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Report for Experiment #1 Density of a Material

Alex Leonelli Lab Partner: Nick Lomohan TA: Your Instructor’s Name 1/14/20

Abstract Our goal was to find the densities of 4

Introduction Measurement is a key component of a wide variety of jobs from construction to chemical engineering, and in this first part of this lab we practiced the basics of it. Measuring with rulers and electronic scales, as well as using water displacement to measure an object. One of the often-overlooked concepts of measurement by the general masses is that measuring almost always has a degree of inaccuracy or uncertainty. For example, if you measure something with a basic ruler there is often a large degree of

uncertainty, especially if the object itself is small compared to that degree of uncertainty. Measuring with a micrometer will make your degree of uncertainty much smaller and accuracy much greater compared to the measurement of the same object with a simple ruler. This understanding and furthermore finding the uncertainty of measurements was one of the key objectives and focuses of investigation 1. Once the measurements and finding the uncertainty was completed, the second focus of investigation 1 was data analysis and derivation. With measurements of diameter, length, and mass you then use those quantities to derive volume and density. Keep in mind that the uncertainties of the measurements translate to uncertainties in the derivations as well. The goal of investigation 1 was to understand and calculate the uncertainty of the measurements taken with a ruler and electronic scale. Four solid metal cylinders were measured in this investigation for their length, diameter, and mass. All length and diameter measurements were taken by hand with a simple ruler. All the mass measurements were taken with an electronic scale that went to one decimal place past grams. One volume measurement was taken, that of the heaviest cylinder using a graduated cylinder. Once all the measurements were taken, we calculated error for all our measurements, then derived volume and density from those measurements. The error or uncertainty was then calculated for the derivations. Investigation 2 of Experiment 1 focused on data collection and analysis, one of the most important fields in todays data driven world. Data analysis explains why ads show up for stuff you recently showed interest in, and how subway systems are optimized to run to the best of their ability. Once again, the basics of data collection and analysis were focused upon on this investigation. Data collection can be as simple as how many customers come in the local sandwich shop and as complicated as how many milliseconds it takes the brain to register an image. Staying with the local sandwich shop, analysis of the data you collect would look at when does business spike, is there a reason different seasons or times of day attract more people. Error and uncertainty also plays into investigation 2 as the data collection is based on time intervals, and human error with stopping the timer as well as the precision of the timer creates uncertainty or a level of error. Recognizing these this uncertainty is another part of data collection as well. The goal of investigation 2 was to get a once again very basic understanding and familiarity with data collection and analysis. For the analysis of the data it was more geared towards presenting data in a way that can be easily displayed and understood and to give a platform for informed analysis of a data set. With that set, creating a graph is the best way to do that. Using the GQ GMCounter PRO software and a Geiger counter 20 data points were collected. The GQ CMCounter PRO software registers each “click” of the Geiger counter that represents the number of radioactive decays occurring in the vicinity around the Geiger counter. The software then counts these clicks and will continue counting until you stop or restart the counter. Twenty one minute trials were run and the number of clicks that occurred within that minute were recorded then the counter was restarted. There was no error in the starting of the counter as the computer software had a built-in timer so human error starting the timer was eliminated. Where error came in was stopping the counter and especially when clicks would occur right around when the minute time limit was reached. These data points were recorded on excel and then a graph was created with those data points.

Investigation 1 The set up was quite easy for Investigation 1. All that was needed was the four metal cylinders, a ruler, an electronic scale, and a graduated cylinder. Pulling up excel to put the data in during the experiment was optional and was easier that putting it on paper then transferring the data to the computer. First all four cylinders were assigned a number 1-4 to keep them separate and not mix up data. Then all four were weighed using the digital scale, making sure the scale was zeroed before using it. Next the cylinders were measured using a ruler for their length and diameter. The error is always ½ of the smallest increment, and this was also recorded when measuring the mass, length and width. The errors were labeled with the ẟ sign on the excel sheet. Once those measurements were taken the volume was calculated using the formula Volume of a cylinder. Eq 1.1

V=

π D2 L . 4

Then the error in the volume was calculated using this formula from Appendix A:

Error propagation Eq 1.2

ẟx x ¿ ¿ ẟy 2 ¿ y ¿ ẟz =√ ¿ z

This equation uses the error in D and L to calculate the error in V. x represented D and y represented L in my calculations and then z represented V. This gives you

