How Things Work - Chapter 9 Demonstrations
Section 9.1 Clocks
Demonstration 9.1.1: Harmonic Oscillators: Pendulums
Description: You swing several pendulums back and forth. Each one has a steady period that doesn't depend on how far it swings. The taller the pendulum, the longer its period.
Purpose: To show that a pendulum has the characteristics of a harmonic oscillator—a restoring force that's proportional to displacement (almost) and consequently a period that doesn't depend on the amplitude of motion (almost).
Supplies:
2 or more pendulums (one should be about 0.25 m tall, from pivot to center of mass/gravity, and another should be about 1.00 m tall).
supports for the pendulums
Procedure: With the 0.25 m pendulum motionless, discuss the fact that it's in a stable equilibrium. Discuss that the restoring force it experiences is proportional to how far it's displaced from its equilibrium position. (Avoid discussing it's slight anharmonicity here—you'll confuse the students.) Note that this proportionality between displacement and restoring force makes the pendulum a harmonic oscillator.
Now displace the pendulum from equilibrium and release it. Time the swings to determine its period. The period should be almost exactly 1 s. Show that swinging the pendulum harder or softer doesn't affect its period (but don't swing it too hard or you'll discover the slight anharmonicity). Now do the same with the 1.00 m pendulum. Discuss how the short pendulum could serve as the timekeeper for a clock with a second hand that advances by 1 s for each full cycle of the pendulum and the long pendulum could serve as the timekeeper for a clock with a second hand that advances by 2 s for each full cycle of the pendulum.
Explanation: The period of a pendulum is equal to 2p times the square root of its length divided by the acceleration due to gravity. A 0.25 m pendulum thus has a period of about 1 s while a 1.00 m pendulum has a period of about 2 s. The increase in period with length is due to a softening of the restoring force—the longer the pendulums arm, the less rapidly the restoring force increases as you displace the pendulum bob from its equilibrium position. Increasing the acceleration due to gravity would stiffen the restoring force and speed the pendulum's motion and period.
Demonstration 9.1.2: Harmonic Oscillators: A Mass on a Spring
Description: A mass hanging on a spring bounces up and down with a steady period, regardless of the amplitude of that bounce. The larger that mass, the longer the period of the bounce.
Purpose: To show that a mass on a spring has the characteristics of a harmonic oscillator—a restoring force that's proportional to displacement and consequently a period that doesn't depend on the amplitude of motion.
Supplies:
1 large coil spring (medium stiffness)
2 different masses
1 support for the spring
Procedure: Hang the spring from the support and attach the smaller mass to it. Allow the spring to stretch until the mass and spring are in equilibrium. Point out that the mass is in a stable equilibrium, that the spring is exerting just enough upward force on the mass to support its weight and that the mass is experiencing zero net force. Show that displacing the mass up or down causes it to experience a restoring force that's proportional to the displacement. Now displace the mass from equilibrium and release it. Time the bounces to determine their period. Show that this period doesn't depend on how large the amplitude of motion is. Repeat this process with the larger mass hanging from the spring. Discuss why the period of oscillation is now longer than with the smaller mass.
Explanation: In this system, gravity merely shifts the mass's equilibrium position because the mass's weight doesn't change with its position. Only the spring exerts a force that changes with position and only the spring contributes to the oscillatory motion. In this case, the period decreases with the increasing stiffness of the spring (stiffer restoring forces cause more rapid accelerations) and increases with the increasing mass of the oscillating mass (larger masses cause slower accelerations).
Demonstration 9.1.3: Harmonic Oscillators: A Ruler Vibrating on the Edge of a Table
Description: You hang a meter stick over the edge of a table and pluck it downward. It vibrates with a period that doesn't depend on its amplitude of motion. As you shift it onto the table and shorten the portion that extends out over the edge, the period of oscillation becomes shorter and shorter.
Purpose: To illustrate yet another harmonic oscillator.
Supplies:
1 meter stick (or another thin, stiff stick)
Procedure: Extend the meter stick out over the edge of a sturdy table and hold the portion that rests on the table firmly against the table. Show that the free end of the meter stick is in a stable equilibrium. Now push the free end away from its equilibrium position and release it. The meter stick will vibrate up and down with a steady period that doesn't depend on its amplitude of motion. Now shift more of the meter stick onto the table and repeat the experiment. Its period will be shorter.
Explanation: The meter stick is a harmonic oscillator with a period that depends on its stiffness and mass. Shortening its free end reduces the oscillating mass and stiffens the restoring force. Since both of these changes make accelerations more rapid, the meter stick's period of oscillation shortens dramatically as you decrease the length of the free end.
Demonstration 9.1.4: Harmonic Oscillators: A Mass on a Hacksaw Blade
Description: A ball of putty attached to a hacksaw blade oscillates back and forth rhythmically. Adding more putty slows the oscillation while shortening the blade speeds the oscillation up.
Purpose: To show that mass-on-spring harmonic oscillators can take many forms.
Supplies:
1 hacksaw blade (a stiff metal-saw blade)
1 ball of putty
1 clamp
Procedure: Clamp the hacksaw blade on the edge of a sturdy table so that it extends far out over the edge of the table. Attach the ball of putty to the free end of the blade. Allow the ball and blade to settle and point out that the ball is now in equilibrium. Show that displacing the ball from equilibrium, either up or down, causes it to experience a restoring force that's proportional to its displacement. Now displace the ball from equilibrium and let go. It will oscillate up and down with a period that doesn't depend on its amplitude of motion. It's another harmonic oscillator. Discuss ways of changing its period (changing the mass of the ball or changing the length of blade that extends over the table). Try these approaches to see what happens.
Explanation: The spring-like blade exerts a restoring force on the putty and the putty's mass resists acceleration. Since the blade's restoring force is proportional to its bend, the system is a harmonic oscillator. Decreasing the mass of the putty allows it to accelerate more rapidly and shortens the oscillator's period. Shortening the portion of the blade that extends out from the table stiffens it and also shortens the oscillator's period.
Demonstration 9.1.5: Harmonic and Anharmonic Oscillators: A Ball Rolling in a Bowl
Description: A ball or marble rolls back and forth in a shallow bowl with a period that may or may not depend on the amplitude of its motion, depending on the shape of the bowl.
