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Torsion Pendulum,[object Object],Tyler Cash,[object Object]
Torsion Pendulum,[object Object],	An object that has oscillations which are due to rotations about some axis through the object.,[object Object]
Damped Oscillations,[object Object],Any oscillation in which the amplitude of the oscillating quantity decreases with time. ,[object Object]
In general, torsion pendulums satisfy,[object Object],J – Moment of Inertia,[object Object],b – Damping Coefficient,[object Object],c – Restoring Torque constant,[object Object],θ-  Angle of Rotation,[object Object]
Solution to the equation yields 3 cases:,[object Object],Underdamped - many oscillations,[object Object],Critically Damped – one oscillation,[object Object],Overdamped – one very long oscillation,[object Object]
Apparatus,[object Object]
Procedure,[object Object],Find natural frequency (no damping) by measuring the period several times,[object Object],Turn on damping current,[object Object],Set pendulum in motion and record angle of rotation after each oscillation,[object Object]
Angle vs. Time Plot,[object Object],Chi-Squared: 2.4,[object Object]
Angle vs. Time Plot,[object Object],Chi-Squared: 1.4,[object Object]
Damping Constants,[object Object],I=204 mA,[object Object],β = .194 ± .004 radians/s,[object Object],I = 448 mA,[object Object],β = .581 ± .018 radians/s,[object Object]
Critically Damped,[object Object],Trial and error found I=1.95 A caused critical damping,[object Object]
Damping Constant,[object Object],Using a fixed displacement and the time for that displacement,,[object Object],Results in,[object Object],β =  2.51 ± .26 radians/s,[object Object]
Results,[object Object],As the damping current increased, the damping constant increased.,[object Object]
Forced Oscillations,[object Object],An oscillation produced in a simple oscillator or equivalent mechanical system by an external periodic driving force.,[object Object]
Apparatus,[object Object]
Procedure,[object Object],Experiment with several driving frequencies in order to find the resonance frequency of the pendulum,[object Object],Record the phase shift between the pendulum and the driving motor,[object Object],Repeat this process over a range of damping currents,[object Object]
Resonance frequency plot,[object Object],Resonance Frequency approximately .54 rad/s,[object Object]
Resonance Frequency Plot,[object Object],Resonance Frequency approximately .52 rad/s,[object Object]
Resonance Frequency Plot,[object Object],Resonance Frequency approximately .51 rad/s,[object Object]
Resonance Frequency,[object Object],From our plots and data, we estimated the following resonance frequencies:,[object Object]
Damping Constants,[object Object],For driven, damped oscillators, ,[object Object],Using this formula, we calculated the damping constant.,[object Object]
Results,[object Object],Our results for Damping Constants are unreliable.,[object Object],Causes?,[object Object],[object Object]
Resonance and natural frequency are so close that errors are multiplied.,[object Object]
Conclusion,[object Object],Overall, our data accurately described the typical motion of a torsion pendulum.,[object Object],To improve our results, we suggest being more careful to take many data points around the resonance frequency.,[object Object]
Questions?,[object Object]

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