ẟz z

or

ẟV V

, which was then

multiplied by V to get ẟV, both or which were recorded in the data table below. Then for the biggest cylinder an additional measurement was taken using a graduated cylinder and water displacement. Normally the uncertainty of the measurement taken from the graduated cylinder would have an error of half the smallest increment, but in this case the relative error is set at 1 ml. This is due to the fact that the video was blurry, and the exact location of the water level was hard to see, as well as the graduated cylinder was lifted off the table and held in front of the camera. This could have resulted in tipping and an inaccurate reading to be taken. On top of this there is two spots for error in measurements, one from the measurement taken with only water in the cylinder and another when the measurement with the brass cylinder in the water was taken. Given the uncertainty of 1 ml during both measurements the total uncertainty could be as high as 2 ml, making it a very inaccurate reading. Therefore the volume value obtained in through formula 1.1 is more accurate. After the volume calculations and volume error calculations, the same thing was done for density, calculating the density and the error of that calculation. The error was calculated using equation 1.2 with the inputs being the errors of mass and volume. The density was calculated with the standard formula for density using volume and mass:

Density Eq 1.3

ρ=

m V

Then the average density was calculated (8.899 cm 3), as all the cylinders were made from the same material. The error in average density was calculated as well. First the error for each individual density was found using equation 1.2. Then the error for the sum of the densities was found by adding together the error of the individual densities then dividing that by four, the number of densities. Table 1.1 measured/calculated values from investigation 1

Average density – 8.899 cm3 Figure 1. Mass vs volume chart

Mass vs. Volume 120 f(x) = 8.19 x + 2.8

100

Mass (g)

80 60 40 20 0

0

2

4

6

8

10

12

14

Volume (cm^3)

A graph of mass vs volume was made with the mass as the y-axis and the volume as the x-axis. Error bars were attached to each of the points to show the size of the error of the measurements. Both axes were labeled, including units. Then a line of best fit was created on the graph and used to find the best value of p. The best value of p from the graph is 8.1944 cm 3, opposed to the value of 8.899cm 3 obtained from averaging the densities. Then the error of the slope was calculated using the IPL website inputting the same values used to create the table. Table 1.2 – IPL straight line fit calculator data

The error that the IPL website produced (error in slope) is equal to the error for the best value of the density. The IPL calculator gave 8.3174 cm 3 as the density, only .123 cm3 off from the density that was derived from the graph. The graphically determined density is much closer to the IPLs value then the value obtained from averaging the densities, which results in a difference of .58165 cm 3. This proves how giving equal weight to all the points regardless of error skews the derivation. The measurements of density were within range of the calculated errors, but the graphically determined density is more precise as it gives less weight to bad data points, or points with a higher error. Therefore, the graphically determined density is more accurate than by averaging the densities. The results of the experiment were the most accurate density from the line of best fit of the graph. This showed how much different methods of measurement and calculation can yield different numbers, especially when multiplying or dividing on measurements with error. Furthermore, how straight averages

can be inaccurate representations as all data points are given the same weight whether good or bad in a straight average. It also gave a basic understanding of measurements and the different values you can derive from those measurements, as well as how to get those measurements off graphs of your data.

Investigation 2 Investigation 2 needed very few materials, only a Geiger counter and the predownloaded computer software GQ GMCounter PRO. Once the Geiger counter was plugged in and hooked up to the computer the trials were ready to begin. The counter was run for 1 minute, the number of clicks were recorded, then the counter was reset. This process was completed 20 times to obtain 20 values or data points. Given the nature of the online classes a pre-recorded set of data was used with 100 counts, which provides a sounder set of data to analysis. The data was entered into excel and then a histogram was created from the data to begin the analysis part of the investigation. The average, n´ , of the 100 trials was taken and recorded. Figure 2.1

The histogram was then used to calculated Sn, the uncertainty. Using full width half maximum, or the full width of the histogram and half of the highest bar, the width of the blocks at half the height of the tallest block was found and recorded. Then using W (width) calculate ẟ n using equation 2.1: Eq. 2.2

ẟ n=

W W ≈ 2 √ 2 ln 2 2.3548

ẟ n = 5.308. This number represents the uncertainty and was found using equation 2.2 with W set as the width of the bins at half the height of the tallest bin. In other words, the width of the bins that were half the height of the tallest bin or taller. Histograms can often be tedious and a bad representation of a large data set. A way to better represent a large spread of data is a “root-mean-square deviation (RMS) or standard deviation as more commonly known. This is a more efficient method as the uncertainty in the average number of counts can easily be found according to the standard error in the mean.

