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Introduction I. Types of Errors There are two types of errors that we encounter as we perform experimental measurements. Systematic errors are mostly due to a problem with the instrument or sometimes how the operator is setting up the instrument. For example, when you measure the width ofa paper using a ruler, the systematic error comes from the spacing between the ticks marked on the ruler. if the spacing between two ticks is larger than 1mm, then all the measurements that you obtain with this ruler will be smaller than the actual value. fths actual width afthe paper is 215.5 mm, you may get 213mm using the r'bad" ruler. Systematic errors do not cancel out as you average the measured values. Random errors occur when you use an instrument to perform the measurement several times. These are caused by the experimenter, the changes in the object you are trying to measure, or changes in the instrument. As you measure the width of the paper with the ruler your reading will depend on how you align the zero with the side of the paper. Also. if you do not look perpendicular at the tick marks, you can have different readings. In addition, if between measurements there is a temperature variation and your ruler is made of stainless steel, that will cause different readings. lfbetween measurements 'd'le humidity changed significantly, the paper width will change. So long as not all of the above changes happen as you repeat your measurements, you will nd that these uctuations in the numbers will cancel out. [L Average and Uncertainty We will now focus on random error and discuss how to include the uctuation in the measured values when we report our results. Consider the following data that was 1-] *3 Dr. Abdullatif Harnacl. Not For distribution. Press I E51: 1 in exit full screen collected hya studentfo - mm 216.0mm, 214.0mm, and 215.5mm. Now we have a question: Which ofthe values should we report? The answer is the average value. The average of N measurements, each of which has a value x,-, is dened as 1 N xave = Zizlxi- {1'1} Therefore, in our case the average width ofthe papers is Wm =215. mm. But how much confidence do we have in this average result? In general, the closer the individual measured values are to the average value, the more condence we have in the measurements. Luckily, we have a tool called the standard deviation of the mean {SDGMJ which gives a numerical value for the uctuation ofthe measured data about the average value. The standard deviation of the mean, which is the uncertainty in the reported average value is defined as d' 1 5x E N = \"(N-1) Ji;|m|l=1[:'x.l xiE'JE [1-2) Here a is the standard deviation. Based on the numbers we have, the uncertainty.F in the measured width of the paper using Eq. [1-2) is 61: = 0.3mm. In general, we report the measured quantity 1: in terms ofthe average value, xave: and its uncertainty, Ex, as I = xve :1: [SI {1'3} Therefore, the width of the paper is reported as w = mm i 5w = 215 i [1.3 mm. [I]. Precision vs Accuracy How do we report an}:r of the individual measurement listed earlier? lCan we write for example the width of the paper as 213 i 0.32 mm? The answer is No. Because the smallest reading, we can do with a ruler is 1mm. The best one can read is '3'; mm using a ruler. So, we are limited to the precision or uncertainty of the instrument. The precision in a measurement done by an analog device like a ruler is 5% the smallest division marked on the instrument. However, for most digital devices the precision is equal to the smallest reading the device can make. The precision tells us about the exactness of the measurement. Thus, the best reading we can report for the width ofthe paper using a ruler is i.5 mm. Now, let us see how we find the accuracy ofour measurement. Accuracy afa measurement tells us now close our measured value to the actual value. This is done using the percent error which is defined as MEGSufE-tiudtul %Errar = x 100 [1-4) Actual 1-2 G Dr. AbdullniiflIarnad. Nol for disiribulion. Therefore, the accuracy ofthe measured width of the paper is 0.23%. 1What about if we want to compare the results of two groups of students performed the same measurement. This is done by using the percent difference. Difference Average rital!)t''erence = I x 1&0 [1-5) The percent difference is also used when you obtain the value of a measurement using different methods. IV. Error or Uncertainty Propagation Suppose we are asked to find the area of the paper. This requires that we measure the length of the paper in addition to its width. if a student measured the length of the paper several times and found that the length is L = 2TB i {1.6: mm. How do we report the area ofthe paper including the uncertainty in the area? Recall the that the area ofa rectangle is given by A = wL. Consider a quantity.r zzy] that depends on two measurable quantities: x and y. The uncertainty in the quantityf depends on how it is related to x and 3'. Below are general cases that are useful and that we will encounter in this course. 1- Uncertain t}! ofsimpie sum or difference of measured quan tities lf ffx,y] = x i y then df = afx)? + (63:)? [1-6] 2- Uncertain ty ofsimpie power ofthe product afrneasa red quan ti ties arm of max 2 n5}: 2 if {(13):}: army" on f[x.;v] = yn then T = (T) + (T) (1-?) 