Purpose: To show that not all oscillators are harmonic. If the bowl has a spherical bottom, this oscillator is resembles a pendulum (without the string) and is nearly harmonic. If the bowl's bottom has a more complicated shape, this oscillator can become decidedly anharmonic.
Supplies:
1 ball or marble
several large bowls, some with round bottoms and others with more complicated bottoms.
Procedure: Place the ball at the bottom of a bowl and show that it's in a stable equilibrium. Discuss how the restoring force the ball experiences depends on its displacement from equilibrium. Now displace the ball from equilibrium and release it. It will roll back and forth through the equilibrium. If the bowl has an approximately spherical shape, then its period will be approximately independent of its amplitude of motion. But if the bowl has a more complicated bottom surface, then its period will probably depend on the amplitude of its motion.
Explanation: Neglecting the rolling process, a ball in a spheric bowl follows the same path that a pendulum bob would and is thus simply a modified pendulum oscillator. The shallower the bowl, the longer the effective pendulum's string and the longer the oscillator's period. But if the bowl's bottom isn't close to spherical, then the restoring force the ball experiences will probably depend nonlinearly on its displacement from equilibrium and the period of the oscillator will depend on amplitude.
Demonstration 9.1.6: Anharmonic Oscillators: A Bouncing Ball
Description: A ball bounces up and down on a hard surface with a period that depends on the amplitude of its motion.
Purpose: To show another form of anharmonic oscillator..
Supplies:
1 ball or marble
1 hard surface
Procedure: Drop the ball onto the surface and listen to its sequence of bounces. They'll get closer together in time as the bounces decay away. The familiar chirp of a marble bouncing on a stone surface is this same decreasing period. The bouncing ball is oscillating about its equilibrium, but since the restoring force it experiences while in the air is constant rather than springlike, the oscillation is definitely not harmonic. Instead, the large-amplitude motions take longer than the small-amplitude motions and the period increases with amplitude.
Explanation: The bouncing ball has the right restoring force to be harmonic when it's below its equilibrium, but the wrong restoring force when it's above its equilibrium. Once it leave the surface on which it is bouncing, the ball experiences only its weight and weight is independent of displacement from equilibrium (for all practical purposes). So the ball's bounces become faster as they become shorter.
Demonstration 9.1.7: Examine a Pendulum Clock
Description: You open a pendulum clock to show how the swinging pendulum controls the turning of the clock's hands.
Purpose: To show how the clock uses its pendulum to time the steps of its second hand.
Supplies:
1 real pendulum clock (not an electronic clock with a decorative pendulum)
Procedure: Time the period of the clock's pendulum and examine its length. It should be 0.248 m long for a 1 s period or 0.996 m long for a 2 s period (from pivot to center of mass/gravity). If you can view the clock's mechanism, watch the swinging pendulum release the toothed wheel that governs the turning of the clock hands. The anchor that tips with the pendulum adds energy to the pendulum and controls the turning rate of the clock hands. Look to see if the pendulum has a temperature compensation system and a length adjustment. What provides the energy that keeps the pendulum swinging?
Explanation: A pendulum clock has a simple mechanism—the pendulum receives small pushes as it swings and it allows the clock's second hand to advance one step with each swing.
Demonstration 9.1.8: Harmonic Oscillators: A Torsional Pendulum
Description: A massive disk hanging from a stiff wire twists back and forth with a period that doesn't depend on the amplitude of its twisting motion.
Purpose: To show that some harmonic oscillators involve restoring torques and angular accelerations rather than restoring forces and accelerations.
Supplies:
1 heavy disk
1 stiff metal wire, thin metal rod, or relatively thin wooden dowel
1 support for the wire, rod, or dowel
Procedure: Attach the wire, rod, or dowel to the center of the heavy disk. Make sure that the disk is balanced and remains horizontal when it's hanging from the wire, rod, or dowel. Attach the wire, rod, or dowel to the support and allow the disk to reach its equilibrium orientation. Show that the disk is in a stable equilibrium orientation—that it's experiencing zero torque but that it experiences a restoring torque whenever it's displaced from its equilibrium orientation. Point out also that the restoring torque it experiences is proportional to its angular displacement.
Now displace the disk from its equilibrium orientation and release it. It will twist back and forth with a period that depends only on the disk's moment of inertia and the stiffness of the wire, rod, or dowel. If you're careful, you can change the moment of inertia with added masses (keep them balanced!) and you can adjust the stiffness of the restoring torque by changing the length of the wire, rod, or dowel.
Explanation: This torsional harmonic oscillator has a restoring torque rather than a restoring force and a moment of inertia rather than a mass. Its period motion involves angular acceleration and angular velocity rather than acceleration and velocity. Nonetheless, its period doesn't depend on its amplitude of motion because the restoring torque is proportional to angular displacement.
Demonstration 9.1.9: Examine a Balance Ring Clock
Description: You open a balance clock to show how the rocking balance ring controls the turning of the clock's hands.
Purpose: To show how the balance ring clock uses the balance ring to time the steps of its second hand.
Supplies:
1 balance ring clock or watch
Procedure: Observe the balance ring rocking back and forth. Identify the spring that provides the restoring torque for this torsional motion. Find the lever and anchor that deliver the tiny pushes that keep the balance ring rocking and that control the advance of the clock's second hand.
Explanation: In a balance ring clock, the balance ring is a harmonic oscillator that experiences the restoring torque of a small coil spring. As the balance ring rocks back and forth, it allows the second hand to advance a small amount for each cycle.
Demonstration 9.1.10: Harmonic Oscillators: A Water Balloon
Description: You strike a hanging water balloon with your hand and its surfaces vibrate in and out symmetrically with a steady period that doesn't depend on their amplitudes of motion.
Purpose: To illustrate the mode of vibration that's used in most quartz crystal oscillators.
Supplies:
1 large water-filled balloon (a large-size latex rubber balloon, filled as full as is practical with water)
Procedure: Hold the water balloon by its nipple in one hand and hit it moderately firmly with the other hand. (Don't break it, of course.) The balloons surfaces will oscillate in and out, with surfaces on opposite sides of the balloon moving simultaneously in opposite directions. Viewed from above, the balloon will first become narrower from left to right and wider from top to bottom and then wider from left to right and narrower from top to bottom. This motion repeats.
