Sooknauth_Experiment1LabReport (4)

Report for Experiment #1 Measurement Diana Sooknauth Lab Partner: Pricilla De Guzman TA: Emily Tsai January 23 rd , 2023 Abstract In this experiment, our overall goal was to learn how to use measurements and account for errors in measurements. For investigation 1, we were trying to figure out what material the four cylinders given to us were made of. We measured the mass and volume of the cylinders and used that data to figure out the density. What we found was the average density of the cylinders was 9.76 ± . 775 g/ cm 3 and therefore the cylinders are most likely made out of brass. For the second part of the experiment, we used a Geiger counter to measure the background radiation count rate. We collected data and found that the average background radiation count rate was 19.5 ± 4 counts/60s.

Introduction In this experiment, our first goal was to figure out the density of the cylinders given to us by using mass and volume. Density is defined as the mass of a unit volume of a material substance and can be found by dividing mass by volume. By finding the density of the material, we would be able to identify it since the density of a material does not change significantly unless it is facing a change in temperature or pressure. Our second goal was to figure out the average background radiation count rate in our lab room. The particles around us naturally contain radiation, which we call background radiation. The air around us contains unstable isotopes and when measured, they are registered as radioactive decay. Radioactive decay occurs randomly and therefore by collecting data, there are noticeable fluctuations in the measurements. The background radiation was able to be measured by a Geiger counter. The Geiger counter has a stable gas in a chamber of the device and when that gas is exposed to radioactive particles, like the ones in the air around us, the gas ionizes and creates an electrical current that the counter records. Our third goal was to learn how to calculate and account for different types of errors in the data since every measurement has limited precision and this will be something we have to do for the rest of our labs. The objectives of this experiment were to get the length and weight measurements and determine their uncertainties/ errors, use that data to obtain values for derived quantities, understand how errors in data propagate, learn how to plot data, and use graphical methods for data analysis, and understand basic aspects of statistics. The different types of errors we had to measure and consider with this experiment are instrumental, systematic, and random. Instrumental errors occur when the instruments and equipment being used are inaccurate. Systematic errors are experimental flaws that cause measurements to be consistently inaccurate in the same way each time they are taken. These could occur due to limitations of the equipment or procedure. Random errors are unpredictable statistical fluctuations; they occur due to chance. Because this experiment involved using data to make calculations, the use of standard deviation and the standard error of the mean were needed. The standard deviation of a data set is a measurement of the amount of variability that the individual data values have compared to the mean, which is the average value of a data set. The standard error of the mean is how far the sample mean of the data is likely to be from the population mean, meaning the average of the rest of the data. The computed standard deviation for investigation 2 in this experiment was 4.00 counts/60s. The computed standard error in the mean was 0.517 counts/60s. Investigation 1 For investigation 1, the equipment needed was four metal cylinders of the same material, a ruler, a digital scale, and a 100 ml graduated cylinder. The ruler was used to measure the length

δL L ¿ 2 2 δD D ¿ 2 + ¿ ¿ δV V = √ ¿ (1) The relative error in volume for Cylinders 1, 2, 3, and 4 were 0.084, 0.112, 0.111, and 0.167 respectively. Unlike with the other measurements, we found the relative error first before finding the error. We used this equation to find the relative error of the volume, but it was also useful to find the error in volume. To find the error in volume we simply multiply the cylinder’s individual volumes by their respective relative errors. By doing this, we found that the error in volume for Cylinder 1 was 0.587 cm 3 . For Cylinder 2 it was 0.327 cm 3 . For Cylinder 3 it was 0.390 cm 3 . For Cylinder 4 it was 0.142 cm 3 . We then conducted another test to find the volume for Cylinder 1. We took the graduated cylinder and put in 60 cm 3 of water. When we placed Cylinder 1 inside, the water rose up to about 70 cm 3 . The displacement of the water was about 10 cm 3 . Calculating the displacement of the water indicates volume, so we knew this meant that the volume given to us by the graduated cylinder was about 10 cm 3 . Compared to the result we got when using the equation to find volume, which was 7.01 cm 3 we believe that the results we got when using the volume equation is more precise. We believe this because there is a difference of almost 3 cm 3 between the two. The graduated cylinder is not as reliable because sometimes it could be hard to read the measurements on it and get an exact measurement. We were not able to get an exact measurement when using the graduated cylinder, which is why we had to estimate that the displacement was about 10 cm 3 . After conducting the test with the graduated cylinder, we then had to start calculating the average density of the cylinders, which was our actual goal for this investigation. To find the average density, we first had to find the density ( ρ ¿ of each individual cylinder. In order to do so we used the equation: ρ = m V , The densities found for Cylinder 1, 2, 3 and 4 were 9.80 g/ cm 3 , 9.81 g/ cm 3 , 9.89 g/ cm 3 , and 10.1 g/ cm 3 respectively. In order to find the relative error in density, the equation below was used: δ V V ¿ 2 δ m m ¿ 2 + ¿ ¿ δ ρ ρ = √ ¿ (2)

