moments of inertia

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Apr 3, 2024

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Lio, Guoquin, Section: 13 Experiment 10: Moments of Inertia 9 November, 2023 Introduction: This experiment's main goals are to find the rotating system's moment of inertia and predict its new moment of inertia after adding point masses to the system. This experiment aims to explore and understand the basic concepts that underlie rotational motion, with a particular emphasis on torque, angular acceleration, and moment of inertia. The experiment involves the use of a rotating apparatus with an axle, onto which a string is wrapped. A hanging mass provides tension to the string, inducing a torque on the apparatus. By manipulating the system through the addition of masses at specific distances, observations and calculations are made to predict the new moment of inertia. Theoretical Analysis: See attached calculations on hard copy. Results: Calculated Data Part 1(a) Part 1(b) Part 2(a) Part 2(b) a 0.014 0.03 0.0016 0.0047 α 0.509 1.09 0.058 0.171 T 0.489 0.977 0.480 0.962 𝜏 0.013 0.027 0.013 0.026 Experimental I 0 : Theoretical I new : Experimental I new : Percent Difference I new :

0.0248 0.138 0.139 1% Discussion of Results: The lab was about figuring out how things spin and comparing what we found to what we expected. Even though the instructions didn't ask for it, I added a comparison. The expected spinning value was 0.138kg m2, and what we actually got was just 1% different from that. While there weren't many errors made in the lab, it was occasionally difficult to measure the beginning height precisely since it was difficult to get the ruler exactly straight and parallel to the table. Furthermore, given how fast we responded, the timing of when everything fell could have been a little bit wrong. In spite of these obstacles, the lab went well and covered the required computations. But why we never got the data needed to calculate the apparatus's moment of inertia in the absence of additional mass is puzzling. If I hadn't missed this section of the lab instructions, this data would have made it easier to compare our dynamic value to an expected one. Post Lab Questions: 1.) What are the units for Torque, Moment of Inertia, and Angular Acceleration? Show all work. Torque is defined as the product of force and distance, and its units can be derived from Newton's equation for torque, which is torque ( 𝜏)=force(F) × distance(r). The SI unit of force is Newton (N), and the SI unit of distance is meters (m). Therefore, the unit of torque is Newton-meter (Nm). The moment of inertia depends on the distribution of mass in a rotating body and is measured in kilogram-square meters (kg·m²) in the SI system. Angular acceleration represents the rate of change of angular velocity with respect to time. The SI unit of angular acceleration is radians per second squared (rad/s²). Mathematically, torque ( τ ) is given by: 𝜏 = I × α

2.) If the Torque applied to a rigid body is doubled, what happens to the Moment of Inertia? Since the mass and radius are used to compute inertia, and applying torque will not alter any of these parameters, the inertia will remain constant. Understanding that torque is equal to inertia times angular acceleration, with inertia being constant as previously said. Torque is directly proportional to the moment of inertia. Therefore, if the applied torque becomes double, the moment of inertia also becomes double. 3.) Why did you need to calculate acceleration to determine I 0 ? Could you have calculated a theoretical I 0 without running any trials? In the context of determining the moment of inertia (I 0 ), the calculation of acceleration isn't directly employed to find I 0 itself but is instead used to measure the tension in the string and subsequently calculate the torque applied to the rotating system. Now, theoretically determining I 0 without running any trials is possible, provided that every pertinent parameter—such as the spinning system's exact size and mass distribution—is well-defined and understood. Nevertheless, in practical applications, particularly with intricate or asymmetrical systems, theoretical computations may incorporate assumptions that diverge from factual experimental findings. 4.) Were any torques ignored in this experiment? What are they? Do you believe they may have significantly altered your results? The ignored torque was friction in this experiment. I do not believe that they were able to significantly alter the results because it is too small. Raw Data/Sample Calculations: See attached calculations on hard copy.

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