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  • G2-26: COUPLED AIR TRACK GLIDERS

    G2-26
    Demonstrate coupled oscillations and normal modes in a system of two identical coupled air track gliders.
    Two moveable air track gliders are connected by three springs to fixed gliders (at each end). Pushing one of the gliders causes motion which rapidly couples back and forth between the two gliders. The two normal modes can be excited by giving the two gliders the same displacements, either (1) in phase, or (2) out of phase, before releasing them from rest simultaneously.
    FS2

  • G2-27: COUPLED SERIES MASSES HANGING ON SPRINGS

    G2-27
    Illustrate coupled oscillations and normal modes.
    Pushing either mass causes oscillations which will couple between the two masses. If the two masses are displaced from equilibrium by the appropriate amount either (1) in phase, or (2) out of phase, the normal modes can be produced. Alternatively, moving your hand up and down at the frequency of a normal mode will excite that mode.

    Check out Erik Neumann's Double Spring simulation here!
    G2

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  • G2-28: COUPLED PENDULA WITH VARIABLE DRIVER

    G2-28
    Show that the maximum coupling occurs between pendula of the same length.
    Adjust the length of the driver pendulum, on the left, so that it will couple to either of the other two. Coupling occurs through movement in the frame.
    FS2

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  • G2-41: WILBERFORCE PENDULUM

    G2-41
    Demonstrate a linear coupling resonance and normal modes.
    Move the bob straight up or down a few inches and release it quickly. It first starts to oscillate vertically, then its energy transfers entirely into rotational motion, then back to pure translation. This motion then repeats. By lifting the mass directly upward and at the same time rotating it (in either direction) before release, the two normal modes of the Wilberforce pendulum can be produced.

    g2-41a

  • G2-42: ELASTIC PENDULUM

    G2-42
    Demonstrate a non-linear coupling resonance and stable fixed points.
    Start the spring oscillating vertically; the energy will then couple back and forth between pendular motion and vertical spring motion. Stationary combinations of these two oscillations (corresponding to normal modes in a linear resonance) can also be produced, by pulling the weight simultaneously down and to the side. Adding an additional weight (attached to the support shaft near the bottom of the picture) to the spring destroys the resonance, resulting in less than total transfer of energy between the pendulum and spring motion.

    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]).

  • G3-02: SHIVE WAVE MACHINE - SUPERPOSITION OF PULSES

    G3-02
    Demonstrate constructive and destructive interference using pulses.

    Starting identical pulses from both ends simultaneously, either in or out of phase, they can be observed as they pass. For two identical pulses, move your hand rapidly down and up at the center of the machine with the two ends fixed. The two pulses created will reflect off the ends (left photograph) and interfere constructively as they cross each other on their return (right). Repeat this with one end clamped to get a phase reversal of the pulse which reflects off that end.
    Engagement Suggestion
    • This is a good opportunity to bring up one or two volunteers from the class to participate, rather than trying to reach both ends simultaneously yourself.
    • Encourage the class to predict what will happen when the pulses pass each other.
    Background
    In a linear medium like this, two waves moving in opposite directions can be seen to pass through each other. The principle of superposition states that when two mechanical waves pass each other in a medium, the net displacement at any point is the sum of the individual wave displacements.
    Ofc

  • G3-03: SHIVE WAVE MACHINE - REFLECTION OF PULSES

    G3-03
    Demonstrate reflection of pulses from fixed ends and free ends.

    A pulse generated at the left end (photograph at left) reflects off the right end. The reflecting end can either be fixed (clamped, center photograph) or free (right photograph).
    Engagement Suggestion
    • Encourage students to make a prediction before each combination as to whether the wave will reflect, and whether that reflection will be upright or inverted. • This can be combined with demonstration G3-05, showing that fixed and free end reflections are the extreme cases of partial reflections due to a change in impedance.
    Background
    Like any wave in a transmission medium, when the medium ends the energy in the wave has to go somewhere. The wave is reflected back from the end. With a free end, the wave reflects identically; with the end clamped, it reflects inverted.

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  • G3-04: SHIVE WAVE MACHINE - STANDING WAVES

    G3-04
    Demonstrate standing waves.

    Standing waves can be generated with either (1) both ends fixed, (2) one end fixed and one end free, or (3) both ends free. You can use either your hand or the motorized drive; your hand possesses better feedback for small adjustments in frequency.
    A metronome is available upon request for comparing the frequencies of various harmonics, or you can time them using the classroom clock or computer.
    Engagement Suggestion
    • Ask students to make predictions before each configuration of the device. Will fixed ends, free ends, or one open and one free end have the longest wavelength fundamental?
    Background
    A standing wave oscillates in time, but the peaks do not travel in space. A standing wave can be created in a mechanical medium of fixed length with an oscillation at that medium's natural frequency, or a multiple thereof.

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  • G3-05: SHIVE WAVE MACHINE - PARTIAL REFLECTIONS

    G3-05
    Show that a wave will be partially reflected at a point where the impedance changes.

    The Shive Wave Machine illustrates transverse waves traveling down a torsional wire. Partial reflection can be produced by
    • • linking the two different segments as shown in the photograph,
    • • adding weights to the end of a central crossarm to produce an impedance glitch, or
    • • attaching the dashpot at a central location and adjusting it for partial absorption of the incoming wave.
    Background
    Changing the arm mass changes the impedance of the medium. This changes the transmission speed; and when a wave passes through the junction, it may be partially reflected. Like a reflection from a free or fixed end (G3-03), this partial reflection can also be upright or inverted. Passing from higher to lower impedance gives an upright partial reflection; passing from lower to higher impedance gives an inverted partial reflection.