Explanation: The balloon acts much like two masses on a central spring. These two masses, effectively the left and right sides of the balloon, move alternately toward one another and away from one another. Their masses and the stiffness of the effective spring depend mostly on the size of the water balloon—the bigger the balloon, the slower its period of oscillation. In a real quartz oscillator, the stiffness is much greater and the period of oscillation is much shorter.
Demonstration 9.1.11: Harmonic Oscillators: A Singing Aluminum Rod
Description: You rub a hard aluminum rod along its length and it begins to emit a clear, high-pitched tone.
Purpose: To illustrate the mode of oscillation used in a quartz oscillator.
Supplies:
1 rod, made of hard aluminum alloy, with a diameter of about 1 cm and a length of about 1 or 2 m
rosin or grip-enhancing spray
Procedure: Apply rosin or grip-enhancing spray to the fingers of one hand. Hold the aluminum rod at its midpoint and gently pull your rosined fingers along one end of the rod. Your skin should slide across the metal with a slip-stick motion, as though you were bowing a violin. When you achieve the correct sliding motion, the rod will begin to vibrate, getting louder and louder as your fingers slide more vigorously along the rod.
Explanation: Your fingers are gradually adding energy to a vibrational mode of the aluminum rod. In this mode, the two ends of the rod are moving in opposite directions and experience restoring forces from the spring-like middle portion of the rod. The period of oscillation depends on the stiffness of the aluminum—its Young's modulus—and on the length (and therefore mass) of the rod.
Demonstration 9.1.12: Harmonic Oscillators: A Tuning Fork
Description: The tines of a tuning fork oscillate in and out with a period that doesn't depend on their amplitudes of motion.
Purpose: To show that a tuning fork is another harmonic oscillator—one that has been used in clocks.
Supplies:
2 tuning forks of different sizes and pitches
1 tuning fork mallet
Procedure: Point out that the tines of a tuning fork and the metal bridge between them forms a harmonic oscillator—displacing the tines causes them to experience restoring forces. Strike one of the tines with the mallet to displace it and cause the tuning fork to vibrate. Note that its pitch (associated with its period) is independent of its amplitude of motion.
Explanation: The tuning fork acts like two masses on the ends of a single spring. The masses (tines) oscillate in opposite directions with a period that increases with their masses and decreases with the stiffness of the spring between them.
Demonstration 9.1.13: A Quartz Crystal Oscillator
Description: An electronic oscillator without a quartz crystal in it produces a current that fluctuates at a moderately steady rate. When a quartz crystal is added to the oscillator, the fluctuations become extremely steady at a particular value—the natural resonant frequency of the quartz crystal itself.
Purpose: To show that the mechanical vibrations of a quartz crystal can be used to control the electric oscillations of an electronic device.
Supplies:
1 quartz crystal
1 conventional electronic oscillator with a natural frequency of oscillation that's very close to that of the quartz crystal. (Many types of oscillators will do and I don't have a particular one to recommend.)
1 frequency meter
Procedure: Use the frequency meter to monitor the frequency of the conventional oscillator. It will drift in frequency with time and temperature and won't hold any specific value for very long. Now insert the quartz crystal into the oscillator in a place where the crystal's natural resonance can affect its frequency of oscillation. If you place it in the proper part of the oscillator, the crystal will begin to oscillate and will pull the frequency of the conventional oscillator into synchrony with its own frequency. The frequency of the crystal oscillator won't drift with time and will remain at a particular value indefinitely.
Explanation: A quartz crystal is a piezoelectric device. When exposed to fluctuating electric fields, it begins to undergo mechanical vibrations. These vibrations are strongest when it's vibrating on its natural resonance. As the crystal vibrates, its piezoelectric nature causes charge to shift on and off its surfaces. In the oscillator, the fluctuating charges in the wires attached to the quartz crystal cause the quartz crystal to vibrate. Once the crystal is vibrating on its natural resonance, it begins to cause large charge fluctuations on its own surface and these charge fluctuations begin to affect the oscillator's frequency. Pretty soon, the crystal's vibrations are determining the oscillator's frequency—the oscillator has become phase-locked to the crystal's vibrations.
Demonstration 9.1.14: Atomic Transitions for Atomic Clocks
Description: A gas discharge is shown to emit very specific wavelengths or frequencies of light.
Purpose: To show that atoms absorb and emit characteristic wavelengths or frequencies of light that can be used as timekeepers for exquisitely accurate clocks—atomic clocks.
Supplies:
1 low-pressure gas discharge lamp (a narrow hydrogen tube is a good choice)
1 transmission diffraction grating (or prism)
1 CCD camera and monitor (optional)
Procedure: Observe the gas discharge lamp through the diffraction grating, or allow the CCD camera to observe the discharge through the diffraction grating and project its image on the monitor. Don't aim the grating-covered camera directly at the discharge; aim it to one side, where it will record dispersed lines of different colors. These are the atomic emission lines with very specific wavelengths and frequencies that are determined only by the characteristics of the atoms involved. Since atoms of the same atomic weights and numbers are indistinguishable, the atoms in this discharge lamp emit the same spectral lines as those in any other similar lamp. If these lines are used as the timekeepers for an atomic clock, the clock won't need to be calibrated, at least in principle.
Explanation: The light being emitted by the discharge lamp has characteristics that are determined by its atoms. Apart from perturbation effects due to electric fields, magnetic fields, and collisions with other atoms, the wavelengths and frequencies of the spectral lines emitted by these atoms will be identical to those in similar discharge lamps anywhere else. These lines can thus be used as precise timekeepers for atomic clocks. (Note that true atomic clocks must use lines that have very narrow intrinsic linewidths—lines that correspond to relatively weak transitions between atomic levels. Those that are easily observed in the visible light from a gas discharge lamp occur too easily and have rather broad intrinsic linewidths.)
Section 9.2 Musical Instruments
Demonstration 9.2.1: The Fundamental Mode of a String
Description: A long rope that's fixed at one end and that's turned by a variable-speed electric motor at the other end is made to vibrate in its fundamental vibrational mode.
Purpose: To display the fundamental vibrational mode of a string.
Supplies:
1 rope (at least 3 m long and about 1 cm or more thick)
1 elevated support for the fixed end of the rope
1 variable-speed (low speed) electric motor
1 short side arm for the motor so that it can swing one end of the rope around in a small circle
1 swivel clip to attach the rope to the motor's side arm
2 clamps
1 strobe system (optional)
Procedure: Attach the side arm to the motor's shaft so that as the motor shaft turns, the side arm swings around in a circle. Attach one end of the swivel clip to the end of the side arm and attach the other end of the swivel clip to one end of the rope. Attach the other end of the rope to the elevated support. Clamp both the motor and the elevated support to a sturdy table. Overall, the rope should be pulled slightly taut between the elevated support on one end of the table and the electric motor and its side arm on the other end of the table.
