Report for Experiment #1 Measurement

Transcription

1 ! Report for Experiment #1 Measurement Abstract The goal of this experiment was to get familiar with the method of measuring with essential tools. For the experiment, rulers, a scale, metal cylinders, graduated cylinder, and Geiger counter was given. It was asked to collect the data, compare results with theory, and then create graphs and tables based on them.

2 Introduction When a particular object is measured, the collected data are not precisely correct because of the imperfectness of the measuring tools. Even though it is common to believe that digital measuring instruments are profoundly accurate, they still have mechanical imperfectness that make their data incorrect for a little percent. There are different types of errors like systematic error (Error that occurs no matter how many times the measurements were done; for example, scales that were not zeroed out), random error (error that occurs because of some extraneous factors, and that can be decreased by repeating the measurements), propagated error (an error that can be found my a number of formulas when two measurements of data are added, multiplied, divided, et al.), uncertainty dn (can be found by dn! = W/2 (2ln(2))), standard deviation, and standard error of the mean (error between mean value and actual measurement). Moreover, there such things like FWHM (full width at half maximum) which are used to track half of the maximum amplitude (the highest value is taken and then divided by two, all values over the line are in FWHM). The goal of this experiment was to make the experimenters familiar with propagated error, data tables, and data depiction in histograms. At the end of the experiment, we learned how to collect data, how to calculate propagated error, how to create tables based on measurements et al. Investigation 1 The goal of this investigation was to fill in the given table and find the densities of the given four cylinders, which had different mass, length, diameter, and volume. First, the rulers and scales were used to find masses, lengths, and diameters of cylinders. These measurements were put into the table with calculated relative errors. Next, volumes were found by the following formula: V! = πr 2 L, where L is the length and r is the radius of the cylinders. Then, the relative error was calculated with formulas from Appendix A and entered into the data table. Meanwhile, the 100 ml graduated tube was used to find the volume of the biggest cylinder. The uncertainty of the gained volumes of the biggest cylinder is its relative error. After having all the necessary information, the densities of the cylinders were found. In order to find the relative error of their densities and average density, formula of propagating error from Appendix A (p. 3293) was used, which is dz! /z = ((d x /x) 2 + (dy/y) 2 ). Hence, we the following formula was found, dz! = z ((d x /x) 2 + (dy/y) 2 ). The average density was found by the following formula: D! = (d 1 + d 2 + d 3 + d 4 )/4, where! d1,! d2,! d3,! d4 are densities of the cylinders.

3 Then it was required to make a scatter of mass against volume. After that, a best-fit-line was asked to be created based on the data.

4 The error of density was a relative error. By repeating the experiment with different objects of the same density a more exact value could be obtained. Moreover, the best fit line depicts that their mass vs. volume values almost create a line (which is in fact density). As many measurements are done, as more correct the slope of the line is, and thus the value of the density is more accurate. Investigation 2 The goal of the second investigation was to collect data on the background radiation, create a table of data, and depict data on a histogram. First, the Geiger Counter was set and restarted. Then after 40 minutes the data were collected and put into the table. Next, bins were sat; average of counts, standard deviation, and bin sizes were found.

5 Since, a histogram had to be created, a new table of data was made. There were defined only bins and their frequencies. After that, the uncertainty error was found with the formula! dn = W/(2 * (2 * ln(2))), where W is equal to 8. Based on that data the histogram was created. Values in FWHM were colored.

