PHYS1146 Lab 1

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Report for Experiment #1 Measurement 09/08/2023 Abstract The density of four cylinders of a material was measured. The average density was calculated to be 9.525 with an error of 0.424. Using a more accurate graphical method the average density was estimated to be 9.645. Background radiation count per minute was measured sixty times. The average count per minute was 17.883 and the uncertainty was calculated using FWHM to be 4.247 which was very similar to the calculated value for standard deviation of 4.762. The standard error in the mean was calculated to be 0.615. Introduction The purpose of this experiment was to study error in scientific measurement which can include random errors, systematic errors, and instrumental error. Random errors are unavoidable and are present in all measurements taken, this is because there are always natural variations in measurement processes. For example, measuring the length of an object using a measure with certain measurement divisions and estimating the value to a precision lower than the division. Systematic errors are usually avoidable by

using correct technique and caused by improper procedures or equipment. They are hard to detect and not only cause measurements to be incorrect but cause all calculations derived from a systematic error to be incorrect. Instrumental errors are a type of systematic error and occur when equipment is not functioning properly and lead to incorrect readings of measurement. Errors can be accounted for, and an estimate of error can be made. A way to find how spread-out values are from the average is to calculate the standard deviation using: (1) Standard error in the mean is a way to measure the uncertainty of the average value calculated by: (2) Random errors can be computed by using (2). Random error can be estimated by halving the smallest division on the measurement device. Computing instrumental error is difficult as it is a type of systematic error which is difficult to detect and introduces a consistent bias. In order to understand if there is an instrumental error the measurement could be taken on several different instruments to test variance in the results. Density is the mass of a substance per unit volume. Density can be calculated by weighing an object and measuring its volume and then using: (3) Background radiation count rate is the measure of the amount of naturally occurring radioactivity without a known source. It is measured using a Geiger Counter which measures the number of radioactive particles that pass through a gas tube that knocks off electrons. These electrons are attracted to a charged wire and counted electronically as decay. The goal of this experiment was to make multiple measurements of the density of a material and background radiation count. The data was analyzed to understand and calculate errors in the data sets. In investigation one, the weight and volume of four cylinders of brass were measured using a weight scale and a ruler; in one sample the volume was measured using water displacement. The average density and error were calculated. In investigation two, the background radiation count was measured for one hour in one-minute increments. The uncertainty was estimated using full width half maximum (FWHM) on a histogram chart. Investigation 1 The experimental setup was made up of four cylinders all made from the same material. Each cylinder's masses, diameters, and lengths were measured, and volumes were calculated. To do this the diameters and lengths were measured with a ruler and the mass was measured with a digital scale. For all measured quantities there were random errors based on the accuracy of the measuring instrument and the σ = ( t 1 − t ) 2 + ( t 2 − t ) 2 + … + ( t N − t ) 2 N δ t = σ N ρ = m V

human factors in taking the measurement. The errors were determined using half the smallest measurement interval for the measuring device. Volume was found using: (4) For comparison the volume of the largest cylinder was found using a 100 ml graduated cylinder filled with water. The cylinders were numbered in order of longest to shortest. Mass was collected for each cylinder using the digital scale, it was first tared and then cylinder weight was measured. By halving the smallest increment of display on the scale, error in mass was found. Relative errors in mass for each of the cylinders were calculated using: (5) The length and diameter of each of the cylinders were measured using a ruler and the error was estimated by halving the smallest increment of measurement on the ruler. Relative errors in diameter and length were calculated using (5). Table 1: Mass, length, diameter, volume, and density data for four cylinders. V = π D 2 L 4 relative error = δ x x Cylinder #1 #2 #3 #4 m (g) 78.900 68.400 28.900 39.500 δ m (g) 0.050 0.050 0.050 0.050 δ m/m 0.001 0.001 0.002 0.001 L (cm) 7.300 6.300 4.780 3.700 δ L (cm) 0.050 0.050 0.050 0.050 δ L/L 0.007 0.008 0.010 0.014 D (cm) 1.200 1.200 0.900 1.200 δ D (cm) 0.050 0.050 0.050 0.050 δ D/D 0.042 0.042 0.056 0.042 V (cm 3 ) 8.256 7.125 3.041 4.185 δ V (cm 3 ) 0.690 0.596 0.339 0.353 δ V/V 0.084 0.084 0.112 0.084 ρ (g/cm 3 ) 9.557 9.600 9.504 9.439

