Pendulum Data and Differential Equations

1
Pendulum Data and Differential Equations

• Tenth Annual Valdosta State University
  Mathematics Technology Conference
• February 25, 2005
• Dr. Thomas F. Reid (thomasr_at_usca.edu)
• Dr. Stephen C. King (stevek_at_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

3
Theoretical Pendulum

• Motion described by where represents
  friction proportional to velocity
• Linear Pendulumfor small ? (say ? lt53)
  sin(?) ?

• Massless bar of length L
• Point mass of M at end of bar
• Angle at time t is ?(t)?

4
Physical Pendulum

• Total mass (M)
• Bar
• Additional mass distributed along some portions
  of bar
• Combined density (?(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

5
ReadySet

• Initial angle
• Initial velocity
• Data in following slides had ?0132.4

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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
  (gt.01)x-intercept is t0

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

8
Try the Linear Pendulum Model

• Same procedure, but using sin(?) ?
• Terrible fit!
• Even playing with the value of L0 will not result
  in a good fit.

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?