Lab 0

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University of Texas *

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Chemistry

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Jun 24, 2024

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Measuring Distance & Time Part 1: Look at the marble, is it a sphere? Method: The goal of this experiment is to determine if the marble is a true sphere. Although initial visual inspection suggests that the marble appears spherical, we aim to verify this through precise measurements. By firstly measuring the diameter of the white small marble with the tape measure, we found it to be 1.4 cm ± 0.1 cm at several different orientations. These measurements stayed consistent within the given uncertainty. Given this information, we could presume that the marble is a sphere. However, to determine the veracity of this claim, it was apparent that we needed to investigate further by taking measurements that would be far more precise than was previously recorded by the tape measure. We chose to use a caliper for its relative ease of use compared to the other provided tools, and, most importantly, its high precision in measuring the diameter of the marbles. We began this experiment by choosing three different marbles, all of which were initially visually inspected to be the same size. Then we placed each marble between the caliper's jaws to obtain immediate and accurate measurements, which was repeated three times for each marble to reduce any possible random error. Any uncertainty that arose in our measurements may be due to the limitations in precision of the caliper or the possible imprecision of the experimenter using said tool. Once all the measurements are recorded, we will take the average diameter of each marble and their respective standard deviations.We will consider the marble to be spherical if the measurements are consistent and fall within the specified uncertainties. Based on this method, we predict that the marbles are indeed true spheres. Data: We conducted measurements for three distinct orientations of each colored stone. Please refer to the table provided below. Table 1: Marble Diameters Measurement of Marble Diameters Small White Marble Small Black Marble Small Blue Marble Trial Diameter (mm) Uncertainty (mm) Diameter (mm) Uncertainty (mm) Diameter (mm) Uncertainty (mm) 1 14.80 +/- .02 14.32 +/-.02 14.24 +/-.02 2 14.00 +/-.02 14.26 +/-.02 14.38 +/-.02 3 14.00 +/-.02 14.24 +/-.02 14.38 +/-.02 Statistical Models for Analysis: Mean: 𝑥 = (1/𝑁) 𝑖 𝑁 ∑ 𝑥 𝑖

Standard Deviation: σ 𝑥 = (1/𝑁 − 1) 𝑖 𝑁 ∑(𝑥 𝑖 − 𝑥 ) 2 Standard Error: δ𝑥 = σ 𝑥 / 𝑁 Best Estimate given as: 𝑥 ± δ𝑥 **Final Best estimate will be reported with biggest uncertainty value (i.e., caliper uncertainty vs. calculated standard error)** Table 2: Marble Diameter Analysis Analysis of Marble Diameters Small White Marble Small Black Marble Small Blue Marble Average Diameter (mm) 14.27 14.27 14.33 Standard Deviation (mm) 0.46 0.04 0.08 Standard Error (mm) 0.27 0.02 0.05 Best Estimate (mm) 14.27 ± 0.27 14.27 ± 0.02 14.33 ± 0.05 The statistical models and parameters listed above were used to calculate these values. Conclusion: In conclusion, the experiment aimed to determine whether the small white, black, and blue marbles are true spheres by measuring their diameters with a caliper and assessing the consistency of these measurements. The small white marble exhibited an average diameter of 14.27 mm with a standard deviation of 0.46 mm, suggesting some variability. The small black marble, with an average diameter of 14.27 mm and a standard deviation of 0.04 mm, and the small blue marble, with an average diameter of 14.33 mm and a standard deviation of 0.08 mm, demonstrated higher consistency and precision in their measurements. The final best estimates for marble diameters were 14.27 ± 0.27 mm, 14.27 ± 0.02 mm, and 14.33 ± 0.05 mm for the white, black, and blue marbles respectively. In the case of the black marble, because its result fell within the specified uncertainty of ± 0.02 mm, it can be concluded that it is a sphere. Conversely, in the case of the white and blue marbles, because their results fell outside of the specified uncertainty of ± 0.02 mm, it can be concluded that they are not spheres. The quantitative results, particularly the low standard deviations for the black and blue marbles, confirm that the caliper is a suitable instrument for precise measurements, and the repeated trials effectively minimized random