Now start the motor turning slowly. The rope will begin to jiggle about at first but when its rotational speed is timed to coincide with fundamental vibrational frequency of the rope, the rope will begin to swing in a wide arc. This motion is the same as that of a normal jump rope. Show that you have to turn the motor at just the right speed or the rope won't vibrate properly. If you increase the tension in the string or shorten its length, you will have to turn the motor more rapidly to excite this fundamental vibrational mode. (If you have a strobe system, time the strobe to fire once per turn of the motor and adjust the phase to freeze the rope while it's an upward or downward curving arc.)
Explanation: The frequency of the rope's fundamental vibrational mode is determined by its mass density, its tension, and its length. The motor's motion can excite this fundamental vibrational mode if it's rotating at just the right speed. This is another case of resonant energy transfer between the turning motor and the swinging rope. Only when they both have the same frequencies of motion is there significant energy transfer from the motor to the rope.
Demonstration 9.2.2: Pluck the String of a Stringed Instrument
Description: You pluck the string of a stringed instrument and it emits a single tone. Changing the length and tension of the string changes the frequency (and pitch) of the tone but the amplitude of the vibration (and the volume of the sound) doesn't affect its pitch.
Purpose: To demonstrate the vibration of a string and to show that it's another type of harmonic oscillator.
Supplies:
1 stringed instrument
Procedure: Pluck the string of the instrument and listen to its tone. Note that the tone starts loud and gradually diminishes, but without changing pitch. Point out that this is evidence that the string is another type of harmonic oscillator. Like many other harmonic oscillators, you can give it a large amount of energy to start with and it will gradually lose this energy as it oscillates or vibrates. Now change either the tension or the length of the string and observe the change in frequency (pitch). Discuss why such a change should occur.
Explanation: The string is vibrating primarily in its fundamental vibrational mode and emitting a single pitch (we'll deal with harmonics later). While the amplitude of the vibration doesn't affect its frequency, the tension and length of the string do. Increasing the tension stiffens the restoring forces and increases the frequency. Shortening the string both stiffens the restoring forces and decreases the mass, again increasing the frequency.
Follwup: Pluck several other things: a wineglass (by tapping it), the air in a bottle (by popping your finger out of the bottle), and even a pendulum (by pulling it back and letting it go suddenly).
Demonstration 9.2.3: The Harmonic Modes of a String
Description: A long rope that's fixed at one end and that's turned by a variable-speed electric motor at the other end is made to vibrate in its harmonic vibrational modes.
Purpose: To display the harmonic vibrational mode of a string.
Supplies:
1 rope (at least 3 m long and about 1 cm or more thick)
1 elevated support for the fixed end of the rope
1 variable-speed (low speed) electric motor
1 short side arm for the motor so that it can swing one end of the rope around in a circle
1 swivel clip to attach the rope to the motor's side arm
2 clamps
1 strobe system (optional)
Procedure: Repeat the procedure needed to demonstrate the rope's fundamental vibrational mode. However, this time continue to increase the rotational speed of the motor until the rope begins to turn as two half-ropes. This second harmonic mode will appear when the motor is turning twice as fast as it was for the fundamental vibrational mode. If you have a strobe system, time the strobe to fire once per turn of the motor and adjust the phase to freeze the rope while it's an S-shaped arc, curving first upward and then downward.
If you increase the motor's rotation rate still further, you'll observe the third harmonic mode (three third-strings), the fourth harmonic mode (four quarter-strings), and so on.
Explanation: The rope's harmonic modes occur when the rope vibrates as several shorter ropes. These harmonic modes occur at multiples of its fundamental vibrational frequency. Although the rope can undergo several different modes of vibration at once, this technique for transferring energy to the rope—resonant energy transfer—only excites one of the modes at a time.
Demonstration 9.2.4: Resonant Energy Transfer in a Pendulum
Description: You give a pendulum a series of carefully timed pushes and cause it first to swing more and more vigorously and then less and less vigorously. Randomly timed pushes do nothing to it on the average.
Purpose: To illustrate resonant energy transfer.
Supplies:
1 tall pendulum (as tall as possible—we use a bowling ball suspended from the ceiling)
1 support for the pendulum
Procedure: Allow the pendulum to settle at its equilibrium position (if it stores energy well, you may have to help it settle). First show that you can give it energy all at once by displacing it from equilibrium and releasing it. In that case, it's energy changes abruptly and it then oscillates at full amplitude. You might point out that as it oscillates, its total energy remains essentially constant, but that this energy transforms back and forth between gravitational and potential energies.
Settle the pendulum at its equilibrium position and this time give it a series of small pushes, timed to coincide with the moments when it's heading away from you. Note that during these moments, you do work on the pendulum by pushing it away from you as it moves away from you. Its amplitude of motion will increase with each properly timed push. You are transferring energy to the pendulum via resonant energy transfer.
Now shift the timing of your pushes so that you push the pendulum when it's heading toward you. Its amplitude of motion will decrease with each properly time push. You are extracting energy from it via resonant energy transfer.
Finally, push the pendulum at randomly timed moments and show that its average amplitude of motion is unaffected. To transfer significant energy to it or from it, you must be in synchrony with it.
Explanation: By timing your pushes to coincide with the cyclic motion of the pendulum, you are allowing energy to flow via resonant energy transfer between two coupled systems with identical frequencies of motion (you are deliberately moving at the pendulum's natural frequency).
Demonstration 9.2.5: Bowing the String of a Stringed Instrument - Resonant Energy Transfer
Description: You bow the string of a stringed instrument and it gradually begins to emit a tone.
Purpose: To demonstrate that bowing is a form of resonant energy transfer that gradually increases the vibrational energy of a string.
Supplies:
1 stringed instrument (ideally a violin)
1 violin bow
Procedure: Slowly draw the bow across the string. Describe the stick-slip process that's occurring as you pull. Whenever the string is moving with the bow, static friction occurs and the bow is able to do substantial work on the string. But whenever the string is sliding against the bow, sliding friction occurs (a much weak force) and the bow does only a tiny amount of negative work on the string. In effect, the bow is exerting carefully timed pushes on the string that always increase the string's vibrational energy. Overall, the vibrational energy in the string gradually increases and it begins to produce significant sound.
