Lab 7_ Moment of inertia (1)

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Apr 3, 2024

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Lab 7 Moment of Inertia Group Members: Hanh Fofana Marian Pena Goals: The purpose of this lab was to become familiar with how the moment of inertia affects the ability to change an object's rotation direction. Another reason is to understand how the geometric distribution of the mass within an object affects the value of the moment of inertia. Lastly, to know the parts of the rotation form of Newton’s 2nd Law: F ɴ E ᴛ = ma and understand how rotation is affected by the moment of inertia. Apparatus: two “barbells” with different mass distribution, a rotary motion sensor with attached ball-bearing disk and pulley, a 5 g mass hanger with masses, a computer with Pasco interface, and Capstone software. Procedure: Part I. Feel by hand the inertia of a rotating barbell. - Barbell 1 has masses attached near the center of the rod, and Barbell 2 has masses attached near the ends of the rod. - Two people hold the barbell side by side and record the observations on how difficult it was for the person with Barbell 2 to mimic the motion of the person with Barbell 1. - Use a meter stick to measure and record r, the distance of the mass from the axis of rotation. Do this for each barbel. Part II. Newton’s 2nd law for rotation. - A rotational motion sensor should be set up. - Hang a 5 g mass hanger to the string to provide torque and rotate the system. - Create a graph display for angular velocity vs. Time. - Place all four of the ball bearings in their outer radius position. Hit Record in Capstone and release the mass hanger. - Repeat data collection after changing the moment of inertia by moving all four ball bearings to the inner radius. - Make an Excel spreadsheet for your data. Errors and Precautions: - String might not be aligned with the pulley. -Angular velocity might need to be recorded correctly. - One source of error could be that the capstone application must be set up correctly. - Error could be in the calculations. -Make sure the capstone application is set up correctly.

-Record data with the ball bearings in their inner and outer radius position. -Make sure when observing 2 barbells' motion there is plenty of room for their rotation. -Make sure the graph titles are correct. Results: * Line 1 is the first trial where the ball bearings are in the outer position. * Line 2 is in the second trial where the ball bearings are in the inner position. 1 2

Questions: Question 1. Both barbells have the same total mass, so what is it about Barbell 2 that makes it difficult to move quickly? a. Both barbells have the same mass, but barbell 2 is challenging to move because of its mass distribution. Moment of inertia = Fr, meaning it depends on the radius for the center of rotation. Because barbell 1 has less r (distance from the center of mass), it needs less moment of inertia than barbell 2, where the r is larger and requires a greater moment of inertia. Question 2. Recall that for translational motion; it takes more force to accelerate a larger mass, which is explained by Newton’s 2nd Law: F ɴ E ᴛ = ma.T This law also holds in rotation form: 𝜏 Net = Ia . In the rotational version of the law, force is replaced by torque , 𝜏 and the acceleration is replaced by angular acceleration a; what takes the places of mass? a. Moment of inertia replaces 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 not affect the ability of the person to rotate the barbell back and forth. Do you agree or disagree? Why? a. We as a group disagree with this hypothesis because the moment inertia is not only R (distance from the center of mass) but also how much mass is there I= mr ² and so increasing mass will increase inertia, making a more significant torque that, in return, will affect the ability of the person to rotate the barbell back and forth making it more difficult. 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? a. The moment of inertia will not be the same as the original value because the equation is I= mr ² , and so increasing the mass and decreasing the radius will result in a minor moment of inertia as the r is squared. Question 5. If you were a tightrope walker, which barbell would you instead be carrying? a. Barbell 2 would be the preferred barbell because the weights are at the ends. Barbell 1 would be harder to manipulate compared to one with weights farther from the center. Additionally, barbell 2 has a higher moment of inertia. Question 6. How did the angular acceleration change with the new moment of inertia? Was your prediction correct? a. The smaller the moment of inertia, the larger the angular acceleration. So, the prediction we made was correct.

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Discussion: As expected, Barbell 1 was easier to rotate than Barbell 2 because of the mass distribution. Because the mass in barbell one was near the center of rotation, there was less moment of inertia, making it require less force to rotate. In barbell 2, because the mass was away from the center of rotation, it needed more force to turn. A higher torque value makes it more difficult to rotate the object. This observation successfully achieved the objective for part one of this experiment. The purpose of the second part of this experiment was to find out how inertia can impact angular acceleration. For this part, everything was the same; the mass was the same force acting on it, except the radius of the ball bearing changed. Our prediction for this part was that the smaller the moment of inertia, the larger the angular acceleration. As we can see, when the ball bearing had less radius, the angular acceleration was larger with a value of 6.32 rad/sec ². When the ball bearing had a larger radius, the angular acceleration was smaller with a value of 2.77 rad/sec ² . The percent difference for both cases is 57.6%, and 44.3% showed that our data and calculations were pretty accurate. The objective for this lab was achieved successfully.

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