Effects of Mass and Amplitude on Simple Harmonic Motion

Simple Harmonic Motion Lab Report Isabelle Farley Phys1051L Section 009 10/23/2023

Lab Records Partners: Jordan Fredrick, Mia Kozlowski, & Katherine Walters Predicted Diagram: The graph created by DataStudio matches the predicted diagram. Experimental Set-Up

Figure 1.1 Spring Constant Spring ID No. Equation for trendline Uncertainty (+/-) 10343 F= 5.31x 0.062 11106 F=6.82x 0.10 13337 F=7.92x 0.63 15244 F=32.9x 0.46 14372 F=45.6x 0.93 16174 F=56.1x 0.83 12250 F=70.6x 1.1 1. The mathematical models are similar for each spring because they all represent the amount of force for the spring constant, the weaker spring has the lowest spring constant. 2. A general pattern is that as the stiffness of the springs increases, the spring constant also increases. 3. The numerical value of F represents the force in newtons, K is the spring constant and it is measured in N/m, describing the amount of newtons of force that are needed to stretch the spring. The x symbolizes displacement meaning how far the string is either stretched or compressed. Factors that Impact Period : Mass, Spring Constant, and Amplitude Amplitude vs Period Experimental Design Table Research Question What affects the period of an oscillating spring/mass system? DV Period of the oscillating spring/mass system IV Amplitude(56.5cm, 58.5cm, 60.5cm, 62.5cm, 64.5cm, 66.5) Control Variables(CV) Spring ID (11106), the mass of the system(50g) Testable Hypothesis The amplitude will not impact the period of the oscillating spring system. Figure 1.2 Amplitude vs Period Data Table Amplitude (cm) Period (s) Angular Velocity (rad/s) Uncertainty (+/-)

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56.5 0.546 11.5 0.013 58.5 0.546 11.5 0.012 60.5 0.546 11.5 0.012 62.5 0.546 11.5 0.012 64.5 0.546 11.5 0.013 66.5 0.546 11.5 0.013 Figure 1.3 Amplitude vs Period As the independent variable, amplitude, increased the dependent variable, the period of the oscillating spring/mass system, did not change. Therefore, there is no correlation between the amplitude (height) a spring is stretched to and the period of the oscillating spring/mass system. The R^2 value of -1 indicates that there is absolutely no correlation between the IV and DV. Error bars are present in the graph but are too small to be seen. The uncertainty values were obtained by the rotary motion sensors and used for the error bars. Mathematical Model: P = 0.546x^-1E-13 Amplitude has no correlation to the period of the oscillating spring/mass system. The mathematical model can be used to support this claim as regardless of the amplitude, the period will remain constant. The mathematical model has been subject to the conditions of this lab. The mathematical model was derived from the data in Figure 1.2. A linear trendline is shown in the graph provided. The mathematical model is from a power trendline. Mass vs Period

Experimental Design Table Research Question What affects the period of an oscillating spring/mass system? DV Period of the oscillating spring/mass system IV Mass (50g,60g,70g,80g,90g,100g) Control Variables(CV) Spring ID(11106), amplitude(57cm) Testable Hypothesis The mass will impact the period of the oscillating spring system. Figure 1.4 Mass vs Period Data Table Mass (g) Period (s) Angular Velocity (rad/s) uncertainty (+/-) 50 0.546 11.5 0.011 60 0.598 10.5 0.0065 70 0.644 9.75 0.0077 80 0.686 9.15 0.0074 90 0.727 8.64 0.0075 100 0.765 8.21 0.0062 Figure 1.5 Mass vs Period As the independent variable, mass, increased, the dependent variable, the period of an oscillating spring/mass system, increased. Therefore, mass has a positive correlation to the period of an oscillating spring/mass system. The R^2 value of 1.0 indicates that there is an extremely strong correlation between mass and the period of an oscillating spring/mass system. Error bars are present in the graph but are too small to be seen. The uncertainty values were obtained by the rotary motion sensors and used for the error bars. Mathematical Model:

P= 0.0819x^0.4851 Mass has a positive correlation to the period of the oscillating spring/mass system. The mathematical model can be used to determine the period with a given mass of the system. The mathematical model has been subject to the conditions of this lab. The mathematical model was derived from the data in Figure 1.4. A power trendline is shown in the graph provided. Spring Constant vs Period Experimental Design Table Research Question What affects the period of an oscillating spring/mass system? DV Period of the oscillating spring/mass system IV Spring constant (6.82, 7.92, 32.9, 45.6, 56.1, 70.6) Control Variables (CV) Mass of system (50 g), amplitude(57cm) Testable Hypothesis The spring constant will impact the period of the oscillating spring system. Figure 1.6 Spring Constant vs Period Spring ID No. Spring Constant Period (seconds) Angular Velocity (rad/s) uncertainty (+/-) 11106 6.82 0.536752137 11.7 N/A 13337 7.92 0.36300578 17.3 N/A 15244 32.9 0.2453125 25.6 N/A 14372 45.6 0.176404494 35.6 N/A 16174 56.1 0.131932773 47.6 N/A 12250 70.6 0.10295082 61 N/A