errors. However, the significant variability observed in the white marble's measurements highlight the limitations of the experiment—possibly due to slight imperfections in the marble or measurement technique. For future iterations of the experiment, it would be beneficial to increase the number of trials to further reduce random errors and improve the reliability of the results. It may also be useful to perform measurements at different points on each marble to assess uniformity more thoroughly. The assessment of the model, which predicted that the marbles would be spherical based on initial measurements and visual inspection, aligns well with the experimental findings to a certain degree of course. Ultimately, the overall inconsistency and imprecision of the diameter measurements do not support the hypothesis that all the marbles are true spheres. Future experiments could explore more detailed questions posed in the lab manual, such as the impact of environmental factors on measurements or the comparison of marbles of different materials. Part 2: Does the color affect how the Rolling Times? Method: The aim of this experiment is to determine whether the color of a marble affects the time it takes to roll down a ramp. Since factors such as gravitational force, surface texture of the ramp, and the shape of the marble are not specifically influenced by its color, we predict that the rolling times will be the same for marbles of different colors. Given that none of these elements are affected by color, we anticipate that the rolling times will be the same for marbles of varying hues. We will use a stopwatch to measure the rolling time of the marbles as they travel down the ramp. The ramp's length and angle will be kept constant throughout the experiment to ensure consistency. The precision of the stopwatch is appropriate for measuring the short durations involved in this experiment, and it provides sufficient accuracy for our needs. We will conduct three measurements of the rolling time for each marble color and thereafter compute the average and standard deviation of the mean. This statistical methodology will assist us in assessing the reliability of our data and determining if the number of measurements is adequate to minimize random uncertainty below the threshold of systematic uncertainty. Data: Tram Setup Measurements: Length of tram: 96.5 cm +/- .1 cm Height of block: 9.0 cm +/- .1 cm Table 3: Rolling Times of Marbles Rolling Times of Marbles Trial 1 Trial 2 Trial 3 White small marble 2.02 s +/- .01 s 1.96 s +/- .01 s 1.80 s +/- .01 s Black small marble 1.80 s +/- .01 s 1.84 s +/- .01 s 1.78 s +/- .01 s Blue small marble 1.87 s +/- .01 s 1.89 s +/- .01 s 1.82 s +/- .01 s

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Statistical Models for Analysis: Mean: 𝑥 = (1/𝑁) 𝑖 𝑁 ∑ 𝑥 𝑖 Standard Deviation: σ 𝑥 = (1/𝑁 − 1) 𝑖 𝑁 ∑(𝑥 𝑖 − 𝑥 ) 2 Standard Error: δ𝑥 = σ 𝑥 / 𝑁 Best Estimate given as: 𝑥 ± δ𝑥 **Final Best estimate will be reported with biggest uncertainty value (i.e., caliper uncertainty vs. calculated standard error)** Table 4: Marble Rolling Time Analysis Analysis of Marble Rolling Times Small White Marble Small Black Marble Small Blue Marble Average Time (s) 1.90 1.90 1.80 Standard Deviation (s) 0.11 0.67 0.02 Standard Error (s) 0.06 0.39 0.01 Best Estimate (s) 1.90 ± 0.06 1.90 ± 0.39 1.80 ± 0.01 The statistical models and parameters listed above were used to calculate these values. Conclusion: The aim of this experiment was to determine whether the color of a marble affects the time it takes to roll down a ramp. After conducting three trials for each color marble (white, black, and blue), we calculated the average rolling times and evaluated the results. The analysis of the data indicates that the color of the marbles does not significantly affect their rolling times. This conclusion aligns with our prediction that the rolling times would be the same for marbles of different colors, given that factors such as gravitational force, surface texture, and shape were constant. The uncertainties in the measurements were small (± 0.01), indicating high precision in our timing method. However, the experiment's limitations include potential systematic errors such as slight variations in how the marbles were released or small imperfections in the ramp surface that were not accounted for. Despite these limitations, the consistency in the rolling times across the different colored marbles supports our conclusion. To further improve the reliability of our results and minimize random errors, future iterations of this experiment could include an

increased number of trials. Additionally, experimenting with different ramp materials and varying ramp angles could provide insights into other factors that might influence rolling times. Conducting the experiment in a controlled environment could also help minimize external factors such as air currents and temperature variations. Part 1 concluded that only the black marble could be considered a true sphere based on its precise measurements, whereas Part 2 demonstrated that color did not significantly influence rolling times, with all marbles exhibiting similar rolling durations within the range of measurement uncertainty. Both parts highlight the importance of consistent measurement techniques and acknowledge potential limitations due to systematic and random errors.