where 𝑡 is the time in seconds and the position ? is meters. Create the time data over the range from 0 to 3 seconds at intervals of 0.1s. Assume the ? position can be measured with a resolution of 1cm (same value for all data points). Create a graph of ? vs 𝑡 with error bars. See details below for creating a properly formatted graph. Checkpoint: Raise your hand and show your properly formatted graph to your TA. It should have the following features: • The ? and y axes should be labeled with a descriptive and units, e.g. for the ? axis “time (s)” and for the ? - axis “position (m)”. The font size should be reasonable (try 14 pt and then adjust).

16 Lab 1. Scientific Literacy and Measurement Apparatus In this lab, you will find and select a peer-reviewed scientific article that your group will read and later, during Lab 3, give an oral presentation on the article. This lab will also give an overview of the available measurement devices in this laboratory and how they function. Part 1: Finding Scientific Articles The UT library system has subscriptions to a wide number of scientific journals. They can be accessed at https://www.lib.utexas.edu/find-borrow-request . Click on Journals> . Type in the journal name (see screenshot below with example of American Journal of Physics). Click on Available Online and you will get the following result. You can search for specific articles or Click on the box with the arrow in it to browse current articles.

17 Your group will be using this general method today to find an article to do a short presentation in a later lab session. The goal is two-fold: to help you become familiar with accessing peer-reviewed scientific articles and to give you a chance to do an oral presentation on a scholarly topic. There are a few peer-reviewed journals that have articles that are meant for a more general audience, i.e. more easily understood by a non-specialist. These include the American Journal of Physics, Nature, and Science. Your task is to find an article in one of these that catches your group’s interest and during the next two weeks prepare a 10 minute talk discussing that article (what it was studying, what it found, how it found it, and its significance). The talk should use Powerpoint slides and each member of the group should talk equally. An example of what the slides should look like for such a talk can be found on Canvas. Checkpoint: Raise your hand and show the article you have chosen to the TA. Be prepared to explain why you have chosen it. You may proceed to the other part of this activity if you must wait for the TA. Part 2: Using the measurement apparatus Measuring distance In lab 0 you used a meter stick or ruler to measure length. This will be our standard way of measuring distance and position. Note the uncertainty associated with this measurement is half of the smallest division on the ruler or meter stick e.g. for a ruler with 1mm divisions the uncertainty is 0.5mm. Typical ruler uncertainty: 0.5mm

19 Quick Start 1. Connect the Motion Sensor to your PASPORT 850 Universal Interface 2. Turn on the interface and make sure it is connected to the computer. 3. Start the Capstone Software and Click one of the available templates. The selected layout appears in the Display Area. 4. Click <Select Measurement> for each display. 5. Select the measurement in the drop-down list e.g. Motion Sensor. 6. Place an object in front of the sensor at least 15 cm away. 7. Click “Record”. 8. Move the object in a straight line directly away from or toward the sensor (recall the results may not be reliable if the object get closer than 15 cm). Click “Stop”. 9. A graph of the Position vs. time is usually the default. By clicking the label of the vertical axis you may select Velocity or Acceleration from the Measurements Menu. Exercise 1: Find the Help tab at the top of the Pasco Capstone window. Use it to figure out how to export your data and then open it in Excel. Make a plot of the position vs. time in Excel (make sure it is properly formatted as described in the previous lab). Checkpoint: When you feel you have learned how to use this sensor let the TA know you are ready to be checked out. You can begin reviewing how to use the next sensor while you wait. Pasco Rotary Motion Sensor

20 The Pasco Rotary Motion Sensor measures the angular motion of a rotating axle. A wheel with fine angular divisions is attached to the axle and light is reflected off the wheel. The variation of the light as the markings pass is converted into angular position data. This wheel and light sensor package is known as an optical encoder. Note the linear motion of a string passing over the pulley can also be found to a high precision. The specifications of the sensor are below: • Three-step Pulley: 10, 29 and 48 mm diameter • Resolution: ±0.09° or 0.00157 radian or 4,000 divisions/revolution o Linear resolution for 10 mm pulley is 7.85 × 10 −6 m. • Maximum Rotation Rate: 30 revolutions per second • Optical Encoder: Bidirectional, indicates direction of motion. Quick Start 1. Connect the Rotary Motion Sensor to your PASPORT 850 Universal Interface 2. Turn on the interface and make sure it is connected to the computer. Start the Capstone Software. 3. Start the Capstone Software and Click one of the available templates. The selected layout appears in the Display Area. 4. Click <Select Measurement> for each display. 5. Select the measurement in the drop-down list e.g. Rotary Motion Sensor. 6. Click or press ‘Record’ to begin recording data. Give the pulley assembly on the sensor a spin. Once it has stopped spinning, Click “Stop”. 7. A graph of the Angular Position vs. time is usually the default. By clicking the label of the vertical axis you may select Angular Velocity or Angular Acceleration from the Measurements Menu. Checkpoint: When you feel you have learned how to use this sensor let the TA know you are ready to be checked out. You can begin reviewing how to use the next sensor while you wait. The Pasco Photogate is an infrared light source mounted in one arm of the U-bracket aimed at a photodetector in the other arm. If something passes between the arms the resulting voltage signal falls from high voltage to low voltage when the light blocked. It can be used to measure the time it takes for an object to pass through the gate (e.g. time from front edge to back edge). Often it is Pasco Photogates