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Lab 11 - Physical Pendulum Procedure In this experiment, we assembled the necessary equipment, including a Vernier Rotary Motion Sensor, a right-angle clamp, a vertical support rod, the LabQuest App, and other related materials. After weighing the aluminum rod and cylindrical weights and measuring the rotation radius (r), we connected the sensor to the interface and adjusted the resolution to X4 mode. In the initial phase, we initiated data collection and used a protractor to displace the rod through 5, 10, 15, and 20 degree angles, subsequently releasing it. We repeated this process for various amplitudes, recording data for each run to determine the oscillation period (T) based on 4 cycles for each graph. Utilizing these values of T, we computed the angular frequency for each oscillation and documented all collected data in a dedicated table for Part 1. Moving to Part 2, we adjusted the weight's position along the rod and measured the length between the pivot point and the weight's center. Employing a 15-degree amplitude, we conducted 8 runs, progressively shifting the weight up the rod by 2cm for each run. All the data from these runs was meticulously recorded in the corresponding data table for Part 2. In Part 3, we introduced an additional cylindrical weight to the rod, determining the distance between the pivot point and this weight's center of mass. Employing the same 15-degree amplitude, we carried out 3 runs, altering the positions of the two weights along the rod. The resulting data was recorded in the designated data table for Part 3.

Data Part 1: Mass of rod: 41.6g Mass of weight (with screw): 80.5g Radius: 34cm L = 38cm X = 3.3cm Run # Amplitude (degrees) Period of oscillation (T) Angular frequency (w) 1 5 degrees 0.866 7.255 2 10 degrees 0.862 7.289 3 15 degrees 0.862 7.289 4 20 degrees 0.864 7.273 1. The number of angular frequency (w) doesn't seem to change with the range of oscillations. The rotational frequency pretty much stayed the same as we raised the amplitude. Over all 4 runs, the average rotational frequency was found to be 7.277. 2. Write the formula for the force that pulls the weight back to where it was before it was released. Torque = Force * Distance from pivot * sin(theta), where theta is the angle between the force and the position vector. 3.Write the Newton's second law equation that shows how the weight will move around the pivot point after it is let go. As a function of time, write the angular acceleration as the second derivative of the angle theta. Sum of torques = moment of inertia (I) * angular acceleration (a) a = d^2(theta)/(dt^2) 4.After dividing your second equation by I, adjust the parts so that the equation equals 0. Congratulations, you have now made a second order differential equation that describes how the pendulum bob moves.

(Sum of torques / I) - (d^2 theta / (dt^2 I) = 0 Part 2: Amplitude: 15 degrees Mass of rod: 41.6g Mass of weight (with screw): 80.5g L = 38cm X = 3.3cm Radius (cm) h (cm) I total (kg*m^3) w theoretical (rad/s) w measured (rad/s) % diff w force (rad/s) % diff 2 34 cm 27.77 cm 1551.74 kg m^3 4.63 rad/s 7.27 rad/s 57.02% 5.08 rad/s 43.11% 32 cm 26.45 cm 1415.71 kg m^3 4.73 rad/s 5.57 rad/s 17.75% 5.08 rad/s 9.65% 30 cm 25.13 cm 1287.56 kg m^3 4.83 rad/s 5.82 rad/s 20.49% 5.08 rad/s 14.57% 28 cm 23.81 cm 1167.31 kg m^3 4.94 rad/s 5.98 rad/s 21.05% 5.08 rad/s 17.72% 26 cm 22.49 cm 1054.94 kg m^3 5.05 rad/s 6.07 rad/s 20.19% 5.08 rad/s 19.49% 24 cm 21.17 cm 950.47 kg m^3 5.16 rad/s 6.42 rad/s 24.42% 5.08 rad/s 26.38% 22 cm 19.85 cm 853.89 kg m^3 5.27 rad/s 6.61 rad/s 25.43% 5.08 rad/s 30.12% 20 cm 18.53 cm 765.19 kg m^3 5.38 rad/s 6.68 rad/s 24.16% 5.08 rad/s 31.50% Part 3: Mass of rod: 41.6g

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Mass of weight (with screw): 80.5g radius (2) = 33cm Amplitude = 15 degrees L = 38cm X = 3.3cm Run # T (period) Positions of 2 weights (cm) w (rad/s) 1 1.14 33 cm 5.51 rad/s 2 0.88 18 cm 7.14 rad/s 3 0.765 8 cm 8.21 rad/s Calculations Angular Frequency: w theoretical = sqrt(Mgh/I) w measured = 2pi/T w force = sqrt(g/L) → w force = sqrt(9.8/0.38) = 5.08 rad/s Moment of Inertia: I = I(point) + I(rod) I = mr^2 + 1/12m(rod) I = mr^2 + 1/2m(rod)L^2 + m(rod)h^2 Mass: M = m + m(rod) M = 122.1g Location of Center of Mass from Pivot: h = (m(rod) * (L/2 - x) + mr) / (m(rod) + m) h (part 2 run 1) = [41.6g (38cm/2 - 3.3cm) + 80.5g (34cm)] / (41.6g + 80.5g) = 27.77 cm Percent Differences: %diff = (absolute value (w theoretical - w measured)) / w theoretical * 100% %diff 2 = (absolute value (w measured - w force)) / w force * 100%

Sources of Error During the whole experiment, there were many things that could go wrong. It's possible that mistakes were made because of problems with the initial setting, such as small changes or misaligned weight places. If the gadget used to collect the data wasn't set up correctly, it could have changed the variation time data that was gathered. It was important that the results from the motion detectors were correct, so any changes in the angle vs. time data could have caused uncertainty. A lot of measurements, like the size of the equipment, were taken by hand, so even small mistakes in these measurements could have changed the results. Also, the large number of formulas needed for data analysis meant that mistakes could happen at different times, which could lead to mistakes in the end results. Conclusion In summary, our experiment taught us a lot about how a real pendulum moves and what changes its behavior when the mass of the system isn't seen as a point mass at the end of a massless string. We used the data we gathered from various runs that looked at changes in amplitude, radius, and mass to find the actual angular frequency (w) numbers for each run. In Part 2, we compared our actual angular frequency to theoretical values and found that they were pretty close, with a difference of only about 21% on average. As we already said, this difference could be due to mistake sources and the settings of the experiment. We also noticed that the first run in Part 2 (radius = 34 cm) didn't match up with the other results. Overall, our experiment went well, and the results were very close to what we thought they would be based on theory. This research helped us learn more about real pendulums that aren't perfect and showed how small mistakes in experiments can change the results.