Measuring Devices & Error Propagation: Lab Report on Errors & Uncertainties, Study notes of Economics

Download Measuring Devices & Error Propagation: Lab Report on Errors & Uncertainties and more Study notes Economics in PDF only on Docsity! Experiment 1: Errors, Uncertainties and Measurement Laboratory Report Clarisse Cuevas, Leanne Curaming, Aline De Castro, Adrienne De La Cruz, Ida Dy Department of Math and Physics College of Science, University of Santo Tomas España, Manila Philippines Abstract The experiment was divided into two parts: the first part made use of the foot rule, vernier calliper and micrometer calliper to measure a sphere of known composition while the second part made use of a foot rule to measure the thumb of different people. The first activity aimed to prove which of the following devices is the most recommendable for measuring substances. The measuring device with the least amount of percentage error was the micrometer calliper. The second activity focused on using constant standards in measurements. After measuring the thumb of each group member, it was confirmed that the thumb cannot be generalized as a measure of an inch. Overall, the experiment explored devices, errors and uncertainties in measurement. 1. Introduction Measurement has become a vital part of our lives. From choosing whether to buy half or whole meal for lunch to preparing the ingredients for baking or Chemistry Lab, the application of measuring is continuous. Measuring requires much attention, although sometimes goes unnoticed because of the habitual practice, and specific devices to attain satisfactory comparison with a given standard. Through the activities in this experiment, students are expected to:  to study errors and how they propagate in simple experimentation  to determine the average deviation of a set of experimental values  to determine the mean of a set of experimental values as well as set of average deviation of the mean  to familiarize the use of the vernier calliper and micrometer calliper  to compare the accuracy of the devices  to determine the density of an object, given its mass and dimensions 2. Theory Propagation of Error is the effect of variables' errors on the uncertainty of a function based on them. Variables have uncertainties due to measurement limitations (e.g., instrument precision). The uncertainty of a measuring device is 50% of the least count. The least count is the smallest subdivision given on the measuring device. The uncertainty of the measurement should be given with the actual measurement.[1] The vernier principle is the difference between two scales. Estimating Uncertainties in Measurements: It is good practice to perform repeated measurements so that a mean value can be estimated together with the standard deviation and/or the standard error in the mean. Uncertainty in Direct Reading Devices: For a direct reading device (like the ruler or caliper) the reading uncertainty is ¼ the smallest division for a single reading. This means ½ the smallest division for a distance.[2] Mean diameter: d= ∑i=1 10 xi n Average deviation: a.d. = ∑ d n Average deviation of the mean: A.D. = a .d . √n Percent error of diameter: % error = A . D . d Volume (cm3): V = 4 3 π r3 Mass (g): M = 28 g Experimental value of density (g/cm3): EV = M V Accepted value of density (g/cm3): AV = 7.8 g/cm3 Percent error for density: % ERROR = |AV −EV AV x100| Legend: d – mean diameter n – number of observations n = 10 a.d. – average deviation A.D. – ave. deviation of the mean d – deviation V - volume ∑ d- sum of deviations π- pi r – mean radius m – mass EV – experimental value AV – accepted value ||−¿ absolute value 3. Methodology The group used a foot rule, vernier caliper, micrometer caliper, electronic gram balance and a sphere. The group compared the accuracy of these measuring instruments (foot rule, vernier caliper and micrometer caliper). They made use of certain formulas to determine the errors of the measuring instruments. They made ten independent measurements for the diameter of the sphere using the foot rule and also determined the density of the sphere given its proportions and mass. 4. Results and Discussion change the system or the measurements made on the system under study. Systematic error affects the accuracy (closeness to the true value) of an experiment but not the precision (the repeatability of results). [4] It is stated at the website, physics.nmsu.edu, that systematic errors may be of four kinds: instrumental, observational, environmental and theoretical.