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Exploring Pendulum Dynamics: Theory and Experimentation for Accurate Modeling

This presentation, delivered at the Tenth Annual Valdosta State University Mathematics Technology Conference, focuses on the dynamics of pendulum motion, contrasting theoretical models with empirical data collection. Dr. Thomas F. Reid and Dr. Stephen C. King explore the linear pendulum model, and its limitations in accurately representing real-world data due to nonlinear effects. Using equipment including a small DC motor and TI-83 Plus calculators, they illustrate how to gather and analyze data, emphasizing the necessity of nonlinear models for precise pendulum motion representation.

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Exploring Pendulum Dynamics: Theory and Experimentation for Accurate Modeling

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  1. Pendulum Data and Differential Equations Tenth Annual Valdosta State University Mathematics Technology Conference February 25, 2005 Dr. Thomas F. Reid (thomasr@usca.edu) Dr. Stephen C. King(stevek@usca.edu)

  2. Equipment • small DC motor • 2 wires w/ alligator clips • ring stand • ring stand clamp • bent threaded rod • solderless lug • Vernier Instrumentation Amplifier • CBL / CBL 2 / LabPro • TI-83 Plus (or better) • GraphLink Cable King/Reid

  3. Theoretical Pendulum • Massless bar of length L • Point mass of M at end of bar • Angle at time t is θ(t)=θ • Motion described by where represents friction proportional to velocity • “Linear Pendulum”:for small θ(say |θ| <53°):sin(θ)≈ θ King/Reid

  4. Physical Pendulum • Total mass (M) • Bar • Additional mass distributed along some portions of bar • Combined density (ρ(x)) • Center of Mass (xcm) • Moment of Inertia (I0) • Center of Oscillation (L0) • Motion described by a theoretical pendulum with point mass M at location L0 King/Reid

  5. Ready…Set… • Initial angle • Initial velocity • Data in following slides had θ0=132.4° King/Reid

  6. Go! • Collect data using TI-83/CBL-2 • Zero the readings • Make sure data collection starts before starting swing • Know starting angle • Easier to estimate time of release (t0): fit line through first few “significant” values (>.01)x-intercept is t0 King/Reid

  7. Compare Data to Nonlinear Pendulum • “Guestimate” value of friction coefficient to get amplitude decay about right • Use maximum amplitude in data to determine scaling factor • Theoretical line goes through max pt King/Reid

  8. Try the Linear Pendulum Model • Same procedure, but using sin(θ)≈ θ • Terrible fit! • Even playing with the value of L0 will not result in a good fit. King/Reid

  9. Conclusion • Clearly shows the need for the nonlinear pendulum model to accurately model the data • Linear friction model seems adequate • Could run with different starting angles • Same friction coefficient works for all? King/Reid

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