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School

University of California, Los Angeles *

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Course

110B

Subject

Electrical Engineering

Date

Feb 20, 2024

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Pages

1

Uploaded by PrivateJellyfishPerson2012

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The standard deviation o of the Gaussian curve is, by definition, {/(Az)?), where the Gaussian PDF is (1/ 27ra)e"‘2/ 20* Using the MATLAB command histfit to fit your his- togram to the Gaussian PDF and determine the value of (Az)2. To get the fitting parameters you actually need to use the fitdist command. 4.1.2 Analysis - method 2 An alternative way to determine (Azx)? is to use the cumulative distribution function (CDF) graph to interpolate the o value from the linear experimental data. See Ref 1 for details. This method is based on the idea that the derivative (slope) of the CDF at z = 0 is inversely proportional to the root-mean square (RMS) displacement: & / . e dg = #e"m"2 e : dz|,._o /- V210 V2ro L 2o 4.1.3 Analysis - method 3 A third choice is the direct calculation of (Az)? from the data: Square each displacement and calculate the weighted average. This estimate is scientifically sound based on the LLN. Once the average square displacement has been calculated for at least four different time intervals, plot 7 vs (Az)? and determine N4 from Eq. (1). Does the outcome of a calculation of (Az)? differ from (Ar)?, where (Ar)? = (Az)? + (Ay)?? If so, in what way? As shown in Eq. (1), the calculation of the Avogadro constant will depend upon the viscosity of water for which the following data are available: T |n T |'n T |n T |n °C|{ 103 Nsm2|[°C| 103 Nsm2|[°C | 103 Nssm2 || °C | 102 N.s.m2 15 | 1.139 19 | 1.027 23 | 0.933 27 | 0.851 16 | 1.109 20 | 1.002 24 | 0.911 28 | 0.833 17 | 1.081 21 | 0.978 25 | 0.890 29 | 0.815 18 | 1.053 22 | 0.955 26 | 0.871 30 | 0.798 Your report should include both the Gaussian distribution fit and direct (Az)? calculation for at least four different time intervals and the cumulative probability curve for at least one time interval. (With the way this experiment is currently setup, acquisition of a 20-30 min long video followed MATLAB analysis of each frame, should yield more than enough data to carry out all analyses.) In the past, students have successfully used 7=15, 30, 45 and 60 s to answer this question. However, at the time, particle tracking was done manually and labor-intensive. Now that we have MATLAB automation of the particle tracking, there is more data available to you. Feel free to use another set of time intervals, as long as you can identify the correct underlying trends. 4.1.4 Questions The following questions should also be addressed in the report: e What are the principal sources of uncertainty in this experiment? 6
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