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ME 3057 - EXPERIMENTAL METHODOLOGY & TECHNICAL WRITING Report/Worksheet/Writing Task: Lab number 2 Full Report: Partial Report: X Lab section: J NAMES: Shameek Agarwal, Date Turned In: 2/4/2012 , , Date Returned by TA: ____________________ Grader: _____________

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ME3057, Spring 2012

Abstract The objective of this lab was to illustrate the concepts while measuring properties of realworld systems. Physical measurements of objects are associated with uncertainty which much be taken into account when analyzing a problem using measurements. Propagation of uncertainty arising from individual uncertainties yields a single combined uncertainty for the calculated value. The lab was structured into three parts; in part one of lab, the diameter of a small piping section 4 was measured using two micrometers, one with a 0.0001” of resolution and other with 0.001” and a caliper with a resolution of 0.001”. Analysis of part showed that maximum uncertainty aroused from instrument with highest resolution and that the best fit for uncertainty is with a standard error using K=3. The percentage error for the 0.001” micrometer, 0.0001” micrometer and the 0.001 caliper were 0.65%, 1.37% and 0.17% respectively. In part 2, a caliper was used to measure the dimensions of a 10mm gage block and error propagation was computed. The blocks density was calculated to be 7780.26 kg/m3. Analysis showed that uncertainties in measurement of individual dimensions cause an error propagation which couples to give a higher uncertainty in the calculated density. The error in the density value computed from the standard deviation was 391.78 kg/m3 while the same computed from the resolution uncertainty was 24.402 kg/m3 In part 3, the time period of a pendulum was measured first using a stopwatch measured over various cycles and then an oscilloscope with help of a sensor. The mean time period calculated for 1, 6, 13, 25 cycles is 0.85s, 0.88s, 0.88s and 0.89s respectively. It was also observed that the relative error decreases with increase in number of cycles, i.e. relative error is minimized when averaging large number of single measurements than few readings over multiple cycles. The relative error in measuring over 1 cycle was 5.12% whereas for 25 cycles was 0.62%.

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MEASUREMENT OF UNCERTAINTY AND ERROR ANALYSIS
Part 1: Experimental Results Attached in Appendix A is multiple measurement s of the diameter of the piping section. The average value of the diameter calculated with the 0.001” Micrometer, 0.0001” Micrometer and the 0.001” Caliper are 6.263 mm, 6.391 mm, 6.339 mm respectively. The values measured are close to the manufactured specified value for the diameter which is 6.35mm with unspecified uncertainty. The measured values by the instruments lie within a range of ± 1.37% from the specified value. Some of the sources of error include variation in the diameter of the plastic tubing along different sections of the tube, human error in measuring could also affect the average result. Part 1: Analysis The mean and the standard deviation are presented in Table 1 and are calculated using Equation 1 & 2 ( ̅) ( ) ∑ √∑
( ̅)

(1) (2)

Table 1: Calculation of mean and standard deviation 0.001" Micrometer (mm.) 6.263 0.1718 0.0001" Micrometer (mm.) 6.391 0.2661 0.001" Caliper (mm.) 6.339 0.06707

Mean Std. Dev.

Resolution uncertainty is half of the resolution and the statistical error is calculated using Equation 3. ( ̅)
√

(3)

Table 2: Calculation of resolution and statistical uncertainty Sample value 0.001" Micrometer (mm) 0.0001" Micrometer (mm) 6.263 6.391
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Resolution uncertainty (+/-) 0.0127 0.00127

Standard Error (+/-) k=1 0.0337 0.05220 k=2 0.0674 0.10440 k=3 0.1011 0.15660

0.001" Caliper (mm)

Sample value 6.339

Resolution uncertainty (+/-) 0.0127

Standard Error (+/-) k=1 k=2 k=3 0.0132 0.0263 0.0395

The mean value listed in Table 1 is the best estimate of the “true” expected value for these measurements. The mean value takes into account errors in the measured values and effectively calculates the “true” measured value. The uncertainty used here, is calculated from standard deviation with K=3. Standard deviation is a better measure of uncertainty than the resolution uncertainty because more values are likely to fall within a larger range, i.e. multiples of standard deviations, than fall in the range of resolution uncertainty about the mean. Table 3: Calculation of relative error 0.001" Micrometer 6.263 6.350 1.370% 0.0001" Micrometer 6.391 6.350 0.6488% 0.001" Caliper 6.339 6.350 0.1730%

