Note that this has been largely replaced by G2-09.

Practice before demonstrating this device. It is a very inclusive machine, and can be used to illustrate most aspects of damped and driven oscillations discussed in intermediate mechanics texts.

Do not exceed the allowable damping voltage for more than a few seconds at a time, or the magnet may burn out.

Of course, this is actually a demonstration of driven resonance - with a bit of practice, via small movements of your hands you can drive any one of the pendula you choose. Encourage your students to analyze pseudoscientific explanations for real phenomena.

For a similar power-driven demonstration, see G3-45.

Invite a student up to measure the pendula to confirm that the responsive one matches in length.

This non-linear coupling resonance occurs when the spring (vertical) frequency is twice the frequency of a pendulum of the length at equilibrium. This is of interest because it is a very good mechanical analog to the v(r) = 2 v(z) resonance in the extraction region of a sector-focused cyclotron, where v(r) and v(z) are the radial and vertical betatron frequencies.

The mass required to be connected to a spring to induce this behavior can be determined as follows by noting the resonant condition:

v(mass on spring) = 2 v(pendulum)

or

sqrt[ k/m ] = 2 sqrt[ g/L ]

so

k = 4 mg/L

This means that you must add a weight so that the increase in length of the original spring is 1/3 of the original spring, or 1/4 of the length of the final spring (spring constant = mg/[L/4]).

Consider combing this with H3-13, and invite students to make predictions about the differences in pitch and volume.

Consider combing this with H3-12, and invite students to make predictions about the differences in pitch and volume.

Invite students to make predictions about the changing resonant frequencies before adjusting, to encourage student engagement.

This is a variation on H3-22, and can be combined with it.

These sorts of globular resonators were used by Helmholtz in the nineteenth century, in the early days of acoustics experimentation. Before the development of spectrum analyzers and similar tools, he developed techniques to analyze the structure of sounds simply by holding a succession of resonators of different frequencies to his ear to pick out the components of complex sounds. The principle behind this is little different from twentieth century analog electronic frequency analyzers, which feed a signal from a microphone into a series of resonant circuits analogous to Herlmholtz's glass globes.

These are the same tubes used in H3-24. Consider showing them to students first and having them make predictions, then using this apparatus to analyze the results.

An audio oscillator and 100 Watt power amplifier are used to drive a heavy-duty horn driver which is mounted in the back of the plastic beaker cavity with the sound emerging through a hole, which can be seen in the photograph. The beaker is positioned on a foam pedestal in front of the speaker hole. A microphone is mounted at 90 degrees from the position of the speaker.

The beaker is marked with its primary resonant frequency, found in advance using digital spectrum analysis of a recording of the beaker ringing after being tapped. Most beakers have two possible resonant modes 45 degrees apart, due to the weight of the spout; the most effective technique is to drive the resonance with the spout facing directly away from the speaker. Set the frequency of the oscillator as shown on the beaker, with an amplitude of around 140mVpp. The oscilloscope will show two waveforms, the input signal and the signal picked up by the microphone. You may need to adjust the frequency slightly to account for changes in temperature or age since the beaker was tested; slowly shift the frequency by tenths or hundredths of a Hertz to find the amplitude peak (do not try to tune by watching for a displacement in the phase relationship, as there is a time delay between the signals introduced by the hardware). This done, set the strobe around 3000 cycles per minute, and adjust it until you can see the sides of the beaker flexing.

This can be used to show the resonance of the beaker. You can also, optionally, shatter it, by increasing the input voltage at resonance. Be careful not to exceed 1Vpp.

After the resonant frequency is found and the amplitude turned up, the oscillation of the beaker can be caused to exceed its elastic limit and thus to shatter. See the video links below to view a slow-motion video of the beaker at the moment it breaks.

Engagement Suggestion

Background