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Figure 1.7 Spring Constant vs Period As the independent variable, the spring constant increased, the dependent variable, the period of an oscillating spring/mass system decreased. Therefore, the spring constant has a negative correlation to the period of an oscillating spring/mass system. The R^2 value of 0.9066 indicates that there is a strong correlation between the IV and DV. Uncertainties and error bars are not shown because there were issues with the computer system and sensor, so the angular velocity values were given by the TA. In an ideal situation, the uncertainties would have been provided by the rotary motion sensors. Mathematical Model: P = 1.505x^-0.589 The spring constant has a negative correlation to the period of the oscillating spring/mass system. The mathematical model can be used to determine the period with a given spring constant of the system. The mathematical model has been subject to the conditions of this lab. The mathematical model was derived from the data in Figure 1.6. A power trendline is shown in the graph provided. Connecting and Comparing Experimental Model to Established Scientific Model The established model is defined by the scientific model with the equation and relationship. The equation is T=2pi square root of m/k meaning, the relationship between the period (T), hanging mass (m) and the spring constant (k). The variables used were spring constant, mass on the spring and amplitude. Experimental Outcomes Organizer

Compare Experimental Model to Established Model The experimental mathematical equations represent the period found of the oscillation and the correlation relationship. This was similar to the established scientific model because the same variables were used and we found it to be the same correlation. The numeral constants were slightly different since within the mathematical model we only had to solve for the period, in the established model the equation was focused on the relationship between the period (T), hanging mass (m) and the spring constant (k) of the spring, since the period was already found prior. Challenge 1: Finding Mystery Mass Actual Weight of Mystery Mass: 202g Calculated Weight of Mystery Mass: 201g (calculations shown below)

Spring ID (15220) Spring Constant Uncertainty (+/-) Trial 1 32.5 0.61 Trial 2 31.2 0.44 Trial 3 33.2 0.78 Average 32.3 - Spring ID (15220) Period Angular Velocity Uncertainty (+/-) Trial 1 0.4986 12.6 0.0055 Trial 2 0.4908 12.8 8.1x10^-4 Trial 3 0.4986 12.6 0.0059 Average 0.496 - - Procedure & Recap of Challenge 1 : In order to find the mystery mass, a spring (#15220) was chosen that had medium resistance. Three trials were conducted to get an average spring constant of the spring. An average period was also found. The mystery mass was hooked onto the spring while determining both the spring constant and period. Ultimately, in order to find the calculated mystery mass, the average spring constant and average period were plugged into the manipulated equation (reference work above). The calculated mass was very close to the mass found using the balance system, therefore, supporting the spring constant and period. Uncertainties in this experiment were useful as they helped to establish a constant spring constant and period. Challenge 2

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Given Period: 0.832 seconds Calculated mass: 0.566 kg Angular Velocity: 7.7 rad/s Procedure & Recap of Challenge 2: In order to prove the given period (0.832 seconds), the same spring and spring constant were used. The spring constant and given period were plugged into the manipulated equation (reference work above) in order to find the mass needed to support the given period with the chosen spring (#15220). The calculated mass was 0.566 kg or 566 grams. This mass was then added to the spring and the angular velocity was found. In order to find the period, 2π was divided by the angular velocity, 7.7 rad/s. The calculated period wawa 0.832 seconds, therefore providing the given period. Uncertainties in this experiment were useful as they helped to come up with a consistent angular velocity. Data from Other Groups

Discussion and conclusion The experiment was used to determine what effects the period of an oscillating spring/mass system. The variables tested were amplitude, mass, and spring constant. The amplitude vs period graph, Figure 1.3, showed that amplitude has no correlation to the period. The mass vs period graph, Figure 1.5, showed a positive correlation the mass has with the period. The spring constant vs period graph, Figure 1.7, showed a negative correlation between spring constant and period. The relationship between period and hanging mass is correlational when spring constant is help constant. The relationship between period and spring constant k is correlational when mass is held constant. Our data should be trusted as our R^2 value is very close to 1 meaning the data is very close to the line of best fit and the error bars on our graph are very small and cannot be seen. The equation determined by Excel should be trusted because it fits the data points well. Our results of the two challenges affirmed our trust in our final model as they prove the known scientific models. While other groups tested different unknown masses and periods, we all used the same equation and our given data which followed the known solutions to the challenges further proving the validity of our data. Limitations of our experiment were the number of springs able to be used. Many of the springs given were too tight to be able to test different weights as they would come off the hook which would ruin the trial we were running. One way this can be fixed is by finding a better way to secure the spring onto the hook. Another limitation was the way the spring started oscillating which in this experiment was using a finger to push down on the mass. This caused a lot of errors as the spring would not always oscillate straight or constantly due to human error. A way to fix this would be to have a constant way to push down the mass to avoid major human error. Somethings that were assumed in this experiment were that as the stretch of the spring increased the force of the system increased. This was proven incorrect by our experiment as the less stretchy spring has more force. Another was that amplitude had an impact on the period of the oscillating system which was also proven to be wrong as amplitude had no correlation to period.

Simple Harmonic Motion Lab Report

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