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Pendulum Data and Differential Equations

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). Equipment. small DC motor 2 wires w/ alligator clips ring stand

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Pendulum Data and Differential Equations

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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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