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The volume of each cylinder was found using (4). Relative error of volume was found for each of the cylinders using: (6) An additional volume measurement for the biggest cylinder was taken by submerging it in a graduated cylinder filled to about 61 ml with water, once submerged the water level rose to around 70 ml . The was volume calculated using (4) and was 8.256 g/cm 3 and volume that was calculated by submersion was 9 g/cm 3 . The error was propagated because there was uncertainty in both the length and the diameter measurements, which were used to calculate the volume. The volume taken by the water method only has a single error, however the precision of that measurement was lower than the precision of the length and diameter methods, so the single error was greater than the propagated error of the calculated volume. The volume taken using (4) was more precise than the water immersion method. The density of the cylinders was calculated by using (3). The error was propagated because the density was calculated by using three measurements which all had their own error. Length and diameter both had an error of 0.050 cm and mass had an error of 0.050 g . Average density was calculated by adding the density values from all four cylinders and then dividing that value by the number of measurements taken, in this case four. Error for average density was found by finding the standard error in mean using (2). Average density was also determined graphically (see Fig. 1 below). The slope of the graph is mass divided by volume which is equal to density. By using the IPL Straight Line Fit calculator, the slope and error in slope was obtained and was 9.639 ± 1.166. Average density was 9.525 and the density from the plot was 9.645, these values were very close to each other. δρ (g/cm 3 ) 0.097 0.113 0.349 0.190 δρ / ρ 0.010 0.012 0.037 0.020 V (ml) 9 x x x δ V V = ( 2 δ D D ) 2 + ( δ L L ) 2

Fig. 1: The cylinders’ mass in relation to the cylinders’ volume. The slope of the best-fit line is shown and is the density of the cylinders. Fig. 2: Slope and error in slope values obtained by IPL Straight Line Fit calculator. Investigation 2 The experimental setup consisted of a GMC-200 Geiger Counter which was plugged into a PC and recorded using GQ GMCounter PRO software. The counter was started, and the program gathered data of the number of counts per minute for an hour. The average value of counts per minute was calculated and this was determined to be the best value for the number of counts per minute. A histogram was made based on the measurements from the Geiger Counter. A bin size of two was used. The

uncertainty of the best value was estimated using FWHM on the histogram created. The uncertainty of the best value was also estimated by finding the standard deviation and standard error in the mean. Standard deviation was calculated using (1) and was an estimate of how spread-out the values are from the best value. Once the standard deviation was found the standard error in the mean was calculated for the number of counts by using (2). Table 2: Counts per minute sorted by time least to greatest. Trial N umber Counts/60s 24 9 45 10 5 11 10 11 40 11 19 12 20 12 25 12 35 12 36 13 59 13 9 14 11 14 23 14 43 14 44 14 12 15 32 15 34 15 39 15 7 16 47 16 8 17 14 17 18 17

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21 17 26 17 28 17 31 17 51 17 3 18 41 18 48 18 56 18 58 18 60 18 15 19 50 19 53 19 1 20 4 20 6 20 13 20 29 20 49 20 22 21 30 21 38 21 46 21 37 22 57 23 27 24 42 24 2 25 33 25 16 26 17 27 55 27

The average count rate was the total count rate for all collected data divided by the number of collected counts and 17.883. Full width half maximum (FWHM) was the width of a data distribution curve at half the maximal value. It was an estimate of uncertainty from the average value. FWHM was calculated by determining where the histogram bins were above half the maximum value and width of the bins that were above half the maximum value. The estimated uncertainty of the average count by using FWHM was 4.247. This was calculated using: (7) Fig. 3: Histogram of amount of background counts per minute with FWHM included. The downward pointing arrow indicates the position of average count. This data was compared to data from another group. The FWHM from the comparison data was 9.9 while the data collected had a FWHM of 10. The FWHM was very similar. The standard deviation of data was 4.762 and the uncertainty calculated from FWHM was 4.247. These are similar values and extremely close in order of magnitude. The standard error in the mean was calculated by using (2) and was 0.615. Conclusion The density of a material was estimated by measuring linear dimensions and weight in four different samples. Random errors occurred as a result of equipment precision and measurement variation. 54 28 52 29 δ n = W 2 2ln2 ≈ W 2.3548

