Moment of Inertia

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Jan 9, 2024

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Title: Moment of Inertia Group 33: Van Tran, Elyse Gallagher, Shrenik Patel Experiment Date: October 24th, 2023 Goals: In today’s lab, we will observe how geometric distribution of the mass within an objects affects the value of the moment of inertia, and we will use Newton’s second law to understand how = = = = = = = = = = = = = = = rotation is affected by the moment of inertia. Procedure: Part I: ● One group member hold Barbell 1 in one hand, and another member hold barbell 2 in one hand ● Both rotate the Barbells back and forth ● Swap out Barbells and repeat the same procedure ● Record observations about the difficulty of barbell 2 Part II: ● Plug rotary motion sensor into the interface and hang a 5g mass hanger to the string ● Turn on Pasco interface, open Capstone, and go to the Hardware Setup menu ● Activate the input for the rotary motion sensor, select medium groove under properties ● Create a graph for angular velocity vs. time ● Spin disk to wind up the string so that the mass hanger is elevated ● Hit record and release the mass hanger ● Repeat data collection and move all four ball bearings to the inner radius position Precautions and Sources of Error : Sources of error in this experiment include analyzing capstone as we have to use our best judgment for the graphs. Another possible error could be that there was too much pressure when releasing the string, however, we tried to be as delicate as possible when releasing it. Data:

Outer Radius Inner Radius Table of Calculations

Inner Radius I 1 =I disk +I 4b 6.12*10 -5 + 4(28)(10) - 3 (2*10 -2 ) 2 1.06*10 -4 kg m 2 6.615*10 -4 /1.06*10 -4 6.241 rad/s 2 y=mx+b y= -5.31x-2.07 ((5.31-6.241)/(6.241))*100 14.9% Outer Radius I 2 =I disk +I 4b 6.12*10 -5 + (4) (28*10 -3 )(4*10 -2 ) 2 2.4*10 -4 kg m 2 6.615*10 -4 /2.4*10 -4 2.756 rad/s 2 y=mx+b y=-2.50x+0.187 ((2.50-2.756)/(2.756))*100 9.29% Questions:

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Question 1. Both barbells have the same total mass, so what is it about Barbell 2 that makes it difficult to move quickly? Barbell 2 because of the distribution of weight being on opposite ends which requires more moment of inertia to move it. Question 2. Recall that for translational motion it takes more force to accelerate a larger mass, which is explained by Newton’s 2nd Law: Net = . This law also holds in rotation form: Net= . In the rotational version of the law, force is replaced by torque , and acceleration is replaced by angular acceleration ; what takes the place of mass? Rotational inertia takes place of mass. Question 3. Another group has hypothesized that adding 100 g extra mass to the center of the barbell where the hand is placed would have no effect on the ability of the person to rotate the barbell back and forth. Do you agree or disagree? Why? Our group agrees because if we add 100g extra mass to the center of rod the moment of inertia of the additional masses would be equal to 0 because r=0 (distance from center of mass) and moment of inertia=Fr. So r=0, moment of inertia would also be 0. Question 4 . If you were to double the mass, but halve the radius of one of the barbells, how would the moment of inertia compare to the original value? The moment of inertia would remain the same because F=ma and I=Fr Question 5. If you were a tightrope walker, which barbell would you rather be carrying? If you were a tightrope walker, you would want to use the barbell with the weights on either end of it because it will be more difficult to move than the barbell with weights towards the center. Question 6. How did the angular acceleration change with the new moment of inertia? Was your prediction correct? Angular acceleration and moment of inertia are inversely proportional, so the prediction was right. The smaller the moment of inertia the larger the angular acceleration. Discussion:

In the first part of our experiment, we examined the sensation of the weights on Barbell 1 and Barbell 2. We reached the conclusion that Barbell 2 presented greater resistance to movement, indicating a higher moment of inertia and, consequently, a greater torque. Conversely, Barbell 1 exhibited easier mobility, signifying a smaller moment of inertia resulting from the proximity of its masses. These findings were substantiated by the calculations presented in the data section above. The expected value for angular acceleration was 6.241 rad/s 2 for inner and our group got an actual acceleration of 5.31. The expected value for angular acceleration was 2.756 rad/s 2 for outer and our group got an actual acceleration of 2.50. Our sources of error in this experiment include analyzing capstone as we have to use our best judgment for the graphs. Our second possible error could be that there was too much pressure when releasing the string, however, we tried to be as delicate as possible when releasing it.

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