Experiment 4 - Non-Linear Models

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Experiment 4 - Non-Linear Models PHYS111L - 003 Group #72 Members: Devan Bidmead (21096248), Likhitha Tananki (21012570) TAs: James ( jdwillis@uwaterloo.ca ) and Reza ( rasadi@uwaterloo.ca ) Professor: Dr. Joey Rucska Date: November 16, 2023 Lab Time: 9:30am - 12:20pm Lab Section: 003 Part 0: Introduction Exploring the Model As seen below, there are two position vs. time graphs for 1) an undamped oscillating system, and 2) a damped oscillating system. There are 4 oscillations in each graph and the red data points on these curves could be used to test the above predictions for the amplitude as a function of time, 𝐴 ( 𝑡 ). Of the system parameters 𝑡? , 𝜔 , 𝜙 , 𝐴 , briefly explain which you think are (Note: Your predictions here don’t have to be correct!): 1) physical properties of the system that depend on physical parameters such as the mass of the hanging mass 𝑚 , the spring constant 𝑘 , and the damping coefficient 𝑏 The system parameter 𝑡? controls the amplitude over time but has the dampening coefficient 𝑏 incorporated into it so it would be dependent on physical properties. 𝜔 would also be dependent on the physical properties because it is related to T, the period. The period would be dependent on the system.

2) mathematical properties that depend more on how the experimenter sets up and records the oscillations. 𝜙 would depend on how the system is set up since it is accounting for where the mass is at t=0. If this is true then 𝐴 would also be dependent on the setup since it relates to the 𝜙 value. Introductory Statement In this experiment, we will test how well the SHM, Exponent Decay and Power-law Decay models describe an oscillating, vertically hanging, spring-mass system. We hypothesize that the Power-law decay will aptly describe the oscillating, vertically hanging, spring-mass system. To test these models we will be using specific data points (the highest point from a set of oscillations) to analyze the data by making graphs for the Exponent Decay, Power Law decay and the linearized best fits for both. We would use these graphs to calculate constants t d and n by using the linearized best fits and re-arranging the equations to solve for these constants. Part 1: Pilot Measurements Bounce Height (m) 2 0.034 5 0.034 10 0.0292 20 0.0261 50 0.0151 80 0.0131 100 0.0106 The manufacturer lists a precision of 0.002 m for the wireless motion sensor, this is an appropriate uncertainty for our measurements. To determine this we calculated the average of two peaks near the end of the dampening that had fluctuations. The peaks were within the uncertainty of the average at all times. As such 0.002m is an appropriate uncertainty. Part 2: Planning To measure the amplitude as a function of time we need both numbers (amplitude and time). Our initial amplitude was 0.065m. We ensured our experiment was as reproducible as possible by setting an initial displacement number from 0. The weight should be pulled down by approximately 3 cm and then the oscillations should be calculated.

The range in time our chosen peaks should be covered is 60 seconds to cover a significant decay in amplitude. As such we will choose one of the first bounces and the one of the ending ones. We will record our data every 10 peaks to prove some consistency between the peaks. Part 3: Execution and Analysis Using the Pre-Lab Excel template we made linear, semi-log, and log-log plots of your amplitude vs. time data. Table 1: Amplitude vs Time Graph 1: Linear Plot

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Graph 2: Semi-Plot Graph 3: Log-Log Plot The semi-log results in the best-fit relationship between the variables (time and amplitude). This can be illustrated by the similarity of the line of best fit to the semi-log According to the fit parameters, we can determine the numerical constant in our best model with uncertainty by calculating the linearized function of each plot (exponential or power decay) and looking at the relationships between the slope and the constants.

Exponential Slope: m= − 1 𝑡𝑑 -0.02861= -1/t d Therefore… t d 34.953 ≈ Uncertainty of t d = |1/m^2|∆m = 0.036651 Y-Intercept: b=ln(A) -3.38145 = ln (A) Therefore… A 0.0339981 ≈ Uncertainty of A = |e b |∆b = 0.000034134 Therefore after completing the above calculations, the constant value t d is approximately 34.953 and A is 0.0340. The final model can then be reported as 𝑥 ( 𝑡 ) = (0.0340 ? − 𝑡 /34.953 )cos( 𝜔𝑡 + 𝜙 ). Part 4: Exploration Making a Prediction 𝑥 ( 𝑡 ) = (0.0340 ? − 𝑡 /34.953 )cos( 𝜔𝑡 + 𝜙 ) 148th peak t=58.28s Measured Amplitude: 0.0077m Predicted Amplitude: 0.00642m Due to the fact that we are calculating the peak of the amplitude cos( 𝜔𝑡 + 𝜙 ) = 1 and as a whole the model is then changed to 𝑥 ( 𝑡 ) = (0.0340 ? − 𝑡 /34.953 ). Inputting the values we have (t), the predicted amplitude x(t) = 0.00642. Using the uncertainty calculated previously (0.002m) the value is correct. As the measured amplitude is 0.0077m and the predicted amplitude is 0.00642m, the uncertainty is then proven accurate. Upper-Bound: 𝑥 (58.28) = (0.0340 ? − 𝑡 /(34.953+0.036651) ) Upperbound = 0.006428 Lower Bound: 𝑥 (58.28) = (0.0340 ? − 𝑡 /(34.953-0.036651) ) Lower bound = 0.006406

Thus, the predicted amplitude is within the uncertainty based on the Upper-bound and the Lowerbound. Since our experiment proved an uncertainty of 0.001m instead of 0.002m we went with the manufacturer's specification to account for any error. Therefore, our predicted amplitude and measured amplitude agree. Conclusion To encapsulate, in our experiment, we found that the Semi-Log Plot showed the most similarity between the linear best-fit line and our data points. This shows that there is an exponential relationship between the amplitude and time. Overall the Semi-Log plot is more accurate in showing our data than the other plots for our experiment.

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