11.) Lab 127_ Torque & Rotational Inertia

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New Jersey Institute Of Technology *

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111A

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Mechanical Engineering

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Feb 20, 2024

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Lab 127: Torque & Rotational Inertia A. Introduction: Theory & Objectives Similar to how angular acceleration is accretion in a circular motion, torque is the force applied to make an object move in a circular motion. As the relation indicates, the force times the radius equals torque. However, like force, torque does not equal ma, it instead equals the angular accretion times a different time of inertia, rotational inertia. This hybrid equation is given below: Στ = 𝐼α Like normal inertia is in a sense the resistance that is felt when accelerating an object, rotational inertia is the resistance felt when angularly accelerating an object. Differently again, the rotational inertia is not just the mass but a compilation of other factors present when in a circular motion. The equations for rotational inertia are below: 𝐼 = Σ𝑚 𝑖 𝑟 𝑖 2 With these two given distinctions from normal linear motion, all the other laws of physics can be applied. To understand the further workings of these elements of circular motion, in this lab, we will explore how the force of tension and force of gravity impact the movement of disks and rings. Compostitly, we will also observe and estimate how that motion changes when the same forces are applied at different locations and at different angles. B. Experimental Procedure 1. With the given values, theoretically calculate the rotational inertia of a disk (center and diameter) and a ring. 2. Measure the diameter of the pulley for further calculations, and place the disk in the shaft, stimulating Figure 4. 3. Place the 50g hangers on the disks, and 200g weight on the pulley. 4. Click record and release the thread, observe the speed vs. time graph, and derive for linear accretion. Repeat two more times and average for a more accurate result. 5. Set up the pulley and disk according to Figure 5 and place the ring into the groove on the disk. 6. Then, once again let go of the strong while recording, and then find the acceleration by deriving the speed vs. time graph. 7. When doing the rotational inertia for the ring alone, the rotational inertia of the disk must be subtracted. 8. Now, one last time takes everything apart and simulates the diagram in Figure 6. Next, let the strong loose and record.

9. Find the accretion with the slope of the speed vs. time graph, and then the rotational inertia by subtracting away some of the previously found properties. C. Results: Data & Calculations Qualities: Disk Ring Mass 1.4035 1.4264 Radius .115 .0525 .0625 Rotational Inertia Center of Mass: .00927 .00475 Diameter: .00464 Case Run Linear (a) Tension Torque Angular ( α ) Total ( I ) Disk 1 0.0414 2.44 0.0305 3.31 .009208 2 0.0415 2.44 0.0305 3.32 .009185 3 0.0414 2.44 0.0305 3.31 .009208 Step-Pulle y & Shaft 1 7.37 0.61 0.00759 589.6 .000013 2 7.39 0.60 0.00753 591.2 .000013 3 7.35 0.61 0.00766 588 .000013 Disk & Ring 1 0.0278 2.44 0.0305 2.22 .013731 2 0.027 2.44 0.0305 2.16 .014139 3 0.0271 2.44 0.0305 2.17 .014087 Disk 2 1 0.0755 2.43 0.0304 6.04 .005031 2 0.0755 2.431125 0.0304 6.04 .005031 3 0.0767 2.430825 0.0304 6.14 .004952 Case Theoretical I Experimental I % Error Disk 1 .00927 .0092003 .752 % Disk 2 .00464 .00500 7.2 %

Disk + Ring .00475 .004716 .7209 % In this lab, we are theoretically and experimentally finding the rotational inertia of various objects. We started off by measuring the rotational inertia of just the disk by measuring its linear speed. By obtaining the linear speed we were able to find the linear acceleration via calculus. Then, with the given accelerations we were able to find the magnitude of the other forces involved in the experiment with Newton’s Laws. Furthermore, with the linear acceleration, we were also able to find the angular acceleration due to their radial relationship. Lastly, with the secondary results of the angular acceleration, we were able to find the Rotational Inertia. We repeated this process of calculations with various setups to see and understand how different weights and weight distribution dynamics change the accelerations and inertia To add, we were in a way able to estimate the rotational inertia of certain components in a system where we knew the total inertia and other components’ inertia as well. This additive property of inertia has allowed us to find the inertia of almost every single in the setup. D. Discussion: Error Analysis/Questions 1.) How good is your experiment result compared to the theoretical one? What could be the possible error of sources? On the large scale of things, both our experimental and theoretical values for all three cases were very similar. In fact, our percent error for the Disk through the center of mass and Ring is about .7%. Our error was so small that it could possibly be assumed as negligible. However, for the disk through the diameter we had an error of 7.2%. This is about 10 times the error we had for the other two. I think this error was actually caused by us experimenters. I think we may have set the experiment somehow wrong, or weren't entirely sure about the placement and angles of all the components of the setup. If given another chance, perhaps we would be more vigilant about the setup for this case and make sure that it is more accurate than the diagram given in the lab manual. 2.) Error Analysis in the previous question. E. Conclusion of Experiment In this lab, I think I was able to use so many different concepts at the same time. When actually doing the lab, I was kind of confused about the whole setup and the overall purpose of the lab. But after completing it, it makes a lot of sense. I personally enjoyed doing all the calculations because I was able to use old concepts like F = ma, and also use recent topics like torque and angular acceleration. Combining all these ideologies and then further calculating the rotational inertia was very intriguing to me. It made me actually think through all the concepts, and understand the indirect relationships. I was able to solidify my knowledge of angular motion a lot more in this lab.

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