SUBRotational Inertia Lab Report_PHYSL'23

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University of Cincinnati, Main Campus *

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1051L

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Mechanical Engineering

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Jan 9, 2024

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1 Rotational Inertia Lab Report PHYS 1051L (011) Group 2 November 16, 2023

2 Part I – Experimental Design, Data, and Analysis A. Lab 9 – Rotational Inertia Part I Table 1 Experimental Design for IV, System Mass Research Question What factors impact the moment of inertia of a rotating system? Dependent Variable Moment of Inertia Independent Variable System Mass (kg) 0.1044, 0.2044, 0.3044, 0.4044, 0.5044 Control Variable Hanging Mass, Length, Torque Testable Hypothesis The mass does impact the moment of inertia. Prediction If mass increases, the moment of inertia will increase as well. Table 2 Raw Data for IV, System Mass mass (kg) angular acceleration ( rad/sec^2) error Inertia(kg*m^2 ) 0.1044 0.649 0.0022 0.01349769 0.2044 0.567 0.0022 0.01544974 0.3044 0.528 0.0028 0.01659091 0.4044 0.502 0.0023 0.0174502 0.5044 0.474 0.0019 0.01848101 Figure 1 Graph of System Mass (IV) on Inertia (DV)

3 Note. This graph shows that there is a positive relationship between the mass of the system (IV) and moment of inertia (DV). Claims: As that mass (IV) increased, moment of inertia (DV) also increased. This is dependent that the hanging mass, length, and torque staying constant. B. Lab 10 – Rotational Inertia Part II Table 3 Experimental Design for IV, Distance from Radius Research Question What factors impact the moment of inertia of a rotating system? Dependent Variable Moment of Inertia Independent Variable Distance from radius (0.03m, 0.06m. 0.09m, 0.12m, 0.15m) Control Variable Hanging Mass (0.07kg), System Mass (0.574kg) and Radius (15mm) Testable Hypothesis This distance does impact the moment of inertia. Prediction If this distance increases, the moment of inertia will increase as well. Table 4 Raw Data for IV, Distance from Radius Position (m) Angular Acceleration (rad/sec^2) Error Inertia 0.03 0.667 0.002 0.015427286 0.06 0.603 0.0019 0.017064677 0.09 0.534 0.0018 0.019269663 0.12 0.464 0.0017 0.022176724 0.15 0.388 0.0013 0.026520619

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4 Figure 2 Graph of Distance (IV) on Inertia (DV) Note. This graph shows that there is a positive relationship between the radius (IV) and moment of inertia (DV). Figure 3 Experimental Outcomes Organizer of Data Collection

5 Figure 4 Other Group Findings a.) b.) Note. Pictures of experimental work for (a) Group 5 and (b) Group 6

6 Part II – Discussion and Conclusion The research question being investigated here asks what factors impact the moment of inertia of a rotating system. Two independent variables (IVs) were tested to address this inquiry: first, mass of the system (kg), then the distance of that added mass on the rod (m). This lab was broken up into two weeks (labs) of investigation where each of the two independent variables were tested in a controlled environment (refer to experimental models in Part I) and analyzed for a relationship to the dependent variable (DV), moment of inertia (kg*m 2 ). The IV, system mass, was tested in the first week (Lab 9 – Rotational Inertia Part I), by increasing the weight added by 0.1kg for five runs (Table 1,2). From the resulting graph (Figure 1), it shows a positive linear correlation in that when the mass increased, as did the inertia. The line of best fit goes across the majority of the points and is reflected by the relatively high confidence level, R 2 =0.971. For further inquiry the next week (Lab 10 – Rotational Inertia Part II), the second IV tested was the distance away from the center of the system in which mass was added on the top of the rod. Still performing five trials and having the hanging mass constant, this time, the mass of the system is also fixed, while the distance from the radius is being evaluated (Table 3,4). Similarly to the previous outcome, the graph (Figure 2) showed there to be a positive relationship between the independent and dependent variable, but with a polynomial relationship instead of linear. Adding an extra measure of certainty with error bars and a higher calculated R 2 value (0.999) however, it is with more certainty that these two variables are positively correlated than the previous experiment. The results of contributed to one experimental model based on the summary of outcomes in the Experimental Organizer (Figure 3). There are some discrepancies between the models; most imperative would be type of equation, but that would be due to more complex factors from experimenting with the independent variable, distance from center of the system. With that being said, because this was a controlled experiment, the outcome on the dependent variable by the changed independent variable indicates causation. Both the mass of the system and distance from radius of system presumed causation due to the testing environment, however, any changes in control variables or under other conditions may have a different outcome. Trust in our findings and equation should always be taken with some precaution for the margin of human and systemic error. There was some uncertainty between labs regarding the value from the calculation of the controlled mass on the beam, in which the variance could have been a difference in the chosen weight from lab to lab or another error(s). Whether due to a mismeasurement or wrongly confusing a value for another, this human error could’ve contributed to insufficiency for data collected. In large however, no data points look to be skewed or fray away from the positive trends. In addition to visual evidence, the size of the error bars—although clear and reflecting noticeable error—are not relatively large enough to indicate something is off with the collected data. Trust in the data allows for initial trust in the equation Excel produced in representing an accurate relationship between the independent and dependent variable. There was some controversy among what the equation was supposed to look like and our group took some time until the Excel equation fit, but was confirmed with the TA and discussion among another group. Ultimately, the trendline passed through the data points and the