5f 7 error which relates the SDDM to the measured value. The quantities f. x. and y in Eq. [1- ?] are the average values or best values. where m and n may be positive or negative values. The quantity is called the fractional Now we are read}.r to calculate the uncertainty in the area of the paper. Using Eq. [14'] we find = tenet 13 '33} Dr. AbdullaiiHInmad. Nol for disiribulion. From our data the area is A = 215 x 2TB = 59??!) nnnz. Therefore, the fractional error in 2 2 the area is [5:30 (i) + (E) .The uncertainty in the paper area is 5:4 = ZETinin2 and we report the area asA = 597TD j; 2.5'Iimni2 orA = 591'? i 2.5?cm2. As an exercise: Suppose a quantity y = x2 where x and its uncertainty Ex are known, either given or you measured them, how do you find the uncertainty in y? Prove to yourself using Eq. {l-T} that dy = 2xdx. 'li'. Linear Graphs in general, graphical methods are the best way to present scientic data. it helps to visualize trends in the data and obtain qualitative ideas about the nature of the relationship between the measured quantities. Each one of the iota activities in this course is designed to test a spedc mode! that reiates two measurahie quantities. Therefore, you will always compare the data to the theoretical prediction. In many cases the nature of the relationship between the variables is linear, however in some cases it is not. In order to analyze the data and be able extract values for the physical quantities in a simple way, you need to represent your data in the form ofa linear graph, which could be tted by y = mx + b [1-9) This is the equation of a straight line that you are the familiar with. The xvalue is usually the independent variable. or the quantity that the experimenter changes. The y-value is the depended variable, which you can think ofas the system response. The value ofm is the slope, which tells us how strongly 3: changes for a given change in x and the value of b is the intercept which is the value of}r when x=il. Let us illustrate how we obtain values for the physical parameters involved in the relationship from the slope and the intercept. Consider the data presented in the table of Fig. 1-1 for the velocity ofan object moving on a flat surface under the inuence ofa force. The object has an initial speed 12,, before the force acted on it. The model that describes such situation is given by i: = at + 120 [ti-lb) In Eq. [1-1), the slope is the acceleration ofthe object, a. The graph of this data shows that it is indeed a linear relationship. [A] How to fit the data Use the Trendl'ine tooi in Excel to fit the data. To do this, rightciick on one ofthe data points on the graph then choose Trendfine 'om the dropdown menu. A window on the right-side of the Excel page wil'i open. Choose linear then click {show equation] and (R?) on chart. These options are at the bottom ofthe side window. The R1 value informs us 1-4 Ci DI. AbdullaiifHamad. Not for distribution. how close the data is to the equation ofthe line we used to fit the data. [flit2 :1 it means that the data is perfectly described by the equation ofa line. The result is shown in Fig. 1-1. The tting equation on the graph is consistent with our model equation which represents the equation of a straight line. Now let us compare our model to the equation that Excel used to fit the data. In this case, the intercept is the value of the initial speed 1,1,, and the slope is the value of the acceleration, :1. Therefore, we obtain no = 3.7"?! mfs and a = 4.34 nifsz. Since the value of R2 is very close to 1, we have condence that our model describes our data. There is If it -- -_ 5\" ',.4_34n..3_ae R' = 0.9944 l {5] Fig. 1-1 Data showing a linear relationship between speed and time of an object under the inuence ofa constant force. another way that we can use to visually determine if the model describes our data well. We can include error bars on the graph. They are constructed from the uncertainty in the measured or calculated quantity, in our case the velocity. The length of the error bar on both sides of the point equals 51: [the uncertainty in 1; not the change in v]. The procedure to include error bars using Excel is presented next. [B] How to include error bars In Excel, click on the graph then Chart Design appears on the top banner. Click on Chart Design, then you will see an icon called Add Chart Element appear at the top- left corner. When you click this icon, a dropdown menu appears. Choose Error Bars and select More Error Bars Options. Now you should see a window on the right side. At the bottom ofthe window, choose custom and click on Specify value. Then choose positive and negative error values from the column that contains the uncertainty in the variable. 1-5 '33 Dr. Abdullatil" Harnad. Nol for distribution. [(1] How to nd the uncertainty in the slope and the intercept To determine the uncertainty in m and h, we use the LINEST function. The LINEST function returns an array that contains the slope, intercept and their uncertainties. To get the values ofthe array you need to highlighta 2x2 ma\"imp?"\"i115ani'i'r'i'i\";il.= =LlNESlenown_ y values, known_x values, 1,1] then press Control + shift + enter. The values presented in the near table is the output of the LINEST function using the data in the table of the Fig.1-1. Recall that in our case the slope is equal to the acceleration, a = 4.8-1.