Lab 06 & 07 - Simple Harmonic Motion Part 1 (1)
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Mathematics
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Feb 20, 2024
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Lab 06: Simple Harmonic Motion Part 1 (2/23/22)
PHYS 2001 Section 021 Team 6: Grant Dunn, Sawyer Cano, Jesmin Castanon Mejia, Kirk Bradley Lab Records _____________________________________________________________________________________________________________________
Research Question:
What affects the period of an oscillating spring/mass system? Prediction: As the force on the spring increases, the resulting stretch will become greater. And as the k constant increases the distance of the stretch will decrease. Figure 1. Sketch of the resulting Stretch vs. Force applied in magnitude.
Figure 2. Sketch of the resulting Stretch vs. K-constant in magnitude as force remains constant. Spring ID #
Equation for the Trendline (in terms of F and x)
Uncertainty*
10304
F=5.29x 0.005
11102 F=7.37x 0.005 13388 F=15.2x 0.05 15126
F=34.3x
0.098
14238 F=51.7x 0.05 16165 F=56.2x 0.05 12258 F=82.3x 0.14 •
For each spring, the mathematical model included a positive slope in terms of x and F. • A
s the relative stiffness of each spring increased, the general value of the slope increased as well.
•
The numerical value of the slope value in the tread line equation represents newtons per meter. The value is then multiplied by x, which is meters, which produces the newtons of force. ____________________________________________________________________________________________________________________________________________________________
Figure 3. Diagram of the experimental set up. Force sensor located at the top of the beam and mass suspended from spring. Experimental Design Template Research Question What affects the period of an oscillating spring/mass system? Dependent variable (DV): Period Independent variable (IV): Spring constant Control Variables (CV): (
include actual values once chosen
) Mass, gravity, amplitude Testable Hypothesis: (must contain IV and DV to be testable) If the spring constant is increased and the mass and amplitude remains constant, the period will decrease. Prediction As the time increases on the x axis, the y-axis period will decrease. Experimental Design Template
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Research Question What affects the period of an oscillating spring/mass system? Dependent variable (DV): Period Independent variable (IV): Amplitude Control Variables (CV): (
include actual values once chosen
) Spring constant, gravity, and mass.
Testable Hypothesis: (must contain IV and DV to be testable) If the starting amplitude is increased and the mass and spring constant remain constant, the period will increase. Prediction As the time increases on the x axis, the y-axis amplitude will increase. Experimental Design Template Research Question What affects the period of an oscillating spring/mass system? Dependent variable (DV): Period Independent variable (IV): Mass Control Variables (CV): (
include actual values once chosen
) Spring Constant, gravity, amplitude Testable Hypothesis: (must contain IV and DV to be testable) If the mass is increased the force will increase, therefore the distance traveled will increase. Prediction As the time increases on the x axis, the y-axis amplitude will remain constant. A
ssumptions: •
The springs are not being over stretched.
•
The amplitude is always the same as recorded. •
There are no outside forces on the system. Graph 1: Spring Constant as the Independent Variable Figure 4. Data table of the period and omega given the independent variable. Figure 5. Graph of the position vs. Time in (cm vs. Seconds) Uncertainty: 0.242813 Claim: As the spring constant is increased, the period becomes shorter. Due to the larger k value. Analysis: The slope of the graph is 2.2409 times the 0.547 exponential. Equation: P=2.2409*S^0.547 S=Spring Constant
Graph 2: Amplitude as the Independent Variable Figure 6. Data table of the period and omega given the independent variable. Figure 7. Data table of the period and omega given the independent variable. Uncertainty: Claim: Changing the initial position of the mass of the spring, from the spring being at its natural length to extended, had no change on the period of the system.
Analysis:
By looking at our data you can see that changing the amplitude of the spring does not change the period of the oscillation.
Equation: P=0.4 Period=0.4
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Graph 3: Mass as the Independent Variable Figure 8. Data table of the period and omega given the independent variable of mass. Figure 9. Data table of the period and omega given the independent variable. Uncertainty: The uncertainty for the spring constant is 0.1
Claim: As the mass is increased, the period also increases. Analysis: The slope of the graph was determined to be P=0.0283m^0.4887, resulting in a positive correlation between the mass and the period.
Equation: P=0.8285m^0.4887 P=Period and M=Mass Conclusion: The only independent variable that had no impact on the period of the system would be the initial release position of the spring. Increasing the mass resulted in a longer period as spring could not return the mass to its original height. Spring constant and mass have an inverse relationship.
Summary After conducting the experiment, it was concluded that the Spring Constant and Mass affected the period of oscillation. The amplitude, however, did not prove to have a relationship to the period. As the spring constant increased, the period of oscillation decreased exponentially given the slope equation of
(P=2.2409m^-0.547). In addition, as the mass increased the tread line showed an increase, with a slope of 2.2409 and exponential factor of –
0.547. We are confident in the numbers collected in our data because we repeated tests that yielded numbers outside of the expected range. The R^2 values of each relation were near one as well. The theoretical model for the spring constant as the IV was P=2.2409m^-0.547
this equation was determined through the appropriate tread line in excel for the recorded value along with the mathematical model of T=2pi((m/k) ^0.5). The theoretical model for mass as the IV was P =0.8285x^0.4887. This was determined through the line of best fit equation along with the mathematical model of T=2pi/ ω. We are confident in the mathematical models provided by excel as the R
2 value .9776 and 0.9847. These values indicate that the model almost perfectly fits the plotted data. The outside factor of air resistance may impact the response to the research question. However, this may be the only significant outside force that might cause slight error. There are not many things that could be improved if we were to repeat this investigation. The exact location that the mass was released would need to be more precise as the recorded value was from an eyeball estimate each time. Taking more test values would also increase the accuracy of our findings.