Measurement_and_Comparison_Lab_V1

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Your Name: ___________________________________ Date: _________________________ Group Members: ________________________________________________________ Measurement and Comparisons Lab Exercise Equipment: Spreadsheet program on computer for data analysis and graphing (usually Excel). 30 cm and/or 15 cm rulers, and calipers Different size circular objects String or a tape measure to measure circumference DVD, Blu-Ray disc, or CD Introduction: Consider two people who measure the width of a table. One person measures it with a meter stick and says its width is 106.7 and the other person measures it to be 1.1. Can they both be right? In fact, from the information given you can't tell if the two measurements agree or disagree. For one thing, the units of the measurement aren't given. If the first person reports the width as 106.7 cm and the second as 1.1 m (or 110 cm), then they are close to being the same value, but we still can't say if if the measurements agree with each other. What's missing is the uncertainty in the measurements. If the two measurements included an estimate of the uncertainty, we could more easily compare the measurements. For example, if the first measurement was reported as 106.7 ± 0.8 cm and the other as 110 ± 4 cm, then we could say that the two measurements agree because we know the uncertainty of the measurements and the first one (106.7 cm) is within the range of 114 cm to 106 cm indicated by the second measurement. This notation is explained later in this exercise . This "Overlap Comparison" is a simple way to compare two measurements if estimates of their uncertainty is available. Using the same units for all quantities; if the first measurement is A and its uncertainty is δA and likewise B and δB, then we can say they agree if B is within the range A−δA to A+δA (The wide (gold) region in the figure below) or A is within the range B−δB to B+δB (The more narrow (violet) region in the figure below) This is a yes/no result, it is more realistic to describe agreement with a criteria that describes the quality or 'goodness' of the agreement of the two measurements. If we consider probabilities of the 'true' value being displaced from the measured value, the probability is not uniform within ±δ and zero for other values. The probability is a smooth function that peaks at the measured value and decreases as we move away from that value. See the figure below. 1

We get a better idea of the agreement of the two measurements using the t′ measure described below. We would calculate that for the case above (106.7±0.8cm and 110±4cm), we get |t′|=0.81, which indicates good agreement. We use three measures of relative difference to describe how much the two measurements differ. The three measures are: Percent Error - The difference, relative to a known or accepted value, Percent Difference - The difference, relative to the average of the two values, and |t′| - The difference, relative to the combined uncertainty. The formula for calculating each of these measures of relative difference are given below. Why Study Measurement and Uncertainty: In most of the lab exercises in this class, we will try to measure a quantity and estimate the uncertainty in our measurement. A major goal of this particular lab exercise is to introduce some of the techniques we use to obtain an estimate of a value and an estimate of the uncertainty in that value. In many cases we will use applied statistics to get an estimate of the value of a quantity and also use statistics to estimate the uncertainty in that value. Statistics can get very tedious if done by hand on a calculator, but we can use Excel to do all the statistics we need. Also, the computer software (Capstone) that reads our electronic sensors can do some statistics for us. Exercise Objectives: You should be able to: 1. Calculate percent difference and percent error and distinguish between them. 2. Calculate the standard error of a mean (also called standard uncertainty of mean) using a calculator, spreadsheet or on- line calculator. 3. Calculate the t ' value for comparing two values with uncertainties. 4. Produce a properly labeled plot of data in Excel. 5. Use Excel's Trendline or Google sheets to fit a line to a set of data. 6. Use Excel or Google sheets to obtain the uncertainty in the slope and uncertainty in the intercept of a line fit to a set of data. Some General Guidelines for Lab Measurements  Read every direct measurement to as many digits as you can . For example, when using a ruler estimate the position between markings, don't just read the nearest marking.  Keep lots of digits in your recorded values and calculations. In our experiments you should keep at least 5 significant digits for these intermediate results.  Keep track of the units you measured and in your calculations. 2

 You only round your results at the end of the analysis, when you put them in VUU format.  Every value on the lab worksheet that has units associated with it must have the units displayed with the value. Reporting Results with Uncertainty: "VUU Format" Use the VUU Formatter or use instructions on http://classes.schmiedekamp.net/PhysicsLabs/DataFitsInExcel-VUU.html to do it manually. (VUU stands for Value , Uncertainty , Units which are the three parts of VUU.) The value and its uncertainty have the same units so the units only need to be given once after the uncertainty. If the value and uncertainty are reported separately then each should have the appropriate units attached. If you are asked to report the value and the uncertainty separately, do not round them off at that point but give the intermediate values with extra significant digits you have been using. Part A - Uncertainty in Repeated Measurements We are going to compare measurements among groups of similar objects. You will be measuring the diameter of a DVD or Blu-ray disc. 1. Let each member of the group measure the diameter in 2 different places and each place with two different devices (rulers and/or measuring tapes), for a total of 4 measurements per person. Try to read the rulers carefully, estimating the fractional distance between markings to get the most precise reading. For a group of 3 students that will give 12 independent measurements of the diameter; with 4 students you have 16 measurements. When you have completed the measurements calculate the mean (average value) of the new measurements. Measurement Number Distance/cm Measurement Number Distance/cm 1 11 2 12 3 13 4 14 5 15 6 16 7 17 8 18 9 19 10 20 Mean (Avg.): 2. Use the Excel spreadsheet or a Google sheet (or a calculator, or the Javascript calculator) to calculate the standard deviation of the values in the table above, the S.U.M and the uncertainty δ. You can also calculate them using a calculator and the formulas provided. 3

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