Explanation: Bowing is a form of resonant energy transfer because the bow's pushes are always synchronized to the vibration of the string.
Demonstration 9.2.6: Resonant Energy Transfer Between Tuning Forks - Via Contact
Description: You start one of two identical tuning forks vibrating. By carefully sliding them against one another, you transfer the vibration from the first tuning fork to the second tuning fork.
Purpose: To demonstrate resonant energy transfer.
Supplies:
2 identical tuning forks
1 tuning fork mallet (option)
Procedure: Strike one of the tuning forks with the mallet or against a firm object (I used the heel of my shoe). Now hold the two tuning forks parallel to one another and touch them together so that the tip of a tine on the non-vibrating tuning fork is touching the base of a tine on the vibrating tuning fork. Now gradually slide the tines along one another so that the tuning forks are soon side by side and then so that their relationships are reverses: the tip of the tine of the initially vibrating tuning fork should now touch the base of the tine of the initially non-vibrating tuning fork. At this point, the initially vibrating tuning fork will not be vibrating and the initially non-vibrating tuning fork will be vibrating—they will have completely exchanged their vibrational energies.
Explanation: The vibrating tuning fork will do work on the non-vibrating tuning fork over and over and will gradually transfer its energy to the non-vibrating tuning fork. By sliding the two tuning fork across one another, you allow them to efficiently transfer their energy. The coupling between them gradually increases as they move toward being side by side and then gradually decreases as they again move toward being widely separated.
Demonstration 9.2.7: Resonant Energy Transfer Between Two Tuning Forks - Via Their Sound
Description: You start one of two identical tuning forks vibrating. By exposing the second tuning fork to the sound of the first tuning fork, you transfer some of the vibrational energy to the second tuning fork.
Purpose: To demonstrate resonant energy transfer.
Supplies:
2 identical tuning forks, mounted on resonant enclosures that assist the tuning fork in producing or absorbing sound (Our tuning forks sit atop rectangular wooden boxes that are open on one side)
1 tuning fork mallet
Procedure: Strike one tuning fork with the mallet and listen to the sound emerging from the resonant enclosure. Now place the two tuning forks side by side, so that their resonant enclosures face one another, and strike one of the tuning forks. After a few seconds, stop the first tuning fork from vibrating and listen to the sound from the second tuning fork. It will have acquired some of the vibrational energy from the first tuning fork.
Explanation: Sound emerging from the first tuning fork and its resonant enclosure has transferred energy to the second tuning fork and its resonant enclosure. The transfer occurred through air in the form of sound waves. The rhythmic pushes exerted on the second tuning fork and its resonant enclosure did work on the second tuning fork and gradually increased its energy. In principle, the two tuning forks will pass the vibrational energy back and forth completely, with a time of transfer that depends on the coupling between them.
Follow-up: Add a small ball of putty to one tine of one of the tuning forks. The mass of the ball will shift the resonant frequency of that tuning fork and the resonant energy transfer will no longer occur.
Demonstration 9.2.8: Resonant Energy Transfer in a Stringed Instrument - Via Sound
Description: A simple stringed instrument has two strings with identical pitches. When one of the strings is plucked, the second string will also begin to vibrate.
Purpose: To demonstrate resonant energy transfer in a stringed instrument.
Supplies:
1 stringed instrument with two of its strings tuned to the same frequency
1 small piece of paper
Procedure: Fold the small piece of paper in half and drape it over one of the two strings. Now pluck the second string. The first string will soon begin to vibrate, as indicated by the jittering of the piece of paper. If you change the pitch of the second string, by shorting it or changing its tension, this resonant energy transfer will no longer occur.
Explanation: Because the two strings are coupled by the musical instrument and by the air, when one string vibrates it exerts tiny rhythmic forces on the other string. Because these tiny forces are timed to coincide with the vibrations of the second string, they transfer energy between the two strings quite effectively.
Demonstration 9.2.9: Resonant Energy Transfer in a Crystal Wineglass - Via Bowing with Your Finger
Description: You draw your wet finger along the rim of a crystal wineglass and it emits a tone.
Purpose: Another illustration of resonant energy transfer to an object with a fundamental vibrational mode.
Supplies:
1 crystal wineglass
water
Procedure: Wet one finger and draw it slowly and gently along the rim of the wineglass. You are trying to achieve a stick-slip bowing effect, in which your finger sticks while the rim is moving with your finger's motion and your finger slides easily while the rim is moving against your finger's motion. With a little practice, you can get the wineglass to vibrate strongly and emit a loud, clear tone.
Explanation: You are bowing the wineglass in much the same way a violinist bows a violin string. With each cycle of vibration in the wineglass, you add a little energy to its motion. You do a little work on it each time its rim moves in the direction that your finger is moving and you do much less negative work on it each time its rim moves in the opposite direction from that of your finger. Overall, you transfer energy to the vibrating wineglass and it vibrates vigorously.
Demonstration 9.2.10: Resonant Energy Transfer in a Crystal Wineglass - Via Sound (Breaking the Wineglass)
Description: A wineglass is exposed to intense sound from a speaker driver. When the pitch of the tone emitted by the driver is just right and the volume is loud enough, the wineglass shatters.
Purpose: To illustrate resonant energy transfer (and to have lots of fun.)
Supplies:
1 crystal wineglass
1 midrange driver, approximately 100 W (the magnet and coil assembly portion of a large horn speaker—available from audio electronics companies)
1 audio amplifier, 100 W or more
1 audio sine wave signal generator
1 small microphone (and power source, if required)
1 oscilloscope
mounting hardware
1 strobe system (optional)
Procedure: Tap the bowl of the crystal wineglass gently and listen to its fundamental tone. This pitch is the one that will eventually break the glass. Stand the wineglass on a table and mount the midrange driver about 1 cm away from bowl so that the sound waves emerging from the driver will hit the side of the glass just below its rim.
Mount the microphone in the same position relative to the wineglass, but a quarter of the way around the glass. If you look down on the arrangement, the midrange driver should be at 3 O'clock relative to the wineglass and the microphone should be at either 12 O'clock or 6 O'clock. Connect the microphone to the oscilloscope—this will be the system that detects when you are using the correct frequency to drive the speaker. Now connect the audio signal generator to the audio amplifier and the audio amplifier to the midrange driver.
