PHY 105M Lab 5

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University of Texas *

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105M

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Mechanical Engineering

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Dec 6, 2023

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PHY 105M Lab 5 Part 1: Hanging Wires Method: Our goal for this lab is to calculate the spring constant, k, for the restoring force given from stretching two different strings by attaching extra masses to them. We chose to experiment with the plastic string and the guitar string on the right (there were two hanging). We hypothesize that guitar string will be easier to stretch and will thus produce greater measured distances as mass and force increase, resulting in overall smaller k values compared to that of the plastic string. The displacement we are measuring is the distance (in meters) in the vertical direction that the string extends from the equilibrium point as a result of adding a certain mass (in kg). The force (in N) that restores this displacement is given by Hooke’s Law as . |𝐹| = 𝑘|∆𝑦| We conducted 3 trials for each string in which we varied the masses of the weights attached to the string (0.100kg, 0.0500kg, 1.000kg). We first recorded the initial measurement of each string without any mass attached by straightening it out (we were careful to not exert extra force on the string) and measuring where the meterstick we are holding horizontally intersects with the meterstick installed on the right side of the device – this value will be the equilibrium point. The position of the meter stick on the right side of the device hanging parallel to the wires will remain constant throughout all trials. Next, we attached our mass to the string and calculated displacement by subtracting the measurement we recorded after attaching the mass to the string from the initial measurement when no mass was attached. We repeated the absolute value of this calculation for each mass we are testing on each string. We then repeated these steps for the second wire, the plastic wire. Gravity (g) in the context of F = mg is equal to 9.8 m/s 2 due to the fact that we are measuring the displacement in the vertical direction (at an angle of 90º from the horizontal beam). The displacement was originally measured in centimeters and later converted to meters in order to calculate the unweighted and weighted spring constants and their associated uncertainties. We calculated the spring constant for each mass tested on each beam by using the equation Next, to find the unweighted uncertainty of the spring constant, k, we used 𝑘 = 𝐹 | | ∆𝑦 | | . the following equation and the associated uncertainties: δ𝑘 = − 𝐹 𝑦 2 • δ∆𝑦 ( ) 2 + 1 𝑦 • δ𝐹 ( ) 2 Uncertainty in Distance ( ∆y) = 0.0005 meters δ *This is the systematic uncertainty associated with the meterstick which is the primary source of uncertainty in this experiment. Uncertainty in Force = 0.005 N δ𝐹 = 𝑔 • δ𝑚 ( ) 2 *Uncertainty in Mass ( ) = 0.0005 kg δ𝑚 *g = 9.81 m/s Data: Plastic String Mass (kg) = 0.0005 kg δ𝑚 Displacement (∆y) ∆y = 0.0005 m δ Force (N) = mg (g=9.8m/s²) 𝐹 = 0.005 N δ𝐹 Spring Constant (k) Uncertainty of k δ𝑘 0.100 kg 0.001 0.98 N 980 490

0.500 kg 0.003 4.90 N 1633 272 1.000 kg 0.005 9.80 N 1960 196 Guitar String Mass (kg) = 0.0005kg δ𝑚 Displacement (∆y) ∆y = 0.0005 m δ Force (N) =mg (g=9.8m/s²) 𝐹 = 0.005 N δ𝐹 Spring Constant (k) Uncertainty of k δ𝑘 0.100 kg 0.002 0.98 N 490 123 0.500 kg 0.005 4.90 N 980 98 1.000 kg 0.008 9.80 N 1225 77 Conclusion: Plastic String k Weighted Average and Uncertainty = 1766.6 +/- 29.7 k weighted = 980 1 490 2 ( ) + 1633 1 272 2 ( ) + 1960 1 196 2 ( ) 1 490 2 + 1 272 2 + 1 196 2 k weighted = δ 1 2 980 1 490 2 ( ) + 1633 1 272 2 ( ) + 1960 1 196 2 ( ) 1 490 2 + 1 272 2 + 1 196 2 Guitar String k Weighted Average and Uncertainty = 1006.2 +/- 22.4 k weighted = 490 1 123 2 ( ) + 980 1 98 2 ( ) + 1225 1 77 2 ( ) 1 123 2 + 1 98 2 + 1 77 2 k weighted = δ 1 2 490 1 123 2 ( ) + 980 1 98 2 ( ) + 1225 1 77 2 ( ) 1 123 2 + 1 98 2 + 1 77 2 Conclusion: Overall, the experiment was successful in that we were able to obtain the weighted spring constant values

