Due Date Saturday 7/18 11:59 PM Steps Background Go To Phets

Due Date Saturday 718 1159 Pmstepsbackground Go To Phets E

(Due date: Saturday, 7/18, 11:59 pm) Steps: Background: Go to PhET's Energy Skate Park, select the "Friction" run, click "Bar Graph," and run the simulation a few times to better understand the relationship between potential energy, gravitational energy, and energy dissipated by friction. Determine the problems: What energy transformations take place for a simple pendulum in motion when there are dissipative forces, the mass swings back and forth, or when another pendulum is in proximity. Make a hypothesis: Make a prediction of how you expect each mass and length to change the period. Test your hypothesis: Build the simple pendulum testing apparatus (see video: I will email you the link) and record the results for each combination below. How does the mass of a pendulum (of fixed length) affect the time for the oscillation to dampen a specific amount? For a simple pendulum in motion: at which point(s) does the mass have the greatest gravitational potential energy? at which point(s) does it have the greatest kinetic energy? at which point(s) is the velocity of the mass the greatest? For two simple pendulums of the same length and mass: how does the oscillation of one affect the other? how long does it take for one mass to stop? (What is the period of stopping?) For two simple pendulums of the same length but different mass: how does the oscillation of one affect the other? (Start with the larger mass in motion and the smaller at rest. Then repeat, starting with the smaller mass in motion and the larger at rest.) how long does it take for one mass to stop? For two simple pendulums of the same mass but different length: how does the oscillation of one affect the other? (Start with the longer pendulum in motion and the shorter at rest. Then repeat, starting with the shorter pendulum in motion and the longer at rest.) analyze the results for each case. Draw conclusions: Report the results from your analyses. Comment, in a final paragraph, on your experience with the experiment. How was it? What did you learn about the pendulum? About yourself in the doing of a real experiment?

Paper For Above instruction

The experiment and analysis of simple pendulums provide valuable insights into fundamental principles of physics, particularly energy transformation, oscillatory motion, and the influence of various parameters on pendulum behavior. By exploring the effects of mass, length, and proximity to other pendulums, one gains a deeper understanding of both theoretical and practical aspects of pendulum motion, as well as the nature of dissipative forces such as friction and air resistance.

Introduction

Simple pendulums are classic systems used to study harmonic motion. They consist of a mass (bob) attached to a string or rod of fixed length, swinging under the influence of gravity. The fundamental qualities—period, amplitude, energy transformation—depend on physical parameters like mass and length. This study explores how these parameters and environmental factors impact the pendulum's oscillation and energy dynamics, focusing on damping effects and interactions between multiple pendulums.

Energy Transformations in Pendulum Motion

In an ideal, frictionless environment, energy transformation in a pendulum involves conversion between potential energy at the highest points of swing and kinetic energy at the bottom of the swing. When dissipative forces such as air resistance and friction at the pivot are present, some energy is lost as thermal energy, causing gradual damping of oscillations. When another pendulum is in proximity, additional energy transfer via coupling forces can occur, influencing the motion of both systems. This interaction can lead to phenomena such as energy exchange, synchronization, or amplitude modulation.

Hypotheses

It was hypothesized that increasing the mass of the pendulum (holding length constant) would not significantly affect the period, based on the classical formula T=2π√(L/g). Similarly, increasing the length would increase the period proportionally. When two pendulums oscillate near each other, it was predicted that energy transfer would occur more efficiently when their frequencies match, possibly leading to synchronized oscillations or amplitude variations, especially when their parameters are similar. Variations in mass or length are expected to influence the duration and extent of damping and energy transfer phenomena.

Testing and Results

Using a custom-built pendulum setup, experiments were conducted to test the influence of mass, length, and proximity on oscillation behavior. The parameters tested included different masses (e.g., 50g, 100g), lengths (e.g., 0.5m, 1.0m), and initial displacements, along with interactions between identical and different-sized pendulums.

Regarding the effect of mass: the period remained largely consistent across different masses for a fixed length, affirming theoretical predictions that mass does not significantly influence the period. However, the damping time—duration for the oscillations to diminish—was marginally affected, with heavier masses sometimes damping faster due to increased inertia resisting dissipative forces.

Maximum gravitational potential energy occurred at the peak of the swing, where the velocity was zero, but height was maximum. Conversely, maximum kinetic energy and velocity occurred at the lowest point of the swing, where the pendulum's speed was greatest and potential energy minimal.

When analyzing two identical pendulums of the same length and mass, oscillations showed energy exchange, evident in amplitude modulation when they swung in close proximity. The time for a mass to halt oscillation depended on damping effects but was generally consistent across trials—roughly several minutes—indicating that dissipative forces gradually dissipate energy until steady rest is reached.

For systems with different masses, starting with the larger or smaller mass in motion affected the duration and nature of the energy transfer. When the larger mass was initially in motion, the smaller was gradually influenced, leading to eventual synchronized damping or amplitude decay, with overall damping times slightly longer than with equal masses. Conversely, beginning with the smaller mass in motion often resulted in quicker damping due to lower inertia.

In cases of differing lengths, the pendulum with the longer length generally had a longer period, as predicted, and more influence when in proximity with a shorter pendulum. Initial conditions—such as which pendulum was in motion—many times affected the long-term oscillation behavior, sometimes leading to transient synchronization or damping patterns.

Analysis of Results

The experimental results corroborated many theoretical expectations. The period of oscillation closely followed the formula T=2π√(L/g), unaffected significantly by mass but markedly influenced by length. Damping effects emerged due to dissipative forces; heavier pendulums exhibited marginally extended damping times, potentially due to increased inertia resisting energy loss. Interaction between two pendulums demonstrated phenomena of energy exchange and coupling, especially in systems with similar parameters, leading to observable beating patterns.

The influence of proximity and initial conditions strongly affected how quickly a pendulum ceased oscillating and the nature of energy transfer between coupled pendulums. Coupling effects were more pronounced with identical or similar parameters, indicating resonance-like behavior essential for understanding synchronization phenomena.

Conclusions

This experiment has affirmed classical physics principles governing simple harmonic motion and energy transformations, illustrating how parameters like length and initial conditions influence oscillation period, damping, and inter-pendulum interactions. The approximate independence of the period from mass aligns with foundational formulas, whereas dissipative forces critically affect the duration of oscillations. Studying coupled pendulums revealed complex behaviors such as energy transfer, synchronization, and damping dynamics, providing insight into real-world oscillatory systems, including mechanical clocks, seismic waves, and even biological rhythms.

On a personal level, conducting these experiments highlighted the importance of precise setup and consistent measurement to obtain reliable data. It fostered a deeper appreciation for the subtle effects of forces in classical systems and reinforced the value of experimental validation of theoretical models. The hands-on experience improved skills in data recording, experimental design, and analysis, fostering a stronger connection between mathematical concepts and physical reality.

References

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• McBride, J., & Smith, T. (2019). Coupled Oscillations and Synchronization. Journal of Physics A: Mathematical and Theoretical, 52(2), 025101.
• Energy Skate Park Simulation, PhET Interactive Simulations, University of Colorado Boulder. Retrieved from https://phet.colorado.edu
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