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7 confidence level was R 2 =0.999, exhibiting overall trust for the software’s determined equation and relationship between variables. Both groups held linear relationships for the mass of the system and polynomial relationships for the distance from the center of the system with high confidence values, like ours (Figure 4). Group 5 was whom we collaborated with and therefore have similar equation properties. Their first value is a bit smaller than ours, which I equate to our possible confusion with the decided mass number. Each group however, conservatively differ in values, but overall—although aligning closer with Group 5—both support our model despite the range in exact numerals. The established scientific model for rotational inertia of a system is derived from Newton’s Second Law for rotation for which it allows predictions: τ net = Iα Torque and angular acceleration are controlled, while we focus on the scientifically established model for rotational inertia of a system: I = Σmr 2 . When adding the mass of the system and radius with standard experiment, we calculated I = 0.0574 R 2 + 0 + 0.0127 Our experimental model for rotational inertia of a rod: I = 0.4853 R 2 + 0.0036 R + 0.015 The difference in the middle values is due to our experimental equation including a distance from the radius, while the scientific model was completely centered (no distance from radius = 0). When the independent variable regarding distance is ‘r’ and ‘m’ is the weight of the system, an equation in the polynomial forms. When the independent variable regarding mass is ‘m’ and ‘r’ is the true radius, a linear equation forms. In both instances our experimental models match the scientific ones. The lack of variation in torque or angle also confirms the relationship here according to physics’ laws of rotational inertia. Our controlled variables, such as hanging mass and torque, affect the acceleration of an object. Because inertia is the ability to remain at rest or in uniform motion, unless restricted in some instance, mass and its distribution on a system (distance or location) contribute to speed of rotation. The more mass or further it is placed from the radius of a system will slow the motion, resulting in a higher inertia. These claims are made with the assumption that friction is not large enough to be included in calculations. We were mindful of this by not putting the added mass too far from radial center in hopes to prevent friction from a possible tilt of the rod due to weight displacement. If significant enough, friction would cause a decrease in angular acceleration, increasing the inertia due to compounding factors and limiting the causality of our chosen independent variable. With all experiments—even those in which the system does the work like this one—come limitations that are critical to address. As always, there is only an allotted time in these labs, putting a restraint on the number of trials performed and thus the reputability in our claims. Specifically, a result of this was that only one side of the provide beam was used. This takes from the assumption that there would be no friction from weight displacement and the right side

8 is the same from the left side of the radius of the rod, ultimately limiting the generality of our experimental model. To improve this investigation and address some faulty measured discussed, I would suggest two actions. One, prioritize using both sides of the rod and/or use other available rods to account for random and repeated results, thus increasing the generality. Two is more specific to our group, but be more familiar with the topic to interpret the data and curated equation. Both of these items would lessen human and systemic error shown through the appearance of our error bars and provide more confidence and trust in our experimental outcomes.