Torsion Pendulum

Tyler Cash

Torsion Pendulum

An object that has oscillations which are due to rotations about some axis through the

object.

Damped Oscillations

Any oscillation in which the amplitude of the oscillating quantity decreases with time.

In general, torsion pendulums satisfy

J – Moment of Inertiab – Damping Coefficientc – Restoring Torque constantθ- Angle of Rotation

Solution to the equation yields 3 cases:

• Underdamped - many oscillations

• Critically Damped – one oscillation

• Overdamped – one very long oscillation

Apparatus

Procedure

1. Find natural frequency (no damping) by measuring the period several times

2. Turn on damping current

3. Set pendulum in motion and record angle of rotation after each oscillation

Angle vs. Time Plot

• Chi-Squared: 2.4

Angle vs. Time Plot

• Chi-Squared: 1.4

Damping Constants

• I=204 mAβ = .194 ± .004 radians/s

• I = 448 mAβ = .581 ± .018 radians/s

Critically Damped

• Trial and error found I=1.95 A caused critical damping

Damping ConstantUsing a fixed displacement and the time for that displacement,

Results inβ = 2.51 ± .26 radians/s

ResultsCurrent (mA) Damping Constant (Radians/s)

204 .194 ± .004

448 .580 ± .018

1950 2.51 ± .26

As the damping current increased, the damping constant increased.

Forced Oscillations

An oscillation produced in a simple oscillator or equivalent mechanical system by an external

periodic driving force.

Apparatus

Procedure

• Experiment with several driving frequencies in order to find the resonance frequency of the pendulum

• Record the phase shift between the pendulum and the driving motor

• Repeat this process over a range of damping currents

Resonance frequency plot

Resonance Frequency approximately .54 rad/s

Resonance Frequency Plot

Resonance Frequency approximately .52 rad/s

Resonance Frequency Plot

Resonance Frequency approximately .51 rad/s

Resonance Frequency

• From our plots and data, we estimated the following resonance frequencies:

Damping Current (mA) Resonance Frequencies

0 .54 ± .09

396 .52 ± .02

800 .51 ± .03

Damping Constants

For driven, damped oscillators,

Using this formula, we calculated the damping constant.

ResultsResonance Frequency (rad/s) Damping Constant (rad/s)

.54 ± .09 .594 ± .435

.52 ± .02 .202 ± .780

.51 ± .03 .143 ± .830

Our results for Damping Constants are unreliable.

Causes?•Not enough data points near resonance•Resonance and natural frequency are so close that errors are multiplied.

Results

• As driving frequency increased, the phase shift increased.

• At low frequencies, the phase shift was zero degrees

• At high frequencies, the phase shift approached 180°

• At resonance, the phase angle was 90°

Conclusion

• Overall, 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.

Questions?