Exp 1 - Lab report

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Feb 20, 2024

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Physics 1AA3 Lab Report Template Experiment 1: The Vitruvian Man Name: Taylor Gronkowski Student number: 400236449 Partner names & student no.: Please fill out your name and student number and then your partner’s name & student numbers above. Use this template to record your answers to the questions for the manual and complete the lab exercise. Please do not add any extra sections/information beyond what is prompted below. Be sure to hand this report in to your TA before you leave.

Question 1 (abbreviated) (1.25 marks) What is the hypothesis (prediction) you will test with your experiment, regarding the relation between foot length and height? Make sure to include a mathematical expression for the relation and explain what any variables represent. Also, be sure to reference where this mathematical expression comes from. In this experiment, it is hypothesized that there is a correlation between foot length and height, where an individual’s height is six times their foot length. This is represented as H = 6 L, where H represents the height of the individual in meters and L represents foot length of the individual in meters. The basis of this expression is derived from Vitruvius Pollo’s rules concerning the proportions of the human body, outlined in his book Ten Books on Architecture , where he states that “the length of the foot is one sixth of the height of the body.”

Question 2 (0.5 marks) What were some ways you attempted to practically minimize uncertainty in your length measurements? One way uncertainty in length measurements was practically minimized was measuring each group member’s height and foot length three independent times, ensuring to reset the setup each time and that a different person read the measurement each time to add reliability to the data. Another way uncertainty in length measurements was minimized was by considering how best to align the ruler with the foot and the top of someone’s head. For example, when taking height measurements, I ensured to always measure from the side of the body, starting at the heel, to the top of the head identified by placing a book on the individuals head to align their height with the ruler. It was also important to acknowledge the impact of foot curvature on measurements, and foot measurements were all taken standing with the foot flat against the ground.

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Question 3 (0.5 marks) What were three sources of variability in your measurement technique? (note: other than distractions) Human subjectivity/variability: different individuals performing measurements may have slight variation in their interpretation of anatomical placement of the measuring tool, such as ruler alignment or the identification of the top of an individual’s head. This also influenced how the measurements were read on the ruler. Instrument Precision: the measurement tools are limited in their ability to provide precise measurements as the smallest incremental measurement was millimeters. This can cause variability in the identification of the smallest incremental value. Anatomical variability: due to differences in curvatures of the body/foot or posture, variations in the ability to measure lengths and heights precisely arise as the measurement tools cannot fit to every curvature accurately, and differences in posture would also alter the accuracy of these measurements for height.

Sample calculation for the average foot length and uncertainty on average, with units and using proper rounding rules for both numbers (round at the end only) (1.0 marks) Foot Length: L 1 = 0.228 m, L 2 = 0.224 m, L 3 = 0.226 m Average Length (L) = (0.228 + 0.224 + 0.226) / 3 = 0.226 m SD = √(1/N-1) Σ (x 1 – x av ) 2 = √( (1/3-1)(0.228 – 0.226) 2 + (0.224 – 0.226) 2 + (0.226 – 0.226) 2 ) = 2.0 x 10 -3 m Final answer: L = 0.226±0.002 m (note: Standard error on the mean (absolute uncertainty on average):) SE = SD/√N = (2.0x10 -3 ) / √3 = 0.00115 ≈ 0.002 m

Question 4 (abbreviated) (0.5 marks) How do the following values compare: the uncertainties on each individual measurement, the uncertainty on the average length of the foot, half of the smallest increment on your ruler? List them, and write a brief comment summarizing whether they differ, or if they are the same. The uncertainties on each individual measurement: the variability in measurements repeated three times for everyone’s foot and height contribute to the uncertainty and between individuals there was some variability due to the inaccuracy of the measurement tool and the person’s ability to read the value. For a single individual, the uncertainties were mainly the same as the differences between the measurements were all very small. The uncertainty on the average length of the foot: the variability in this measurement depends on the variability of the three repeated measurements taken to produce an average. The uncertainty between individuals average foot length varies because of differences in curvature and size. Half of the smallest increment on your ruler: The ruler used was the same across all measurements and individuals and half of the smallest increment on the ruler is equal to 0.5mm. This was the same across all measurements but limits the precision of measurements.