The standard deviation was calculated using the STDEV function on excel. Both the standard deviation and the value of ẟ n should be similar as they both represent the mean of the data set, which is a relatively small one. Figure 2.2 – average and standard deviation of the data set.

Figure 2.2 shows the average, n´ , of the counts/min as well as the standard deviation of the data set. The standard deviation and ẟ n should be similar as they both represent the spread of data about the mean. In this case the numbers were remarkably similar, as the ẟ n was 5.308 and the stdev was 5.195. Finally, the standard error was calculated in the mean number of counts ẟ n´ .This shows how if you were to repeat the experiment again, you could obtain a slightly different average value for n. The standard deviation was found using the STDEV function on excel. The standard error was found by taking the square root of the standard deviation Eq 2.3 Standard error, ẟ n´ : stdev / √ N

where N is the number of counts recorded.

Standard error = .53

The result of the counter data was very straightforward, the higher number of counts gave a better data set to work with, the standard deviation and FWHM calculations came out to be nearly exactly the same, which was what was expected. This part of the lab gave a great introduction into analysis of larger sets of data using different methods. Error could easily be found in human reaction time and having to make the call on whether a count should or should not be included in the data set when it was right on the 60 second mark, and not being able to really pinpoint if that count was before or after the minute time window.

Conclusion In Investigation 1 measurements of four cylinders of identical substance were taken with rulers and electronic scales and a single volume measurement with a graduated cylinder. Using those values the volumes and densities of the cylinders where derived and the error for all the measurements and calculations were propagated. Then the mass and volume measurements were used to create a graph and then a line of best fit which gave the best value of the density available in the investigation and proved to be better than the value calculated by a standard average. Investigation 1 in the beginning was driven by hands on data collection and the study of how that data collection is only as good as the instrument by

which you measure, and now error is ever present. Using electronic scales with somewhat decent accuracy and standard rulers are not super accurate forms of measurement and this really showed when the densities of the four cylinders were calculated. Despite all cylinders being made of the same material, all the densities calculated were different from each other, which really showcased how prominent error in measurement can be and how it translates to error in calculations based upon those measurements. In the future improvements upon this lab could be using more accurate rulers and electronic scales with a higher degree of accuracy. The higher the accuracy of the instrument you use the smaller the error should be. In Investigation 2 the data collection was quite simple and straight forward, run the Geiger counter and corresponding software for a minute, record the number of counts, and repeat another 19 times. Given the online circumstances this part was voided, and 100 counts were pre-recorded. Then once the data was collected a histogram was made to analyze the data. Standard deviation, ẟ n, and standard error was all found from the histogram. Utilizing full width half max showed how it can be more accurate than using traditional standard deviation, but the numbers were similar. Possible improvements to the experiment could have been finding a way to be more exact with the time cut off on the minute trials. The fact that there were 100 counts instead of 20 made the data and analysis much more reliable, and the standard deviation smaller.

Questions 1. If the scale had not been zeroed out in Investigation 1 it would have completely skewed every measurement except the length and diameter measurements, which are completely independent of the mass measurement. Depending on how far off the scale was from 0 before weighing the cylinders could have resulted in a catastrophically inaccurate data set or a slightly off data set. Either way the data would have to be refuted and would be unable to be used 2. p=m/V

V=(pi(D^2)L)/4

a. p=8.1944g/cm^3 b. V=m/p

m=250G

D=10cm

V=250g/(8.1944g/cm^3)

c. (4V)/(piD^2)=L 3. Vs=(4/3)(pi*r^2)

(p=density)

V=30.51cm^3

(4*30.51)/(pi10^2)=L

L=.388cm

p=m/V

a. R=10cm p=8.1944g/cm^3 b. Vs=(4/3)(pi*10cm^2)=420cm^3 c. pV=m

8.1944g/cm^3*420cm^3= 3441.6g=m

4. One random error could be that the speedometer rounded up and therefore put you over the speed limit when if a more precise speed was given you would’ve been given your true speed which was under the speed limit. A systematic error that could have occurred was your speedometer was

damaged by the “repair guys” and no longer properly zeros out when the car is at rest and therefore adds a couple of mph on the speedometer to your actual speed. 5. The standard deviation of two Geiger counters will be smaller than the standard deviation of one Geiger counter as there is more data and therefore a smaller standard deviation.

Acknowledgments...