You're ready to begin. Turn everything on and adjust the signal generator's frequency and the amplifier's volume so that the midrange driver begins to emit a gentle tone with the same frequency that you heard when you tapped the wineglass. The wineglass will begin to oscillate weakly and the microphone will detect this oscillation in the wineglass and display a fluctuating voltage on the oscilloscope. Carefully adjust the frequency of the audio signal generator to find the wineglass's precise resonance. When you reach it, the wineglass's vibration will increase dramatically and the microphone and oscilloscope will detect this enhanced vibration.
When the frequency of the audio signal generator is perfect, you're ready to go. If you have a strobe system, time it to flash almost—but not quite—in synch with the audio signal. You will see the rim of the crystal wineglass undergo a remarkable quadrupole oscillation in which two opposite sides—say east and west—will move toward one another as the other two opposite sides—say north and south—move away from one another.
To break the glass, turn up the volume. The tone will probably have to become unpleasantly loud before the wineglass finally breaks. It's amazing how far wineglasses can move before they shatter. How they shatter depends on the wineglass and on your luck. Sometimes they break beautifully into little pieces and sometimes they just crack undramatically.
Explanation: The rhythmic pushes from the sound waves emerging from the midrange driver gradually add energy to the vibrating wineglass. When its vibration exceeds the wineglass's elastic limits, it shatters. Without the oscilloscope, you would have enormous difficulty hitting the resonance accurately enough to break the wineglass. It's extremely unlikely that a singer could hit the required note accurately enough, long enough, and loud enough to break the glass without electronic assistance of some form.
Demonstration 9.2.11: Resonant Energy Transfer in Two Harmonic Oscillators
Description: Two floppy posts can vibrate back and forth independently until you stretch a rubber band between them. Then they transfer energy back and forth between them.
Purpose: To demonstrate resonant energy transfer in a pair of harmonic oscillators.
Supplies:
2 hacksaw blades clamped into a base and tuned with small clamps at their free ends so that they vibrate slowly at the same frequency
1 rubber band
Procedure: Show that the two hacksaw blades vibrate slowly back and forth when you pluck them and that their motions are essentially independent. Then stretch the rubber band between them so that they are now coupled. If you now pluck just one of the blades, it will gradually transfer its vibrational energy to the other and then this energy will flow back and forth. You can also show that there are two time-independent modes of vibration for this coupled system: symmetric and antisymmetric: pluck them both in the same direction for symmetric and opposite directions for antisymmetric. Both modes should not evolve with time.
Explanation: Because the two harmonic oscillators vibrate at the same frequency, they can experienced resonant energy transfer if something passes energy between them.
Demonstration 9.2.12: Resonant Energy Transfer in a Mass and Spring
Description: A weight hangs from a coil spring that's attached to a rope. When the rope is given rhythmic jerks at just the right frequency, the weight begins to bounce more and more vigorously.
Purpose: To demonstrate resonant energy transfer.
Supplies:
1 coil spring (long and soft, if available)
1 weight
Procedure: Hang the weight from the bottom of the coil spring and hold the top of the spring in your hand. Gently pull the top of the spring upward in a rhythmic fashion. Show that if you choose a beat at random, the weight won't move very much. However, when you pull upward in synchrony with the weight's bouncing, you can get it to bounce more and more vigorously. Be careful the weight doesn't bounce off the spring and fall.
Explanation: When the upward movements of your hand are timed properly, they always do work on the spring and mass, and add energy to that harmonic oscillator.
Follow-up: Rather than using your hand, you can suspend the coil and spring from a device that supplies the rhythmic upward movements. We use a cord that runs over a pulley and is then attached to a variable-speed electric motor. As the motor turns, the cord is gently jerked and the spring and mass are similarly jerked upward. Selecting the right frequency causes the weight to begin bouncing wildly.
Demonstration 9.2.13: Resonant Energy Transfer Between a Drill and Some Hacksaw Blades
Description: Three hacksaw blades are clamped to a board so that they project outward from the board by different amounts. A variable-speed electric drill with a bent nail in its chuck is also attached to the board. When the drill is turning at just the right speed, one of the hacksaw blades begins to vibrate strongly.
Purpose: To demonstrate resonant energy transfer.
Supplies:
3 hacksaw blades
1 board (about 30 cm on a side)
1 small board (to hold the hacksaw blades against the other board)
1 variable-speed electric drill
1 large nail, bent at a right angle
2 clamps
Procedure: Place the 3 hacksaw blades on the board, so that they extend outward from its edge by different amounts. Place the small board on top of the blades, along the edge of the larger board and clamp the two boards together so that the blades can't slide or move. Put the bent nail in the chuck of the drill and clamp the drill to the board. Make sure that the free end of the bent nail won't hit either you or the board as the drill chuck rotates.
Start the drill rotating. As you slowly increase the drill's rotational speed, the hacksaw blades will begin to move slightly. When you reach the resonant frequency of the longest hacksaw blade, it will begin to vibrate strongly. Keep increasing the rotational speed until the middle length blade and finally the shortest blade exhibit their resonances.
Explanation: The rotating nail is transferring energy to the hacksaw blades via resonant energy transfer. Only when it's turning at just the right rate will it be able to push on one of the hacksaw blades in synchrony with that blade's vibrational motion.
Demonstration 9.2.14: Helping a Tuning Fork Produce Sound
Description: By itself, a tuning fork produces very little sound. But when you hold a cardboard frame around one tine of the fork, its volume increases substantially.
Purpose: To show that narrow vibrating objects (e.g. violin strings) aren't very good at producing sound.
Supplies:
1 tuning fork
1 piece of cardboard, with a slot cut in it just a little wider than the side width of the tuning fork's tines
1 tuning fork mallet (optional)
Procedure: Strike the tuning fork with the mallet or against a firm object (again, I use the heel of my shoe). Point out how weak its sound is. Now repeat this procedure, but hold the vibrating tuning fork up behind the cardboard sheet so that one of its tines vibrates in and out of the slot in the cardboard sheet. The volume of sound emitted by the tuning fork will increase dramatically.