and their associated uncertainties for each string we tested. In addition, the experiment proved our hypothesis correct in that the weighted spring constant for the guitar string and its associated uncertainty were smaller than that of the plastic string. The weighted spring constant and the uncertainty for the plastic string was calculated to be 1766.6 +/- 29.7 and that of the guitar string was calculated to be 1006.2 +/- 22.4. The largest source of uncertainty in this experiment comes from the meter stick due to the fact that it is our primary source of measurement. We were not able to calculate any t-scores comparing our weighted k values to the other group’s because they inaccurately calculated the spring constant. Part 2: Nathan Ngo + Arkira Jaruves 1) Were the instructions clear/legible/readable? What was good, and what could be improved, in this respect? a) The instructions were clear and legible, and in general we were confident in how to do the experiment in the way they conducted it by reading their directions. One thing that could be improved is including their final k-value at the end of their instructions. We were not given this value, and had to find the group who wrote the instructions in the room to ask for it so we could compare our data as directed by our TA. I also wish they had been more specific about which masses and wires they used, because each wire has a different k value because it is a different material. Also, although theoretically the k value should be the same for any force you apply to the string using any mass, one thing we have learned over the past few weeks in lab is that often our models are not perfect, and our results are only applicable to a certain range of measurements. They just told us to select 2 different wires and test 3 different masses on each, and this could be a major source of error in my opinion. I also wish we could have used a level on the meter stick that we positioned horizontally to the meter stick that was hanging parallel to the strings. This could have improved our accuracy in measuring the change in the length of the wire by making sure the tool we were lining up with the mark on the string was completely straight, especially since the difference in length is so small. 2) Were the instructions adequate to reproduce the results? What was good, and what could be improved, in this respect? a) When we asked for the values that the group who did this lab last week got for comparison, they gave their entire part 1, including all the data and calculations they did to get to their results. After reviewing this, we found a lot of errors in their calculations. First, they majorly mixed up units. Some calculations were in centimeters, others were in meters, with no conversion step. They also used the wrong uncertainty for length. Because the meter stick is an analog tool measuring with a precision of 1 mm (0.001m), the uncertainty in length should be 0.5mm (0.0005m), or half of the smallest unit. They used 0.05m as their uncertainty in their calculations of the uncertainty of k. Also, at the end of their calculations, instead of reporting two different k values for each wire, they combined them to report a single value. I’m not sure why they did this, since it makes more sense to report their values individually since they are two different materials. 3) What is your assessment of the groups methods and how they did the experiment? Would you have done it differently, or done some aspects differently? Can you think of ways to improve it? a) We would have (and did in our Part 1 for this lab) used 0.0005m for our uncertainty in length (Δy) as this is the correct uncertainty. We also made sure to have consistent units, and made sure to keep our units straight throughout our calculations. We also reported separate k values for each wire; one for the guitar string and one for the plastic wire. There were two guitar strings hung, and we used the one on the right. I assume the set up did not change from last week, so I would have specified

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which strings we used in our experiment to ensure the next group was measuring the k values of the same strings, otherwise you can’t compare them. I also would have given the weighted k values at the end of the instructions so the next group could compare their results. Because the group before has had so many errors, we were unable to compare our results using t-scores, and this was verified by our TA in class.