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Sample calculation for the uncertainty for the slope and intercept, with units, if applicable. (0.5 marks) Slope:  Best fit: 5.18  Max: 5.78  Min: 4.88  |best – max slope| = |5.18 – 5.78| = 0.6  |best – min slope| = | 5.18 – 4.88| = 0.3 Y-intercept:  Best fit: 0.459 m  Max: 0.338 m  Min: 0.544 m  |best – max y-int.| = |0.459 – 0.338| = 0.121 m  |best – min y-int.| = |0.459 – 0.544| = 0.085 m H = 5.2(±0.6)L + 0.5(±0.2) m Report/write down the full equation of the linear fit, with units and uncertainties below (0.25 marks) H = 5.2(±0.6)L + 0.5(±0.2) m

Question 5 (abbreviated) (1.5 marks) Do the results from your graph and Table 2 support or refute your hypothesis, and how do you know? If necessary, give plausible causes for disagreement between your result and the Vitruvian model. You can think about the idea of this model and if it leaves room for the diversity of proportions of the human body or not. Additionally, comment briefly on the value of your uncertainties and what this means for how one should interpret your results. The results from the graph and Table 2 do not support my hypothesis that H = 6L. The results of the experiment, as denoted from the data and graph, was: H = 5.2(±0.6)L + 0.5(±0.2) m. The experimental slope goes from 4.6 to 5.8, however the hypothesis slope is 6 and so therefore they do not agree. The experimental y-intercept goes from 0.3m to 0.7m, whereas the hypothesis y- intercept is 0m, so therefore they also do not agree. Since both experimental values, with the consideration of their uncertainties, do not agree we can reject the hypothesis that height is equal to six times foot length (H = 6L). Additionally, the Vitruvian model that derived this hypothesis might not account for the natural variation in human measurements, however, the data with uncertainties considered did not include the expected values regardless. This may be due to the small sample size.

Question 6 (0.25 marks) How might the sample size affect your measurement of the relation? Be specific. Our sample size was 4, with every measurement repeated 3 times. Only having 4 data points greatly reduces the reliability of our data, and with more data we would likely have a more reliable standard deviation which would also contribute to a normal distribution. The sample size significantly affects the measurement of the relationship between variables in these specific ways: 1. Larger sample sizes allow for more precise measurements for the relation. It provides a better representation of the population as a whole and reduces random variability in individual measurements impacting the data. This often also leads to narrower confidence intervals regarding slope and y-intercept. 2. Smaller sample sizes reduce the statistical power of the data analysis. Thus, it reduces the ability to identify significant relationships between the variables measured. 3. Larger sample sizes increase stability in the results. Small sample sizes are more susceptible to the influence of outliers, and each individual data point impacts the data more drastically.

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Question 7 (1.25 marks) What is the height and its associated uncertainty of the mystery person that you predict from the mystery footprint using the relation from you got from your linear fit equation? Refer to Section 3.3 in the reference material manual for the rules of propagating uncertainties. Show your work. H = 5.2(±0.6)L + 0.5(±0.2) m Average Foot Length: 0.209 m SD (uncertainty): 0.002 m H = 5.2(±0.6)(0.209)(±0.002) + 0.5(±0.2) m Propagating uncertainties: Multiplication Relative uncertainty = √((0.6/5.2)2 + (0.002/0.209)2) = 0.116 Absolute uncertainty = (5.2*0.209) * 0.116 = 0.126 ≈ 0.2 Propagating uncertainties: Addition Uncertainty = √((0.126) 2 + (0.2) 2 ) = 0.175 ≈ 0.2 Therefore, the height of the mystery person, including uncertainty, is 1.6 ± 0.2 m. H = 1.6(±0.2) m

Digital Submission (2.5 marks): Figure 1 (1.0 marks), Figure 2 & 3 (1.0 marks), Capstone workbook completion (0.5 marks)

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