Explanation: The wavelength of the sound that the tuning fork emits is much larger than the width of the tuning fork's tines. As a result, air has plenty of time to move around the tines during each cycle of vibration. Thus instead of pushing the air toward and away from your ear as it vibrates, each tine tends to push the air back and forth around its surfaces. Blocking the path around the sides of the tine prevents the air from flowing around it and helps the tine push the air toward and away from your ear. You hear much more sound as a result.
Follow-up: Listen to a very small, unenclosed speaker playing music. Then place the speaker against a hole in a broad sheet of cardboard. Again, the volume of the speaker will increase dramatically when the air can no longer flow around its sides from front to back and must instead form compressions and rarefactions that travel as sound to your ears.
Demonstration 9.2.15: Helping a Music Box Produce Sound
Description: By itself, a music box produces very little sound. But when you touch it to a hard surface, its volume increases substantially.
Purpose: To show that narrow vibrating objects (e.g. the music box's teeth) aren't very good at producing sound.
Supplies:
1 music box without its case
1 hard surface
Procedure: Play the music box in the air, without touching anything hard. Point out how weak its sound is. Now touch it to the hard surface and notice that the volume of sound emitted increases dramatically.
Explanation: The teeth of the music box are too thin to compress and rarefy the air effectively. The teeth simply vibrate back and forth ineffectually. But when you touch the music box to a surface, the vibrations of the teeth cause the surface to vibrate and that surface projects sound nicely.
Demonstration 9.2.16: Air and Helium Vibrating in a Bottle
Description: You blow gently across the lip of a bottle and it emits a tone. Adding water to the bottle raises the pitch of that tone. The amplitude of vibration (and the volume of the sound) don't affect its pitch.
Purpose: To show that a column of air can vibrate as a harmonic oscillator.
Supplies:
1 beverage bottle with a narrow neck
water
helium
Procedure: Place your lips against the mouth of the bottom and blow gently across the mouth of the bottle. Air from your mouth should be directed so that it can flow either over the far edge of the bottle mouth or against that far edge. When you aim the air correctly, it will cause the air inside the bottle to begin vibrating and the bottle will emit a tone. Adding water to the bottle will shorten the air column inside it and raise the frequency and pitch of the tone. Point out that the amplitude of the vibration (and the volume of the tone) don't affect the frequency (and pitch) of the tone—you have another harmonic oscillator. Replacing the air in the bottle with helium will raise the pitch (briefly -- the helium quickly escapes; holding the bottle upside-down helps) because helium has a smaller density than air.
Explanation: Air from your mouth is adding energy to the vibrating air in the bottle. The pressure in the bottom of the bottle is fluctuating up and down, and the velocity of the air in the neck of the bottle is fluctuating in and out. Air from your mouth joins air vibrating in and out of the neck of the bottle, doing work on that vibrating air at just the right times to cause resonant energy transfer. By adding water to the bottle, you shorten the air column, stiffening its restoring forces and decreasing its mass. As a result, its frequency of oscillation (and its pitch) increases. Putting helium in the bottle lowers the density while leaving everything else unchanged so that system vibrates faster and at a higher pitch.
Demonstration 9.2.17: Air Vibrating in an Organ Pipe
Description: As air blows through the whistle of an organ pipe, the pipe emits sound.
Purpose: To demonstrate how an organ pipe makes sound.
Supplies:
1 organ pipe (or a penny whistle or a recorder, which are effectively small, shrill organ pipes)
1 air blower (or your mouth)
Procedure: Connect the organ pipe to the air blower and start it emitting sound. Point out that air is vibrating in and out of both ends of the pipe—the open top and the open hole in the whistle at the base of the pipe. The air being blown across the whistle is adding energy to the air vibrating in the pipe. Note also that changing the volume of the pipe doesn't change its pitch—it's a harmonic oscillator.
Explanation: In the organ pipe's fundamental vibrational mode, air is flowing into or out of both ends of the pipe at the same time and the air pressure near the middle of the pipe is fluctuating up and down around atmospheric pressure.
Demonstration 9.2.18: Air Vibrating in a Carpet Tube
Description: A giant cardboard tube is lowered over a large gas burner. A loud, low tone soon emerges from the tube.
Purpose: A fun demonstration of resonant energy transfer.
Supplies:
1 or more carpet tubes (large, sturdy cardboard tubes placed at the centers of wall-to-wall carpet when it's delivered to a carpet store. Other wide pipes will also work.)
1 large gas burner (e.g., a Fisher burner)
matches
water (to extinguish a burning carpet tube, if necessary)
Procedure: Light the burner and place it on the floor. Slowly lower one end of the open carpet tube over the burner. It will emit a low, loud tone. This tone may actually extinguish the burner, so be careful. Also be careful not to start a fire. The longer the tube, the lower its pitch.
Explanation: Hot, rising air from the flame tends to add energy to the air vibrating up and down near the lower end of the tube. Through resonant energy transfer, the flame gradually increases the strength of this vibration.
Follow-up: We have a shorter metal tube (about 1 m long and about 4 cm in diameter) that contains a piece of stainless steel gauze near one end. When that gauze is heated red hot by a burner and the tube is then held vertically, with the hot gauze at the bottom, it emits a tone. Tipping the tube horizontally stops the tone but the tone reappears when the tube is returned to its vertical orientation.
Demonstration 9.2.19: Blow Across an Open and Closed Straw
Description: You blow across the end of an open straw and it emits a tone. When you block the bottom of the straw with your finger, the pitch suddenly drops by an octave..
Purpose: To show that the air vibrating in a straw has different node structures when that straw is opened and closed.
Supplies:
1 drinking straw
Procedure: Hold the straw vertically and blow across the top to produce a tone. Block the bottom of the straw as you continue to blow and the pitch will drop by an octave.
Explanation: The tone you hear when the straw end is opened is the fundamental vibration of an open-open air column. .The pressure antinode is at the middle of the straw. But when you block the bottom of the straw, it emits a tone corresponding to the fundamental vibration of an open-closed air column and the pressure antinode is at the blocked end.
Demonstration 9.2.20: Harmonic Vibrations in a Plastic Tube
Description: You hold an open plastic tube by one end and swing it in a circle. It emits a tone that changes in discrete steps as its speed changes, like the tones of a bugle.
Purpose: To show that the air vibrating in a container can exhibit harmonic vibrational modes.