Measured value (mm) Manufacturer value (mm) Relative error

As seen from table 3, the least error in measurement of the pipe diameter comes from caliper and the maximum from the micrometer. This could be attributed to the fact that a micrometer has many sources of operator error compared to a caliper, for every measurement made by a micrometer a different measured value could be expected because of lack of experience in handling the micrometer. The variable application of force at the end of the ratchet and the biases in reading the measurement could have caused a larger relative error in measurement by the 0.001” micrometer. A PARKER 8MSEL6N-S pipe fitting has a diameter of 12.7mm (0.5”) but the measured value of our piping section has a diameter of approximately 6.3mm, hence out piping section will fit into the PARKER 8MSEL6N-S but this will not be a tight fit, i.e. the pipe section will slide into the fitting. The potential sources of random for all the instruments is the same as the environmental conditions do not change with measurements. But systematic errors are not the same for each instrument as the operator error is different because of different ways of measurement, also the offset introduced for each instrument is different in each case.
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Yes, the standard deviation are consistently related to the instrument’s resolution such that higher the instrument’s resolution, higher the deviation from the mean of the measured data and vice-versa. As shown in Table 1, the above statement is in approbation with the theory. The standard deviation for the 0.001” micrometer, 0.0001” micrometer and the 0.001” caliper is 0.1718, 0.2166 and 0.06707 respectively. In case of no operator error in using the 0.0001” micrometer the measured value should have the most accuracy and hence deviation of the each measured value from the mean to be the lowest, as the resolution is far greater than the other two instruments. Part 2: Experimental Results Attached in Appendix B are the measurements of the dimensions of the block and the mass of the block. Regardless of using a caliper with a resolution of 0.001” the last set of measure values, mass and length, significantly increase the standard deviation. The mean value of dimensions of the 10mm block obtained is 9.877mm for width, 30.665mm for length and 8.795mm for thickness. The mass of the block is measured to be 20.643g. Two data points, one from the length and other from width measurement were completely off the mean value and hence were not considered in the data analysis of this part of the lab. Part 2: Analysis The mean and standard deviation of the block’s dimensions and mass is shown in Table 4. Table 4: Mean and Standard deviation of block dimensions and mass Width (mm) 9.964 0.162 Length (mm) 30.067 0.251 Thickness (mm) 8.795 0.148 mass (g) 20.5 0.088

Mean Std. Dev.

The density of the block is found using Equation 4 and substituting values from Table 4. ( )

(

)

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The propagated uncertainty is calculated using Equation 5,

(

)

√∑

(

)

(5)

Using N=4, the number of measured variables, and expanding Equation 5 gives Equation 6 and Equation 7.

√( √( )

) (

(

) )

( (

) )

( (

) )

(6)

(7)

Case A: Error propagation of measurement uncertainty using ½ instrument resolution

Plugging the above values in equation 6 gives

Case B: Error propagation of measurement using twice the standard deviation

Plugging the above values in equation 6 gives

The propagated uncertainty values are represented in Table 5 Table 5: Comparison of error propagation obtained from resolution & standard deviation Δl (mm) Δw (mm) Δth (mm) Δm (g) Δg (kg/m3) 0.5 * Resolution 0.127 0.127 0.127 0.05 24.4 2 * Standard deviation 0.502 0.324 0.296 0.176 393.37

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As seen from Table 5 the propagated uncertainty values obtained using the resolution and the standard deviation methods are a little different. The uncertainty value for the dimensions in case of standard deviation is nearly twice the resolution uncertainty; this could be attributed to the fact that the large values calculated from the standard deviations have a greater influence on the uncertainty density. No, two quantities with identical uncertainties will not always contribute equally to the uncertainty of a calculated quantity, because equation 6 requires a differential and if the function depends on 2 quantities differently the value will change significantly. As a percent of the mean, uncertainty of density and uncertainties of calculated quantities is compared in Table 6 below using Equation 8
(8)

Table 6: Comparison of uncertainty as a percent of mean value 0.5* Resolution Measured value Percentage 0.127 1.27% 0.127 0.42% 0.127 1.44% 0.05 0.24% 24.402 0.31% 2 * Standard deviation Measured value Percentage 0.502 3.24% 0.324 1.67% 0.296 3.37% 0.176 0.86% 391.783 5.03%

Δl (mm) Δw (mm) Δth (mm) Δm (g) Δg (kg/m3)

The uncertainty in the density is in close relation with the uncertainties in the individual measure quantities as seen in Table 6. Although there is a little variation in the percentage of uncertainty in length and thickness but this could be attributed towards biases in measurement and human error in measuring the dimensions. Part 3: Experimental Data Attached in Appendix C are measured data for this part of the lab. Few values were significantly off the mean expected measured value and were omitted in the calculation performed in the analysis. These data points are represented in red font in Appendix C. The average measured period of the pendulum for different cycle numbers is 0.88s, the time period measured by the oscilloscope is 0.89s. The time period measured by the stopwatch is less compared to oscilloscope; this could be as a result of operator error such as reaction time to calculate the right time period.
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Part 3: Analysis The Geometry of the pendulum is given in Figure 1

Figure 1: Geometry of the pendulum The period of the pendulum is calculated using Equation 9 ̈ For θ