Sooknauth_Experiment1LabReport (4)

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Jan 9, 2024

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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

and diameter of each cylinder, the digital scale was used to weigh each cylinder, and the graduated cylinder was used as another way to figure out the volume of each cylinder. To begin our investigation, we had to collect data for mass, density, and length as well as their errors and relative errors. To collect the data for mass, we used a digital scale. We made sure the scale was zeroed before we began the measurement. We started measuring each cylinder and recorded their respective masses in grams. We kept track of the cylinders and their data by numbering them 1-4. Cylinder 1 had a mass of 68.7 g. Cylinder 2 had a mass of 28.7g. Cylinder 3 had a mass of 34.6g. Cylinder 4 had a mass of 8.6g. After documenting the cylinders’ mass, we had to calculate the error in mass, which is represented by δm. To do so we just used the fact that a good estimate of the error is half of the smallest increment you are measuring. In this case, the error in mass was determined to be 0.050 grams for each cylinder, since .100 g would be the smallest increment when using grams. We then had to find the relative errors in mass. In order to find the relative errors in mass, represented by δm/m, we divided each cylinder’s respective error in mass by their mass. Cylinder 1 had a relative error in mass of 0.001. Cylinder 2 had a relative error in mass of 0.002. Cylinder 3 had a relative error in mass of 0.001. Cylinder 4 had a relative error in mass of 0.006. Once we finished collecting data for mass, we now had to collect data for the diameter and length of each cylinder. We used a ruler to help us collect data for this. We measured each cylinder’s diameter with the ruler. Cylinders 1, 2, 3, and 4 had a diameter of 1.20 cm, 0.900 cm, 0.900 cm, and 0.600 cm respectively. Similar to finding mass, we had to find the error in diameter ( δD) which we also found by taking half of the smallest increment of a centimeter. This gave us an error of .050 centimeters for each cylinder. We then had to find the relative error as we did with mass. The relative error in the diameter of Cylinder 1 was 0.042. The relative error in the diameter of Cylinder 2 was 0.056. The relative error in the diameter of Cylinder 3 was 0.056. The relative error in the diameter of Cylinder 4 was 0.038. After finding the diameter of each cylinder, we measured the length of each cylinder with the ruler. Cylinders 1,2,3, and 4 had lengths of 6.20 cm, 4.60 cm, 5.50 cm, and 3.00 cm respectively. The error in length for each cylinder (δL) was 0.050 cm, which makes sense as we used centimeters for the diameter and had the same error. The relative error in length (δL/L) for Cylinder 1 was 0.008. For Cylinder 2 it was 0.011. For Cylinder 3 it was 0.009. For Cylinder 4 it was 0.017. We now had all the data needed to move forward. We now had to find the volume of each cylinder. The equation used to find the volume of each cylinder with the data of the diameter and length we had was: V = π D 2 L 4 . For Cylinder 1, we calculated a volume of 7.01 cm 3 . Cylinder 2 had a volume of 2.92 cm 3 . Cylinder 3 had a volume of 3.50 cm 3 . Cylinder 4 had a volume of 0.848 cm 3 . After finding this data, we then had to calculate the relative error for volume. The equation given to us to find the relative error in volume was:

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:

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