Average density was calculated using all samples and was 9.525. The propagated error of density was calculated to be 0.424. A graphical method was then used to calculate average density and was 9.654. These average values were very similar. The graphical method was considered to be more accurate because it considered higher quality points more than lower quality points. In addition, one sample had its volume measured with water displacement. The error of this measure was estimated to be higher than the propagated error of the linear measurements due to the large increments of volume measurement. There may have been systematic errors such as a bent ruler or a weighing scale which was not properly tared. These are very hard to detect. The accuracy and precision could be improved by increasing the number of samples, increasing the number of times each sample was measured, and having more accurate measuring equipment. The background count of radioactive decays was counted by using a Geiger Counter. The device counted this for one hour in one-minute increments because of this many measurements were taken. A histogram of the data was prepared and FWHM was determined. The FWHM was used to calculate the uncertainty, which was 4.247. This was extremely close to the calculated standard deviation of 4.762. The standard error in the mean was calculated to be 0.615. This is a naturally occurring and inherently random process and therefore subject to random error. The accuracy and precision could be improved by increasing the number of times the count was taken. Questions 1. If you had forgotten to zero-out (tare) the scale before weighing the cylinders in Investigation 1, how would it have affected your data? What type of error would this have introduced into your calculations? a. If the scale was not tared the weight recorded would be an inaccurate weight of the cylinder. The calculations derived from the cylinders mass would also be inaccurate. This would have caused a systematic error in the calculations because the equipment was incorrectly set up by not taring the scale. 2. A cylinder of the same material as the one you used in your experiment has a mass of 250 g and a diameter of 10 cm . What is its length? a. Average density calculated = 9.639 g/cm 3 b. c. For a cylinder , therefore d. 9.639 g/cm 3 = , therefore = 0.3302 cm ρ = m V V = π r 2 l ρ = m π r 2 l 250 g π ( 10 2 cm ) 2 l l = 250 g π ( 10 2 cm ) 2 ( 9.639 g / cm 3 )

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3. A sphere of the same material as the one you used in your experiment has a radius r = 10 cm . What is its mass? (Hint: V sphere = r 3 ) a. and b. = 40375.7 g 4. Suppose you receive a traffic ticket for speeding and want to contest it in court. Come up with two arguments, one using systematic error and the other using random error, that you could use to challenge the speed given by either your speedometer or the radar gun. a. One argument could be that my speedometer is measured in increments of 5 mph and so my true speed could have been ± 2.5 mph from the speed I read on my speedometer, which would be random error. b. Another argument could be that the radar gun had not been calibrated correctly and was reading higher than the true speed which would be a systematic error. 5. If the data from two Geiger counters are combined, how will the standard deviation of the new data set compare to that of each of the individual Geiger counters? a. The standard deviation represents the spread in the values of decay counts per minute. If the standard deviations from two Geiger Counters were added, the pooled standard deviation would depend on the individual standard deviations and the number of observations for each data set. The equation to calculate combined standard deviation is: b. Assuming the number of observations and time collected is the same, combining the data from two Geiger Counters the standard deviation of the pooled data set would be between the two individual standard deviations. The number of observations in the combined data set would be much higher than the individuals. Honors questions 3 3. In honors question 2, how many additional measurements will you need to take to decrease the error in the mean by half? 4 3 π ρ = m V ρ = 9.639 g / cm 3 m = 9.639 g / cm 3 × ( 4 3 π ( 10 cm 3 ) ) σ pooled = ( m − 1) σ 2 1 + ( n − 1) σ 2 2 m + n − 2

a. You would need to take 60 additional measurements. Acknowledgements References [1] Hyde, Batishchev, and Altunkaynak, Introductory Physics Laboratory, pp 5-11, Macmillan Higher Education, 2022. [2] IPL Lab Report Guide, https://web.northeastern.edu/ipl/wp-content/uploads/2017/09/IPL-Lab-Report- Guide.pdf [3] Guide for Exp. 1 – Measurement lab report, https://web.northeastern.edu/ipl/wp-content/ uploads/2018/09/Guide-Exp-1.pdf [4] IPL Straight Line Fit Calculator, http://www.northeastern.edu/ipl/data-analysis/straight-line-fit/ [5] Giancoli, Physics Principles with Applications, Pearson, 2021. [6] Pooled Standard Deviation, https://www.statisticshowto.com/pooled-standard-deviation/