Physics Lab 9

Physics Lab 8 Names: Shri Patil, Sean Kim, Ethan Pereira Introduction: In this lab, we will be determining the moment of inertia for the IOLab device and two tape rolls as it rolls down the ramp. The determined value from this lab will be compared to the one found in Lab 8 to see how close the two are. The comparison of the two values will be done with the use of a t-score. In order to compare the two moments of inertia accurately, the IOLab and tape rolls will both be modeled as a simple cylinder, as shown in the diagram below. However, this model will most likely overestimate the moment of inertia for the system, as in reality it doesn’t form a perfect cylinder. To determine the moment of inertia for this lab, conservation of energy will be used. The IOLab will be rolled down a ramp, starting with a set amount of potential energy. By the time it rolls down to the bottom of the ramp, it will only have kinetic energy. Because the IOLab is rolling as it moves down the ramp, it will have both translational and rotational kinetic energy. This means its kinetic energy can be written as ½ * m * v 2 + ½ * I * ω 2 . In this setup, we are assuming that there is friction present when the IOLb rolls down the ramp, so it will be rolling without slipping. The potential energy can be written as m * g * h. Setting these two energies equal to each other will allow us to solve for I and give us an approximate value. We plan to avoid any sources of error by letting the device roll down the ramp on its own and only using recorded values from when the device rolls down the ramp and not when it bumps into the end of the table. Additionally, we will extend the height of the ramp to a reasonable level so that the device won’t immediately roll off the table. Figure 1 Methods: 1. Attach two masking tape rolls to the ends of the IOLab, so that the IO Lab can roll (see figure 1)

2. Setup cardboard ramp to be propped up on one end by an IOLab box 11.5 cm from the end of the cardboard ramp (see figures 2 and 3) Figure 2 3. Place IOLab with its tape rolls attached at the top so that the front wheel is 17 cm away from the higher edge of the cardboard ramp (see figure 1) 4. Measure H, the height from the ground to the bottom of the wheel of the IOLab when it is sitting on top of the cardboard ramp 5. Release the IOLab from rest, and let it roll down the ramp 6. In IOLab desktop app, record ω and t for the IOLab to roll down to the bottom of the ramp Figure 3 7. Use v = r*ω to find transational v 8. Using magnitude of |ΔKE| = |ΔU|, calculate value of I using equation |ΔU| = ½ m*v 2 + ½ I*ω 2 where |ΔU| is given by mgH 9. Using t’= |μ A - μ B | / sqrt ( ( A 2 - B 2 ), calculate t’ using I value calculated from this experiment as A, and I value calculated in Lab 8 as B, as well as the respective uncertainty values 10. If t’ is below 3, then both methods are within reasonable range of each other. Calibration

𝜔 1 = 20.243 rad/s 𝜔2 = 20.292 μ = (20.292 + 20.243) / 2 = 20.2675 BB + = 0 ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆ ∆ = 0 − G∆ ∆ ℎ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ = 1/ ∆2 ∆2 ∆2 ∆2 ∆2 ∆2 ∆2 ∆2 ∆2 ∆2 ∆2 ∆2 ∆2 ∆2 ∆ 2 2 + 1/ 2 2 2 ∆ℎ = 6.2 ± 0.05 cm = .062 ± .0005 m μ = 20.2675 BB x = 43.7 ± 0.05 cm = .437 ± .0005 m M = 222 ± 0.5 grams = 0.222 ± 0.0005 kg G = 9.81 m/s^2 V = r r r = 4.075 ± 0.025 cm = .04075 ± .00025 m V = .04075 ± .00025 m * 20.243 V = .8249 m/s ± .00506075 - = ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆∆ ∆ ∆ 0 − G∆ = 1/ ℎ2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 + 1/ 2 2 2 (0.222 *9.81 * .062) = (0.5 * .222 * .8249 2 ) + (0.5 * Iy * 20.2675 2 ) 0.1350248 = (0.07553 + 10.134 Iy) Iy = 5.871 * 10 -3 kg*m 2 σ = (2 0.292 - 20.243)/2 = 0.0245 δI By = .0245 / √ (2) = .01732

δI Ay = 1.2045 * 10 -6 T’ = |μA - μB | / √ (δA 2 - δB 2 ) = |1.8 * 10 -4 - 5.871 * 10 -3 | / √(1.2045 * 10 -6 - (.01732) 2 ) T’ = .3292 Discussion: In this lab, we modeled the IOLab device and tape rolls as a perfect cylinder. However, this is not the case, since the mass isn’t uniformly distributed throughout the IOLab device and the tape rolls. The model as a whole is better represented as a cylinder with two hoops at either end. Because of our misrepresentation, it is likely that the moments of inertia calculated from this lab and lab 8 are much greater than the actual moment of inertia for the tape rolls and IOLab device. Additionally, the actual energy transferred in the experiment would be slightly lower than what we have calculated since there is friction present between the tape rolls and the ramp as the IOLab device rolls down the ramp. Conclusion: Based on the results of the experiment, we can conclude that the two methods used to find the moments of inertias for the IOLab device and the tape rolls are both equally valid methods. When calculating the t’ score for the two moments of inertia, we calculated a value of 0.3292. This is much lower than 3, which suggests that the two moments of inertia fall under the same parent distribution. Because the t-score is so low, we can conclude that that simply using the formula for the moment of inertia of a cylinder(½ * m * r 2 ) or using conservation of energy to find the moment of inertia of the IOLab device and tape roll system will both produce reliable results. Work done by each person: Shri: introduction, discussion Ethan: methods,