Supplies:
1 flexible plastic tube with two open ends (about 2 or 3 cm in diameter and about 1.5 m long)
Procedure: Hold one end of the tube and swing it rapidly in a circle. Keep the end that you're holding relatively still. A tone will soon emerge from the tube. If you swing the tube relatively slowly, the tone will be low. But as you swing the tube faster and faster, you'll hear a series of higher pitched tones.
Explanation: The lowest tone that you hear at low speeds is the fundamental vibrational mode of the tube—the air at the tube ends flows inward and outward together and the air at the middle of the tube experiences up and down pressure fluctuations but no velocity fluctuations. The higher tones correspond to harmonic vibrational modes, in which there are two or more regions within the tube that are experiencing up and down pressure fluctuations but no velocity fluctuations.
Demonstration 9.2.21: Vibrational Modes of a Surface - Chladni Plates
Description: You sprinkle sand on a metal plate that's supported at its center and then bow its edge with an cheap violin bow. The plate emits a tone and the sand forms interesting patterns on the plates surfaces.
Purpose: To show that surfaces can also act as harmonic oscillators, with pitches that don't depend on their amplitudes of motion, and to show that the pitches of their harmonic vibrations don't occur at simple integer multiples of their fundamental pitches.
Supplies:
1 Chladni plate (a hard metal plate that's supported at its center by a rigid post and clamp)
1 cheap violin bow
sand
Procedure: Mount the Chladni plate on a sturdy table so that it's surface is horizontal. Sprinkle sand lightly on its surface. Now bow the edge of the plate gently. Try to keep the bow in the same spot on the plate edge as you bow. The plate will begin to vibrate and the sand will begin to move. When you excite a strong vibrational mode of the surface (and hear a clear tone), the sand will move into the vibrational nodes of the mode—the portions of the plate that don't move while the plate is experiencing that vibrational mode.
Experiment with bowing at different places around the plate and you'll find the fundamental vibration and various harmonics. Note that the harmonic frequencies of this vibrating two-dimensional surface aren't simply integer multiples of the fundamental vibrational frequency. Only in some one-dimensional oscillators such as strings and organ pipes are the harmonic frequencies all integer multiples of the fundamental vibrational frequency.
Explanation: Surfaces have complicated vibrational patterns and don't vibrate as "half-surfaces or third-surfaces" the way strings do. As a result, their harmonic vibrations have pitches that aren't integer multiples of their fundamental pitches and they have interesting patterns of vibrational nodes and antinodes for any given harmonic vibration. The sand tends to accumulate in the nodes, so that you can see these patterns. Just how you bow the plate (and where you might be touching it as well) determines which vibrational mode is excited by the bow and which pattern the sand adopts.
Section 9.3 The Sea
Demonstration 9.3.1: Water Sloshing in a Tank
Description: You move your fingers gently back and forth in a rectangular tank of water. When you move your hands rhythmically at just the right frequency, the water begins to slosh vigorously.
Purpose: To show that resonant energy transfer can excite a natural resonance in the water.
Supplies:
1 rectangular water tank (a glass or plastic aquarium)
water
Procedure: Fill the tank half full of water and allow the water to settle. Insert your hand into the water and jiggle it back and forth randomly. Point out that the water acquires relatively little energy from this random motion. Now move your hand back and forth rhythmically so that you excited the fundamental sloshing mode for the water in the tank. Pretty soon the water will be sloshing vigorous back and forth and may even slosh out of the tank.
Explanation: The water has a natural resonance in which it travels back and forth from one end of the tank to the other. You are exciting that resonance by pushing it forward as it sloshes forward and backward as it sloshes backward. In the tank, this resonance has a relatively high frequency of perhaps once per second. But in a huge channel, it may have a period of 12 hours and 26 minutes so that it can be excited by the tide.
Demonstration 9.3.2: Traveling Waves on a Long Slinky (Transverse)
Description: A long Slinky stretches from your hand to a fixed point far away. Quick shifts of your hand cause ripples—transverse waves—to travel along the Slinky.
Purpose: To illustrate transverse waves that are similar to water surface waves.
Supplies:
1 long Slinky (or a loose spring or even a rope)
1 fixed support
Procedure: Attach one end of the Slinky to the fixed support. Hold the other end of the Slinky in your hand and stretch it just enough to lift the middle well off the floor. Now jerk the end that you have in your hand upward and then back to its starting place. An upward heading ripple will head out across the Slinky. When it reaches the fixed end, it will reflected back toward you. The speed with which the wave travels increases with the tension in the spring and decreases with the spring's mass density. That's why a loose but massive Slinky makes it possible to have slow moving transverse waves.
Explanation: The Slinky behaves like a very massive, low-tension string. While it could easily be made to exhibit standing waves, such as its fundamental vibrational mode, it can also be made to exhibit transverse traveling waves.
Demonstration 9.3.3: Traveling Waves Between Magnets (Longitudinal)
Description: A set of repelling magnetic rollers rests in a track. When one of the rollers is suddenly displaced, it initiates a longitudinal wave that passes through the whole collection of rollers.
Purpose: To illustrate longitudinal waves.
Supplies:
1 set of magnetic rollers on a track (available from a scientific supply company)
Procedure: Turn the rollers so that they repel one another and allow them to settle so that they're even space on the track. Now push the last roller toward its neighbor and watch the longitudinal wave travel from roller to roller all the way along the track.
Explanation: At rest, all of the rollers are in equilibrium. Displacing one upsets the equilibrium of the next, which in turn upsets the equilibrium of the next and so on. The speed with which the wave travels through the collection of rollers depends on their masses and on the stiffness of the forces between them.
Demonstration 9.3.4: Traveling Waves on a Wire (Torsional)
Description: A wire that's supporting a long collection of torsion beams rests on a support. When one of the torsion beams is twisted, it initiates a torsional wave that pass down the wire and twists each of the torsion beams in turn.
Purpose: To illustrate torsional waves.
Supplies:
1 wire with torsion beam attached (available from a scientific supply company)
Procedure: Set up the wire and allow the torsion beams to settle to a horizontal orientation. Now displace the beam at one end of the wire and then return it to its original orientation. A torsional wave will travel along the wire from torsion beam to torsion beam.
Explanation: At rest, all of the torsion beams are in equilibrium (they're experiencing zero torque). Displacing one beam upsets the equilibrium of the next and so on. The speed with which the wave travels through the torsion beams depends on their moments of inertia and on the torsional stiffness of the wire that connects them.