Experiment eleven_ Torque

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Alexis Ortiz Experiment 11 A20532815 Experiment eleven: Torque 1. Introduction: The purpose of this experiment is to study torque, the rotational force, and its relationship with mass, moment of inertia, and angular acceleration. The moment of inertia and the angular acceleration were found to have an inverse relationship. The mathematical equation used to calculate the magnitude of the torque of the system is: (1) τ = 𝐹 × 𝑟 Where is the magnitude of torque acting on a system, F is the force acting on τ the system, and r is the radius of rotation. The mathematical equation used to calculate the net torque acting on the system is: (2) τ ??𝑡 = 𝐼α Where is the net torque acting on the system, I is the moment of inertia of the τ ??𝑡 system, and is the angular acceleration of the system. α The mathematical equation used to calculate the moment of inertia of the system is: (3) 𝐼 𝑡?𝑡𝑎? = 𝑖 ? ∑ ?𝑟 2 Where is the total moment of inertia of the system, n is the number of parts 𝐼 𝑡?𝑡𝑎? in the system, m is the mass of a part of the system, r is the distance to the rotational axis, and is the summation of each individual moment of inertia. ∑ The mathematical equation used to calculate an individual moment of inertia (parallel-axis theorem) is: +m (4) 𝐼 𝑇?𝑡𝑎? = 𝐼 𝐶 ?? ? ? 2 Where is the total moment of inertia of an object, is the moment of 𝐼 𝑇?𝑡𝑎? 𝐼 𝐶 ?? ? inertia of the center of mass of an object, m is the mass of the object, and d is the distance the object is from the center of mass. 2. Experimental Procedure:

Alexis Ortiz Experiment 11 A20532815 Materials and equipment needed for the experiment are: a rotating platform, an attachable rail, assorted masses, a pulley, a string, a balance, a ruler, and the PASCO computer software and force sensor. Part 1: 1. The mass of the rotating platform and large masses were weighed and recorded with the balance. The widths of the large masses, radii of the bobbins, and length of the rail were measured with a ruler. 2. The pulley system was set up with the string about one of the bobbins on the rotating platform. 3. The PASCO software was turned on with a quadratic fit and the force sensor was connected to the rotating platform. 4. A set mass was connected to the bottom loop of the string. 5. The mass was released once taut and the angular acceleration was recorded on a graph. 6. This process was repeated 2 more times. 7. Steps 2-6 were repeated for 2 different radii. Part 2(a): 8. With the pulley set up from part 1, the lowest bobbin position was selected for all trials. 9. When the string with the mass at the end was wrapped around the final bobbin and taut, the mass was released and its angular acceleration was recorded. 10. This was repeated 2 more times. 11. Steps 9 and 10 were repeated for two different total masses. Part 2(b): 12. With the pulley set up from part 1, the lowest bobbin position was again selected for all trials. 13. The large masses were fastened to the sides of the rail. This distance was measured as the radius of the rail. 14. When the string with the mass at the end was wrapped around the final bobbin and taut, the mass was released and its angular acceleration was recorded. 15. This was repeated 2 more times. 16. Steps 13-15 were repeated 2 more times for 2 more distances between the masses. 3. Results, Analysis, and Questions: (Part 1) Table 1: Table 1 shows radius of the bobbin and angular acceleration of the system. Radius (m) Angular Acceleration (rad/s^2)

Alexis Ortiz Experiment 11 A20532815 0.015 0.870 0.015 0.878 0.015 0.896 0.024 0.826 0.024 1.28 0.024 1.34 0.036 2.20 0.036 2.22 0.036 2.21 Based on Table 1, graph 1 shows the relationship between the radius of rotation and angular acceleration of the system. From graph 1, the slope of the line was found to be 64.631, which represents the force acting on the mass divided by the moment of inertia of the system (based on equations 1 and 2 where ). As opposed to the calculated value of the moment of inertia of α 𝑟 = 𝐹 𝐼 59.930, the 7.88% error suggests that there were some small errors in the procedural

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Alexis Ortiz Experiment 11 A20532815 calculations. The coefficient of determination being 0.8911 also suggests that the trials had some errors. Potential sources of error include the apparatus not being fully balanced, the string being twisted (creating motion in the x and z directions), and the quadratic fit of the highlighted data being less accurate than the linear fit. (Part 2(a)) Table 2: Table 2 shows the hanging mass on the pulley and the angular acceleration of the system. m (kg) Angular Acceleration (rad/s^2) 0.1498 2.22 0.1498 2.19 0.1498 2.19 0.1897 2.78 0.1897 2.81 0.1897 2.8 0.2297 3.43 0.2297 3.45 0.2297 3.42 Based on table 2, graph 2 shows the relationship between mass acting on the system and angular acceleration.

Alexis Ortiz Experiment 11 A20532815 From graph 2, the slope of the graph was calculated to be 15.436, which represents the gravitational constant times the radius of rotation divided by the moment of inertia of the system (based on equations 1 and 2 where ). Compared to the calculated value α ? = ?𝑟 𝐼 of the slope of 15.1, the 2.23% error suggests that the data was highly accurate and only contains some minor errors. The high coefficient of determination also supports this assessment. Some potential errors may be due the apparatus being improperly balanced or the string having motion in the x or z direction. (Part 2(b)) Table 3: Table 3 shows mass acting on the system, the radius of rotation, the angular acceleration of the system, and the inverse value of the moment of inertia, Mass (kg) Angular Acceleration (rad/s^2) Radius (m) 1/ Moment of Inertia (kg*m^2)^-1 0.08 2.62 0.036 9645.061728 0.08 2.61 0.036 9645.061728 0.08 2.6 0.036 9645.061728 0.16 1.55 0.036 4822.530864

Alexis Ortiz Experiment 11 A20532815 0.16 1.55 0.036 4822.530864 0.16 1.55 0.036 4822.530864 0.2 1.18 0.036 3858.024691 0.2 1.18 0.036 3858.024691 0.2 1.19 0.036 3858.024691 Based on table 3, graph 3 shows the relationship between the inverse value of the moment of inertia of the system and the angular acceleration. Based on graph 3, the relationship between the angular acceleration and moment of inertia is proven to be a proportionally inverse relationship. This is seen as the angular acceleration and inverse of the moment of inertia increase at a given rate. The high coefficient of determination suggests that the data points were fairly reliable. A small source of error may be due to friction between the string and the pulley. 4. Conclusion: In this experiment, torque and its reliant variables were studied. The inverse relationship between the moment of inertia and angular acceleration was also studied and positively corroborated with the data. The moment of inertia of the system was found to be 64.631. To improve accuracy, all angular acceleration should be calculated with a

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Alexis Ortiz Experiment 11 A20532815 linear fit in the software and the string should be kept straighter. Friction did not have a major effect on the data, but also may account for some of the resultant error. 5. Data Sheet: see below

Alexis Ortiz Experiment 11 A20532815

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