[5] Random errors are positive and negative fluctuations that cause about one-half of the measurements to be too high and one-half to be too low. [4] Random error affects the precision of an experiment, and to a lesser extent its accuracy. [5] 3. Sketch a) a vernier caliper that reads 5.08cm and b) a micrometer caliper that reads 2.55cm. Please refer to attached paper for the sketches. 4. A student weigh himself using a bathroom scale calibrated in kilograms. He reported his weight in pounds. What is the percentage error in his reported weight if he used this conversion: 1kg=2.2pounds? The standard kilogram is equal to 2.2046 pounds. % error=|SV −EV SV |×100 % error=|2.2046−2.2 2.2046 |×100 % error = (2.026546312 × 10-3) × 100 % error = 0.208654631 % % error = 0.21 % 5. In an experiment on determination of mass of a sample, your group consisting of 5 students obtained the following results: 14.34g, 14.32g, 14.33g, 14.30g and 14.23g. Find the mean, a.d. and A.D. Supposed that your group is required to make only four determinations for the mass of the sample. If you are the leader of the group, which data will you omit? Recalculate the mean, a.d. and A.D. without this data. Which results will you prefer? D= ∑ of values number of values = 14.34 g+14.32 g+14.33 g+14.30 g+14.23 g 5 D=14.304 g deviation1=¿ Di−D∨¿ deviation1=¿14.34−14.304∨¿ deviation1=0.036 deviation2=¿ Di−D∨¿ deviation2=¿14.32−14.304∨¿ deviation2=0.016 deviation3=¿ Di−D∨¿ deviation3=¿14.33−14.304∨¿ deviation3=0.026 deviation 4=¿ Di−D∨¿ deviation 4=¿14.30−14.304∨¿ deviation 4=0.004 deviation5=¿ Di−D∨¿ deviation5=¿14.23−14.304∨¿ deviation5=0.074 a .d .= ∑ d n ¿ 0.036+0.016+0.026+0.004+0.074 5 a.d. = 0.0312 A . D.= a .d . √n ¿ 0.0312 √5 A.D. = 0.013953064 % error=|SV −EV SV |×100=|a . d .−A . D. a . d . |×100=|0.0312−0.013953064 0.0312 |×100 % error=55.28 % When the value 14.23g is removed D= ∑ of values number of values ¿ 14.34 g+14.32 g+14.33 g+14.30 g 4 D=14.3225 g deviation1=¿ Di−D∨¿ deviation1=¿14.34−14.3225∨¿ deviation1=0.0175 deviation2=¿ Di−D∨¿ deviation2=¿14.32−14.3225∨¿ deviation2=0.0025 deviation3=¿ Di−D∨¿ deviation3=¿14.33−14.3225∨¿ deviation3=0.0075 deviation 4=¿ Di−D∨¿ deviation 4=¿14.30−14.3225∨¿ deviation 4=0.0225 a .d .= ∑ d n ¿ 0.0175+0.0025+0.0075+0.0225 4 a.d. = 0.0125 A . D.= a . d . √n ¿ 0.0125 √4 A.D. = 0.00625 % error=|SV −EV SV |×10 ¿|a . d .−A . D. a .d . |×100 ¿|0.0125−0.00625 0.0125 |×100 % error=50% We would prefer the latter result which resulted to a 50% percentage error. It is more accurate and acceptable than the former result’s percentage error of 55.28%. The lower the percentage error, the more accurate the results are. 7. References [1] Lepla Org. (nd). Errors and Statistics: Instrument Uncertainty and Least Count. Retrieved from the World Wide Web on December 4, 2012 [http://www.lepla.org/en/mod ules/Activities/p04/p04- error4.htm] [2] Uregina. (nd). Experiment 109-1: Measurements, Uncertainties and Errors. Retrieved from the World Wide Web on December 4, 2012 [http://uregina.ca/~szymanss/ uglabs/p109/Experiments/109 -1Meas&Error08.pdf] [3] Merriam - Webster. (2012). Error. Retrieved from the World Wide Web on December 4, 2012 [http://www.merriam- webster.com/dictionary/error] [4] O.S.U. (nd). Experimental Error. Retrieved from the World Wide Web on December 4, 2012 [http://chemistry.osu.edu/~co e/research/documents/experi mental_error_new2.pdf] [5] N.M.S.U. (nd). Types of Experimental Errors. Retrieved from the World Wide Web on December 4, 2012 [http://www.physics.nmsu.ed u/research/lab110g/html/ERR ORS.html]