- i 0.13 m/sz. The intercept is equal to the initial velocity, 13,, = 3.?9 i [LEI] mfs. What do we do if our data doesn't follow a linear relationship? This means that the model describing the data is not linear. There are different ways to represent and analyze the data. In this course [Phys 131L} we will always try to linearize our data, i.e. we try to make the relationship between the variables looks linear by reossigning one or both of the variables. Let us clarify this by an example. The vertical position. y. ofan object that dropped from an initial height, ya, as a function oftime. t. is given by 1 y =ya _EHt2 [1-11) Where 9 is the acceleration due to gravity. Clearly this relationship is not linear, it is a quadratic equation. Let us consider a set of data collected by a student for a situation described by the model in Eq. (1-11]. If we plot the position as a function of time, we obtain the result shown in Fig. 1-2[a}. Clearly the plot shows that the data doesn't follow a straight-line relation. So, how we are going to find the values p and ya? One solution to this problem is to plot [y vs. H] or [ t2 vs. y}, which gives a linear graph as shown in Fig. 1-2fb]. The reason why we obtained a linear plot is because we considered t2 is the variable rather than t. This makes Eq. [1-11} represent a linear relationship like Eq. [1- 9]. Thus, we do expect data that follow Eq. [1-11] to show a linear behavior ifwe plot [y vs. :2] or [ :2 vs. 3:]. The error bars on Fig. 1-2[b] represents the uncertainty in t2 not t. Following the discussion presented toward the end of the error propagation section, we find the 6(9) = 2tdt. Therefore, the uncertainty in t2 depend on the value oft and its uncertainty. 1-6 (Ci- Dr. Abdullniil'I-lamad. Nol for disiribulion. $95.1 us: Fist] vim} nus} nlt'Hsl} \"J\" m [LEE oars 15 am 0.14 i" '\" ' "Inn . _:;__ _ 1o: 1.011] 1:. our use _ p.94 5 1.43 "m " '+ f 1-50 _IIEIE . ___L_: +' 1.35 3.423 2 our 0.3: + 2-05 _IIEIEEI =-r-' Fig. 1'2 Data table and graphs for an object dropped from an initial height Fa {a} plot of}I vs. 5 shows that}! is not linear as a function oft. [b] Plot oftz vs. y shows a linear relationship. Now, let us find the values ofg and ya. In order to compare the equation obtained by Excel to the model equation, Eq. [1-11], we need to rearrange the model equation to have 2 on one side and the rest ofconstant and y on the other side. After rearrangement Eq. [1-11] becomes t2 = 33:0 3y. Notice that y is the independent variable and t2 is the dependent variable. Compare this result to the equation of a line. we see that the intercept is represented by b = Syn and the slope is equal to m =3. Therefore, the value ofgI is 2 = 9.31 mxsz and y, = 1933 m. slope calculated from g = In order to determine the uncertainty in g and ya, we use the LINEST - {1.204 4.022 function. The results obtained for the data ofFig. 1-2 are shown in the - 0.00? 0.091 near table. Since 9 is related to the slope by g = i, then using the 2 5m quadrature formula Eq. [1-1], we find that the uncertainty in g is given by g = F' From the table, 6m = DDT, therefore dig = 0.3-1- nails2 and we write a = 9.31 i 0.34m s2 as our final result. Recall that yo is related to the intercept, b, by b yEr =31: = E" Therefore, the uncertainty in y,, is influenced by 5,9 and 5h. Using the quadrature formula Eq. (1-?) we obtain Eye = yo [ugfgjz + [ubfb}2. Dr in terms of the uncertainty in b and m we get (Syn = y,, (dmfnz + (Elijah)? Using the result from the LINEST table we obtain, from either formula, Syn = 0.81 m and the nal value for y,, is written as ,, = 19.13 i [1.31 "g 1-? 3:} Dr. Abdullaiil'HarnaLi. Nol for disiribulion. Experimental Procedure Task 1: Reproduce plots in Fig 1-2 [a] and [b] {a} Use Excel to reproduce plot in Fig 1-2 [a] and [h]. Be sure to calculate the columns for t2 and 6&2] not just entered as numbers. (b) On your graphs show the error bars. (c) Use the Trendiine tool to fit the data, show the tting equations and the R2 values. (d) Reproduce the results of the LINEST function for the data. Task 2: Density and its Uncertainty To find the density ofaluniinum. a student used an aluminum cube. She found that its mass is 53.3 1 0.1g and its volume is 21.30 i 0.3?cm3. Recall that the density of an object is given by p=Mfli where M is the mass and His the volume. {a} Find the aluminum density and its uncertainty. Use Eq. [1-27] to nd the uncertainty. [b] If the accepted value for aluminum density is 2.?1gfcrn3, nd the percent error in the measured value. Clearly show all steps of'your Work on paper then take a picture ofyour calculations and paste it into the spreadsheet that you used for Task] . Task 3: Final Velocity and its Uncertainty [a] Find the final velocity and its uncertainty for an object in free fall if the object was dropped from a height of 2.45 i [I114 m. Consider the acceleration due to gravity is 9.81 i 0.05 mfsz. Recall that the final velocity in term ofdisplacement is given by V? = \"'2 + Egy, where vfis the nal velocuity,vf is the intial velocity. andg the acceleration due to gravity and y is the displacement. Hint: since the object was dropped, its initial velocity is zero. Use Eq. [it-T] to find the uncertainty. Clearly shov.r all steps ofyour Work on paper then take a picture of your calculations and paste il into the Spreadsheet that you used for Tasks l . 1-3 'E Dr. Abdullatil'Harnad. Not for distribution