PC141 Moment of Inertia Lab

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Conestoga College *

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Feb 20, 2024

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Determining the β for Moment of Inertia Ishika Sharma 190720720 PC141 L10 Lab Instructor: Hasan Shodiev Lab IA: Ariana Vesnina December 8 th , 2021

Purpose: The purpose of this experiment is to understand and use the experimental procedure in order to find the coefficient of the moment of inertia, which is shown as β. By determining this shape constant, we can find the actual yield of the product and use that to compare it to the theoretical yield given in this experiment. Another factor of this experiment is to realize the factors that contribute to the effect of the incline in order to make sure the object does not slip off the ramp. We use basic calculations to find the uncertainties and repeat instrument measurements. This experiment consists of 4 trials for each of the 5 different objects (spherical shell, solid sphere, cylindrical shell, solid cylinder, frictionless cube). Experimental Results: Part I: Trials (Table 1) Instrument name timer/iPhone stopwatch units seconds precision measure 0.01 zero error 0 Trials 1 2 3 4 Spherical Shell 4.33 4.27 4.35 4.3 Solid Sphere 3.93 3.98 4.01 3.99 Cylindrical Shell 4.69 4.8 4.84 4.75 Solid Cylinder 4.13 4.2 4.16 4.21 Frictionless Cube 3.39 3.33 3.29 3.28 Part II: Data Processing of Time (Table 2) Time Spherical Solid Cylindrical Solid Frictionless Shell Sphere Shell Cylinder Cube average 4.3125 3.9775 4.77 4.175 3.3255 sigma 0.035 0.034 0.0648 0.0369 0.0499 alpha 0.0175 0.017 0.0324 0.0184 0.0249 unc. in avg t 0.0175 0.017 0.0324 0.0184 0.0249 Part II: Data Processing for Beta (Table 3) Beta Spherical Solid Cylindrical Solid Frictionless Shell Sphere Shell Cylinder Cube average 0.48 0.26 0.81 0.39 -0.12 uncertainty 4.3 +- 0.41% 3.9 +- 0.45% 4.7 +- 0.64% 4.175 +- 0.44% 3.32 +- 0.75%

Table 4 – Values and % Difference Determined Accepted Difference Value Value in % Spherical Shell 0.48 0.67 33% Solid Sphere 0.26 0.4 42% Cylindrical Shell 0.81 1 21% Solid Cylinder 0.39 0.5 25% Frictionless Cube -0.12 0.16 1400% Data: The uncertainties in average times in table 2 are all the same as the alphas because the alpha is larger than the precision measure, so the uncertainty stays the same as well. The most average trial for the spherical shell would be trial 4, for solid sphere it is trial 2, for cylindrical shell it is trial 4, for the solid cylinder it is trial 3, and lastly for the frictionless cube it was trial 2. All the values can be easily located in the experimental results section of the lab report in table 1 and table 2. As you can see, trials 2 and 4 were the most accurate when comparing to the average of that specific object. These trials were most likely the most accurate because we as a group got more precise as the trials went on, the first trial may have been the least accurate due to it being the first round out of 4. By working together and each taking on the task to time one of the five objects, in order to make sure everyone participated it made it easier to get through the experiment and get proper results as questions could be asked throughout the process. When looking at all the trials as a whole, the objects and their specific times are in the same range which tells us that the experiment was done correct and with accuracy. Now when we talk about table 4, this is the most crucial data as the whole purpose of the experiment was to find the determined value and compare it to the accepted value to get the percent difference between them. The percent difference for the frictionless cube seems to be out of range from the other 4 objects, this is because the determined value was calculated to be -0.12, most values are not negative. The percent difference shows how accurate we were to the accepted value, meaning it tells us if the experiment was done incorrectly or correctly. We put the angle of inclination at 17 o , the height was 5.26m and the length from top to bottom was measured at 18m. Sample Calculations:

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Percent Difference equation: A − B ( A + B 2 ) x 100% Spherical Shell: 0.67 s − 0.48 s ( 0.67 s + 0.49 s 2 ) x 100 = 33% Solid Sphere: 0.4 s − 0.26 s ( 0.4 s + 0.26 s 2 ) x 100% = 42% Cylindrical Shell: 1 s − 0.81 s ( 1 s + 0.81 s 2 ) x 100% = 21% Solid Cylinder: 0.5 s − 0.39 s ( 0.5 s + 0.39 s 2 ) x 100% = 25% Frictionless Cube: 0.16 s −(− 0.12 s ) ( 0.16 s +(− 0.12 s ) 2 ) x 100% = 1400% Relative Uncertainty for Spherical Shell: = (absolute uncertainty/best estimate) x 100% = (0.0175s/4.3s) x 100% = 0.41% Solid Sphere = (0.017s/3.9s) x 100% = 0.45% Cylindrical Shell = (0.03s/4.7s) x 100% =0.64% Solid Cylinder = (0.0184s/4.175s) x 100% = 0.44% Frictionless Cube: = (0.0249s/3.32s) x 100% = 0.75%

Discussion: The biggest systematic error that could have altered with the results would be a human error that consists of the starting and stopping of the stopwatch, this error is most likely the reason as to some of the times are not 100% accurate. Since 4 experimenters took turns timing the 4 different objects, there was bound to be some slight changes in when the stopwatch was started and stopped. This could have been somewhat avoided if one person took charge of the stopwatch, as the timing would be accurate. Overall, we did not come into any difficulty when doing the problems or completing the experiment itself, this was highly because the experiment and instructions were read beforehand in order to build knowledge and understanding of the experiment itself. Regardless of this human error, one systematic error could have been the angle of the incline and this could’ve been a possibility of error if the labs were in person as the angle could easily be hindered if not checked after every trial, in the online virtual version this was not a big error that could occur as the angle of 17 o used was saved into the program and so it did not have to be changed at all throughout the experiment process. Now, in table 1 all the trials and times are recorded, we used those times to complete table 2, which had the averages of the objects as well as the sigma, alpha and uncertainty in times. Table 3 shows the average time and uncertainty in beta and lastly table 4 brings all those numbers together to find the determined values, accepted values and percent difference between those two to know if the experiment was done correctly. By calculating the average times with their uncertainties for the four objects, you would not be able to tell whether they will produce significantly different values for β before you calculate β because the low values and beta on its own are inversely proportional to speed, which means that when speed increases, beta decreases. If the angle of the ramp was increased this would definitely increase the height itself and would also increase the shape factor beta. Now to apply some knowledge and understanding, if there were a set of solid cylinders with each being a different length, but constant radii and density, we now know that the longer cylinder would roll faster because beta will decrease, this happens due to these two things being inversely proportional. If spherical shells with different radii but constant length and density were given, the large sphere would roll faster as a larger radius amounts to shorter length, which ultimately increases beta as the sphere with the larger surface area would roll faster. With all this information, the percent difference was calculated, and these calculations can all be seen in the sample calculations portion of this lab, each individual object had a different percent difference; spherical shell: 33%, solid sphere: 42%, cylindrical shell: 21%, solid cylinder: 25%, and frictionless cube: 1400%, these values can also be found in table 4 above. In table 3, the uncertainties for beta were also calculated which were shown as; spherical shell: 0.41%, solid sphere: 0.45%, cylindrical shell: 0.64%, solid cylinder: 0.44%, and frictionless cube: 0.75%. this experiment was done using group help and every individual played their part and participated. The procedure was done with as much accuracy as possible and all numbers were agreed upon by each individual included, to reduce any misinformation the numbers were written down in the excel tables immediately after having retrieved them, making the calculation process much more efficient and easier.

Conclusion: In conclusion, the experiment done was to find the moment of inertia of 4 different objects by conducting trial based timed experiments, to complete this experiment individuals were asked to access and online virtual website that all the work was done on. I believe this experiment was a success as the percent rates were all below 50% for all 4 of the objects used. The calculations were to show these percent differences that can be seen in the discussion portion of the lab; spherical shell: 33%, solid sphere: 42%, cylindrical shell: 21%, solid cylinder: 25%, and frictionless cube: 1400%. The uncertainties were also calculated during this lab; ; spherical shell: 0.41%, solid sphere: 0.45%, cylindrical shell: 0.64%, solid cylinder: 0.44%, and frictionless cube: 0.75%. When doing the experiment there could have been slightly hindered results as every group member had to time a different object, making room for more error and less accuracy. Overlooking that aspect the rest of the experiment went smoothly with no interruptions or errors. The data calculated is in correlation with the purpose of the experiment and the numbers are in the right ballpark range, making this experiment a success. References: PC141 Lab Manual 2021 by Terry Sturtevant, revisited by Hasan Shodiev

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