6 As it turned out, the obtained results of the average count and uncertainty of n, which was equal to approximately 3.4, were very similar to the results obtained by other teams which were 23.85, 20.65, and 24.4 counts; and ~1.7, 5.9, 5 uncertainties of n. The difference in results of counts and uncertainties of n among groups happened because of a relative error. Conclusion This experiment is made of two investigations. In the first one, we used basic measuring tools to determine measurements of four metallic cylinders, fill out the given table and then find the relative errors by using formulas from Appendix A. It was allowed to switch some measurements into SI measuring system. Then we were asked to create a plot with the best-fit line. In the second investigation, it was required to set up and turn on the Geiger Counter minutes before the experiment, since each count is in 60 seconds. Next, we collected the data, put them into the table. Later, a second plot was made that consisted of Bins and Frequencies. We found the uncertainty error and created a histogram. In the first investigation, it was found that densities of the cylinders are very similar. Moreover, it was found that the relative error of the average error is approximately kg/l, while the density was approximately kg/l. In the second investigation, the average count was found, which was equal to 21.8 per 60 sec. Standard deviation and bin size were calculated and added to the table and were equal to and In the future experiments, in order to prove that all three cylinders have the same density, an equal amount of weight could be taken from the cylinders and liquified. Next, liquified pieces could be put into the graduated scale to see if they have the same volume. The relative error that could be negotiable in mass and took into account in volume will give us a more exact density of the metals. Questions 1. If zeroing out the scale was forgotten before weighing the cylinders in Investigation 1, the results of mass would all be higher for an equal amount. The relative error of the masses would not change. 2. If a cylinder, that is similar to the four cylinders from Investigation 1, had a mass of 250 g and diameter of 10 cm, then it can be assumed that it has a mean density of these four objects. Ergo, its volume would be equal to approximately !. 3. The same material sphere with radius equal to 10 cm would have a volume of m^3 and mass of approximately 40 kg. m 3

7 4. If I was in a court and would want to prove my innocence by giving examples of random and systematic error, I would postulate the following ideas. First, random error is an error that occurs because of extraneous factors. For example, a radar gun could give a wrong number because of radar gun casual error, or some other wave coming from my direction, or because of some object flying in front of the car. Random error can be reduced by repeating the experiment, and if it was possible to repeat the experiment, the radar gun would give the other result. However, if the error was systematic, then repeating the experiment would not reduce the error. A systematic error could occur if the speedometer was not synchronized with the engine, and thus shows the seed lower than the actual one. A systematic error also could occur if the car was broken and always resulted in a wrong value of speed. 5. If results of two Geiger Counters are combined and the width of the bins is not changed, then the standard deviation will not change. However, if it was increased, then the standard deviation will also be increased, and vice versa. Honors Questions (extra points) 6. The volume of a hollow cylinder with length L, radius r, and the thickness of the walls can be defined as V! = πl(d 2 + 2rd ). That formula was found by subtracting volume with the smaller radius from the volume with the larger radius, e.g. V! = V R V r, where R = r + d. A hollow sphere with the wall thickness of 3 cm and density of 9306 kg/! m 3 will have a mass of approximately kg. It was found by using the previous formula, e.g. V! = V R V r, where R was equal to 10 cm and r was equal to 7 cm ( 10 cm - 3 cm = 7 cm ). 7. If 20 measurements were made and a mean value n was calculated with the standard deviation dn, both quantities can be substantially changed only if the following measurements significantly differ from the first twenty. If the following measurements are much lower than the first twenty, the mean value will decrease, while the standard deviation will increase, and vice versa. 8. In order to decrease the error in the mean in by half, by using a propagated error formula it can be assumed that two times more measurements will have to be done. 9. Propagated error formulas for division and multiplication are the same because they are the same types of arithmetic action. It is the same as why the propagated error formula for subtraction and addition are the same. When an object is measured, its data are written in a±b, for example, 7.34±0.005 cm. The same principle is with division and multiplication. Acknowledgements

8 I would like to thank my lab partner, Alejandro Hervella, for his help in this experiment and our TA, Vivan Nguyen, for helping us understand the propagated error formulas and explaining how we should make this experiment. References A. Hyde, O. Batishchev, and B. Altunkaynak, Introductory Physics Laboratory, p , Hayden-McNeil, 2017.

9 ! Report for Experiment #3 Motion in One and Two Dimensions Abstract The goal of this experiment was to study to time dependence of displacement, velocity, and acceleration and one and two dimensions, and show that motions is directions of the motions can be independent. In this experiment an air table, puck, spark timer, sheet of paper, ruler, and wooden block were used. At the end of the experiment the acceleration of the puck in m m investigation 1 and 2 were obtained, and equal to 0.597±0.0120! and 0.484±0.0112!. Moreover, it was learned that motion in OX direction is uniform, though motion in OY direction has a free fall acceleration. s 2 s 2