The relative densities for Cylinder 1, 2, 3 and 4 were 0.084, 0.112, 0.111, and 0.168 respectively. Similar to how we found the error in volume, we were also able to find the error in density (δρ). The error in densities for Cylinder 1, 2, 3 and 4 were 0.821 g/ cm 3 , 1.10 g/ cm 3 , 1.10 g/ cm 3 , and 1.70 g/ cm 3 respectively. All of the calculated data can be seen in Table 1 below. To find the average density and error in density, we added the densities and divided them by four; we did the same thing for the error in densities. The average density was 9.91 g/ cm 3 and the average error in density was 1.18 g/ cm 3 . Cylinder 1 2 3 4 m(g) 68.7 28.7 34.6 8.6 δm (g) 0.050 0.050 0.050 0.050 δm/m 0.001 0.002 0.001 0.006 L (cm) 6.20 4.60 5.50 3.00 δL (cm) 0.050 0.050 0.050 0.050 δL/L 0.008 0.011 0.009 0.017 D (cm) 1.20 0.900 0.900 0.600 δD (cm) 0.050 0.050 0.050 0.050 δD/D 0.042 0.056 0.056 0.083 V (cm3) 7.01 2.92 3.50 0.848 δV (cm3) 0.587 0.327 0.390 0.142 δV/V 0.084 0.112 0.111 0.167 ρ(g/cm3) 9.80 9.81 9.89 10.1 δρ(g/cm3) 0.821 1.10 1.10 1.70 δρ/ρ 0.084 0.112 0.111 0.168 Table 1: Data collected for investigation 1. After we collected our numerical data, it was time to express our data graphically to obtain the best value of density. Using the mass data for the y-axis and the volume data for the x-axis, Graph 1 was plotted:

2 26 3 25 4 18 5 13 6 21 7 18 8 19 9 17 1 0 20 1 1 17 1 2 15 1 3 16 1 4 21 1 5 13 1 6 19 1 7 16 1 8 19 1 9 27 2 0 27 2 1 22 2 2 21 2 3 19 2 4 28 2 5 20 2 6 19 2 7 17 2 17

8 2 9 20 3 0 15 3 1 23 3 2 16 3 3 16 3 4 19 3 5 21 3 6 26 3 7 25 3 8 12 3 9 17 4 0 13 4 1 24 4 2 17 4 3 27 4 4 20 4 5 19 4 6 17 4 7 17 4 8 20 4 9 20 5 0 20 5 14

Graph 2: Histogram created for Investigation 2. Conclusion The purpose of this experiment was to get used to measuring and calculating errors and relative errors of data collected. We learned about the different types of errors and how to account for them. In investigation 1, we used measuring devices to find the mass, diameter, and length of four different cylinders. We then used this data to calculate the volume of each cylinder and their densities. We calculated the average density, but also learned how to graph it with a scatterplot. From graphing, which we found was more precise, we concluded that the average density of the cylinders were 9.76 ± . 775 g/ cm 3 . An improvement to investigation 1 that I believe would help the results come out more precisely is to use a more precise measuring device for finding the diameter and length of the cylinders. A digital caliper would be beneficial as it is a hand held device that students can easily use, and gives measurements up to .01 millimeters. Using a caliper is also beneficial because in many manufacturing and engineering jobs, a caliper is used which will help students learn how to use an important tool they may need in their future jobs.

In investigation 2, we used the Geiger counter to find the average background radiation count rate. After conducting a trial for an hour, we found that the average background radiation count rate is 19.5 ± 4 counts/60s. We also learned how to graph our data using a histogram. An improvement to this experiment that would make the resulting data more accurate would be to run the counter for longer, maybe 2 hours instead of 1. Having more data will help to determine the accuracy of the average. Questions Question 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? If I had forgotten to zero-out the scale before weighing the cylinders, this would cause our data to be construed and inaccurate. This would most likely cause the first or first few cylinders we measured to have an inaccurate reading, which would result in inaccurate computations when calculating the average density of the cylinders. This would introduce an instrumental error in our calculations since the error is due to an inaccurate reading from one of our measuring devices. Question 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? The length of the cylinder is 0.326 cm. I found this by first using the equation ρ = m V . I knew that ρ = ¿ 9.76 g/ cm 3 because of the density found in investigation 1, and I knew that m= 250g. I then found volume which was 25.6 cm 3 . I then used the equation L = 4 V π D 2 . I inputted V=25.6 cm 3 and D= 10cm. Question 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? The sphere has a mass of 40800 g (when accounting for significant figures). I started with the equation given: V = 4 3 π r 3 . I found that V= 4180 cm 3 when given r=10 cm. I then used the equation ρ = m V . I used ρ = ¿ 9.76 g/ cm 3 and used V= 4180 cm 3 . Question 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 the speed given by either your speedometer or the radar gun.