  • G3-06: SHIVE WAVE MACHINE - IMPEDANCE MATCHING

    G3-06
    Show that no reflection occurs when the impedance of the load (absorber at right) matches the impedance of the wave machine.
    Adjust the impedance of the absorb-o-matic to eliminate any partial reflection of the incoming wave. The impedance of the absorb-o-matic can be changed by adjusting the string tension.

  • G3-07: SHIVE WAVE MACHINE - TAPERED TRANSFORMER

    G3-07
    Useful for obtaining a reasonably good impedance match between the two large segments of the Shive machine.
    Connect the tapered transformer between the two large wave machine segments. Inserting this reduces the impedance mismatch and allows a continuous sine wave to pass from one medium into the other with minimal reflection, as seen in the photograph at the left. At the right is a close-up of the tapered transformer.

  • G3-08: SHIVE WAVE MACHINE - FABREY-PEROT INTERFEROMETER

    G3-08
    Demonstrate the mechanical analog of the optical Fabrey-Perot interferometer.
    Pairs of small weights are connected to two arms six inches apart in the center of the machine. The dashpot is attached to the end of the machine opposite the wave generator to prevent reflections (It must be adjusted.). Measure the difference in amplitude of the transmitted wave by measuring the amplitude of the oscillation of the dashpot, and compare that with the amplitude of the incoming wave as measured by the amplitude of the generator. The maximum transmitted wave occurs when the reflected wave is minimized, that is, when the two arms with the weights are one-quarter wavelength apart.

  • G3-09: SHIVE WAVE MACHINE - FREQUENCY FILTERING

    G3-09
    Demonstrate frequency filtering through a "filter" consisting of four weighted crossarms and the interference from partial reflections they produce.
    Four sets of weighted small weights are positioned on crossarms at equal intervals along the Shive machine, with the generator at one end and the dashpot at the other end. When the frequency of the generator is adjusted so that the wavelength is twice the spacing of the weights the reflected wave will be minimized and the transmitted wave maximized.

  • G3-10: SHIVE WAVE MACHINE - BRANCHING

    G3-10
    Demonstrate branching and recombining with two wave machines in parallel.
    This is the mechanical equivalent of the Quincke tube interference demonstration for sound (H2-25) or of a parallel LC circuit. At the first connecting point the amplitude and phase are directly coupled (i. e., the same) and waves continue down both machines. Due to the different wave velocities in the two machines, when the transmitted waves arrive at the second coupling they may be out of phase. If they arrive 180 degrees out of phase a node is created and no further energy is transmitted. In analogy with pinching one of the Quincke's tubes, one of the connectors can be removed to allow energy to pass. Crossarm weights are used to counterbalance the alligator crossarm connectors.

  • G3-11: SHIVE WAVE MACHINE - RESONANCE ABSORPTION

    G3-11
    Demonstrate resonance absorption of wave energy by a mass-on-spring system.
    Sending a wave along the machine drives the spring-mass attached to one of the crossarms. The greatest effect will be at the resonant frequency for the mass on the spring. A well chosen driving frequency will result in almost complete absorption of the wave.

  • G3-23: TRANSVERSE WAVES ON A LONG SPRING - FREE END

    G3-23
    Show reflections at a free end.
    A string holds one end of the long tight spring to the clamp on the lecture table. Because the string is long, and light compared to the spring, this forms a free end for the spring, allowing the end of the spring moves when a wave approaches.
    G3

  • G3-25: SLINKY ON LECTURE TABLE - IMPEDANCE MISMATCH

    G3-25
    Show partial reflections and dependence of wave speed on density of the medium.
    A string (running inside the SLINKY) connects one end of a SLINKY with a point about 3/4 of the way from that end, with the end taped to one end of the lecture table. When the SLINKY is extended it has regions with two different densities, causing two different wave speeds. A wave started at the free end of the SLINKY (right side in photographs above) will experience an impedance change; it may produce (quickly attenuated) partial reflections at the boundary. The wave moves more slowly in the section at the left, as seen in the photograph at the right.
    G3

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  • G3-31: Spring and Horns

    G3-31
    To audibly illustrate transmission of energy in a mechanical wave
    This long spring has a horn mounted on each end, which amplifies the vibrations from the spring as they pass into the air. Invite a student volunteer to hold each end, and show how energy is transferred through a wave and produces sound at the far end of the spring.

    This demonstration was donated by Prof. William Dorland.

    G3

  • G3-42: TORSIONAL WAVES

    G3-42
    Demonstrate wave phenomena such as traveling waves, standing waves, and reflection of waves.
    Waves can be started from either end. One end can be clamped to show reflection from free or fixed ends. Both ends can be clamped at the same time or one end can be free and the other fixed to show standing waves.
    G3

  • G3-44: WAVE-DRIVEN BUMPER JACK

    G3-44
    Demonstrate that waves transmit energy.
    Waves are sent along the stretched spring from your hand to the other end, which is attached to the handle of a bumper jack. If you send an appropriate frequency of wave, the energy transmitted to the bumper jack will lift the 7 kg mass, demonstrating that the wave is actually transmitting energy.