Mark this Report - Uncertainty Lab

Material Identification by Density Course: PCS-211 Section 7 Instructor: Dr. Eric Da Silva TA: Diana Ha Date of Submission: September 17, 2023 Performed by: Ammar Siddiqui and Shayan Hajizad

Introduction The purpose of this lab was to identify an unknown material which was given in the form of a cuboid block. The volume of the material was measured using three distinct methods. Each different method yielded three different volume values. These volumes were used to calculate three different densities. By comparing the average density to a table of known values, the unknown material was identified. This lab also accounted for the uncertainty of each measuring method. The uncertainties were utilized to determine the consistency of the results. Theory There were two core physics concepts investigated in this lab, uncertainty in physics, and density of materials. Uncertainty In physics, it is important to show that the tool used has some sort of uncertainty associated with it. When expressing a measured value, x, then the measured value should be written in the following way, , 𝑥 ± ∆𝑥 where is the associated uncertainty. ∆𝑥 Any measuring device is limited by the process it was manufactured. No process of manufacturing is perfect. Due to this, when reporting measurements taken with instruments, the associated uncertainty is the smallest increment of that tool divided by two. Calculating the mean value of measurements makes sure that the results are consistent. In this lab, the mean value of the densities is calculated using the following formula, (µ) (1) µ = ??? ?? ???????? ?????? ?????? ?? ?????? When performing calculations with uncertain measurements, in this case addition/subtraction followed by multiplication/division the following two rules are followed, (2) ∆? = ∆? 2 + ∆? 2 (3) ∆? ? = ( ∆? ? ) 2 + ( ∆? ? ) 2 where , are the measurements and is the result of the operation. , , and are the ? ? ? ∆? ∆? ∆? associated uncertainties with their respective measurements. Density of Materials Density is a measure of how much mass is present in a given amount of volume. The SI unit of density is kg/m 3 (kilograms per cubic meter). The closer the particles in a substance are packed together, the higher its density. Materials with uniform properties exhibit clearly defined densities, making them distinguishable based on their density. The quality of uniform materials

Results and Calculations It is known that when dealing with water, and Volume, (2) . 1? = 1?? 3 ? = 𝐿 * ? * 𝐻 Table 1: Summary of dimensions and mass of block, including calculations for volumes, density, and average density. [See Appendix A for data.] Metre Stick Uncertainty Vernier Caliper Uncertainty Volume Displacement Uncertainty Volume (cm 3 ) 12 0.6 10.8 0.05 12 0.5 Density (g/cm 3 ) 9.5 0.4 11.0 0.05 9.9 0.4 Average Density (g/cm 3 ) 10. 1 Sample Calculations (Metre Stick): ● Volume ? = 𝐿 * ? * 𝐻 ? = 3. 1 * 3. 1 * 1. 3 ? = 12. 493 ?? 3 ● Uncertainty Using equation (3) ∆? = ( 0.05 3.1 ) 2 + ( 0.05 3.1 ) 2 + ( 0.05 1.3 ) 2 * 12. 493 ∆? = 0. 55864546 ?? 3 Therefore ? = 12 ± 0. 6 ?? 3 ● Density Using equation (4)

ρ = 118.5 ? 12.493 ?? 3 ρ = 9. 485311775 ?/?? 3 ● Uncertainty Using equation (3) ∆? = ( 0.05 118.5 ) 2 + ( 0.55864546 12.493 ) 2 * 9. 485311775 ∆? = 0. 4241705156 ?/?? 3 Therefore ρ = 9. 5 ± 0. 4 ?/?? 3 ● Average Density Using equation (1) µ = 9.5 + 9.9 + 11.0 3 µ = 10. 13 ● Uncertainty Using equation (2) ∆? = 0. 9 2 + 0. 05 2 + 0. 8 2 ∆? = 1. 205197079 Therefore the average density which is used for conclusions, ρ = 1. 0 ± 0. 1 * 10 1 ?/?? 3 By using the average of the density values obtained using multiple methods, accuracy and precision can both be addressed. Accuracy is improved because the potential biases and errors introduced by each measurement method is reduced. This makes the final result closer to the actual true value. Precision is improved because the consistency of results obtained from different methods indicates that the measurements are reproducible. It also checks for the uncertainties of each of the different measurement methods. (3)

because it assesses the accuracy of the measurement tools and methods. A high percent error indicates that there are significant errors in the experiment (5) . The average density found in this experiment is . ρ = 1. 0 ± 0. 1 * 10 1 ?/?? 3 This is closest to the accepted value for the density of silver which is [see ρ = 10. 49 ?/?? 3 Appendix B]. Due to time constraints, there was only one trial performed for each of the measurement methods. However, looking at the percent error for each method, it seems that the vernier caliper is the most accurate method of measuring the dimensions of the block which had a percent error of only 4.86% . To conclude, the unknown block was silver.

References: (1) Britannica, T. Editors of Encyclopaedia (2023, July 4). density. Encyclopedia Britannica. https://www.britannica.com/science/density (2) Aqua-Calc. (n.d.). Weight to Volume Calculator for Water. Aqua-Calc. https://www.aqua-calc.com/calculate/weight-to-volume/substance/water (3) Accuracy, Precision, and Significant Figures. LibreTexts. https://chem.libretexts.org/Courses/Ontario_Tech_University/OTU-_Chemistry_1010/01 %3A_Chemical_Tools-_Experimentation_and_Measurement/1.09%3A_Accuracy_Precis ion_and_Significant_Figures (4) Helmenstine, A. (2016, November 3). Error in Science. Science Notes. https://sciencenotes.org/error-in-science/ (5) Jim Frost. (n.d.). Percent Error. Statistics by Jim. https://statisticsbyjim.com/basics/percent-error/