Torsion Pendulum
- 2. Torsion Pendulum An object that has oscillations which are due to rotations about some axis through the object.
- 3. Damped OscillationsAny oscillation in which the amplitude of the oscillating quantity decreases with time.
- 4. In general, torsion pendulums satisfyJ – Moment of Inertiab – Damping Coefficientc – Restoring Torque constantθ- Angle of Rotation
- 5. Solution to the equation yields 3 cases:Underdamped - many oscillationsCritically Damped – one oscillationOverdamped – one very long oscillation
- 6. Apparatus
- 7. ProcedureFind natural frequency (no damping) by measuring the period several timesTurn on damping currentSet pendulum in motion and record angle of rotation after each oscillation
- 8. Angle vs. Time PlotChi-Squared: 2.4
- 9. Angle vs. Time PlotChi-Squared: 1.4
- 10. Damping ConstantsI=204 mAβ = .194 ± .004 radians/sI = 448 mAβ = .581 ± .018 radians/s
- 11. Critically DampedTrial and error found I=1.95 A caused critical damping
- 12. Damping ConstantUsing a fixed displacement and the time for that displacement,Results inβ = 2.51 ± .26 radians/s
- 13. ResultsAs the damping current increased, the damping constant increased.
- 14. Forced OscillationsAn oscillation produced in a simple oscillator or equivalent mechanical system by an external periodic driving force.
- 15. Apparatus
- 16. ProcedureExperiment with several driving frequencies in order to find the resonance frequency of the pendulumRecord the phase shift between the pendulum and the driving motorRepeat this process over a range of damping currents
- 17. Resonance frequency plotResonance Frequency approximately .54 rad/s
- 18. Resonance Frequency PlotResonance Frequency approximately .52 rad/s
- 19. Resonance Frequency PlotResonance Frequency approximately .51 rad/s
- 20. Resonance FrequencyFrom our plots and data, we estimated the following resonance frequencies:
- 21. Damping ConstantsFor driven, damped oscillators, Using this formula, we calculated the damping constant.
- 22. ResultsOur results for Damping Constants are unreliable.Causes?Not enough data points near resonance
- 23. Resonance and natural frequency are so close that errors are multiplied.ResultsAs driving frequency increased, the phase shift increased.At low frequencies, the phase shift was zero degreesAt high frequencies, the phase shift approached 180°At resonance, the phase angle was 90°
- 24. ConclusionOverall, our data accurately described the typical motion of a torsion pendulum.To improve our results, we suggest being more careful to take many data points around the resonance frequency.
- 25. Questions?