The students will witness several examples of periodic motion and develop some basic mathematical concepts and definitions used in the description of periodic motion.
The instructor could provide several of the following devices as examples: grandfather clock, torsion pendulum clock, bouncing ball (super ball works well), ball bearing in a spherical mixing bowl, a weight hanging from a string as a pendulum, a weight bouncing on the end of a spring, a tuning fork, or better, one end of a yardstick held tightly on the surface of a table and the other end "plucked" so that it oscillates, rocking chair, beating heart, clarinet reed, line dancing. The instructor may be tempted to use circular motion as an example but he should not yet.
By recalling the scientific method, the instructor should encourage the students to observe, measure, hypothesize and test. The instructor should allow the students wallow in each of the demonstrations but offer hints so the students can understand that periodic motion is a very general concept. In this session, we hope the students can come to some very general mathematical expressions concerning the definition of periodic motion. When considering what quantities are appropriate to measure, the students should quickly recognize to some of the quantities listed below.
The period of periodic motion is usually denoted by the symbol T.
The frequency of periodic motion is usually denoted by the symbol f or ν.
Period and frequency have an inverse relationship: T = 1/ν Discuss units.
The amplitude, A, of periodic motion is often defined as the half the difference between an object;s maximum and minimum position (or angle). However, there are instances when the whole distance between maximum and minimum can be used as the amplitude.
We hope that the students can state the following definition of periodic motion at the end of their investigation.
We say periodic motion occurs with period T when the position (or angle, or magnitude, or any other measurable quality of a phenomena), F, obeys the following relationship for all values of time, t: F(t) = F(t + T) We should probably also require that F be a non-constant function.
The students will discover the relationship between Hook's Law and harmonic motion.
Again, the instructor should review the scientific method. The instructor should demonstrate simple harmonic motion with a weight and spring where the weight is much heavier than the spring. Vertical motion makes a good demonstration.
The students should soon understand that the two relevant physical laws are Newton's second law of motion and Hook's law.
F = ma
F = -kx
The instructor may have to guide the students through some
intricacies
of the scientific method. In this case we have theory
described in the two equations above. Students do not have the
mathematical
tools to take full advantage of the theory because the two
equations form a differential equation whose solution
describes
harmonic motion quite well. That is,
is exactly equivalent to
Students can, however, observe the motion of the system and make good guesses about how the mass moves in time. Encourage the students to draw a a graph or devise a function of the form x = F(t). The "correct" answer is x = A sin (bt + c) where A, and c are constants that depend on the amplitude and "phase" of the motion. The constant b is defined by the solution to the differential equation. Students will investigate A, b and c in the next lesson.
After finding a graph or function, encourage the students to build a another graph that shows the velocity as a function of time. The students will need to remember that the velocity of the mass, at any instant in time, is the slope of the line showing the position of the mass as a function of time. The students can continue in this vein by building a graph showing the acceleration of the mass as a function of time. Finally, the students should compare their acceleration graph to the physical laws described in points 1 and 2 above.
Students will investigate the mathematical details of harmonic motion, especially relationship between the spring constant, the mass and the period.
This lesson requires the instructor to give a small presentation about two common mathematical techniques used in physics, that is, normalization and dimensional analysis. The outline of the presentation follows:
As we learned in the last lesson, harmonic motion can be described as x = A sin (ωt + δ).
A is the amplitude of the motion. It is a positive number and it corresponds to the maximum distance the weight moves from the equilibrium position.
δ is a constant "phase". It corresponds, roughly, to an offset in the starting position of the mass or, equivalently, an offset in the starting time. For example, if δ is 90°, then we could describe the motion as x = A cos(ωt).
More interesting is ω. We know that the period of a specific instance of harmonic motion is given by T. We can "normalize" the time, t, by dividing it by the period. As t increases from 0 to T, the value of t/T increases from 0 to 1. (The value of t/T is called the normalized time.) However, we know the the sin function goes though a full period as its argument goes from 0° to 360°. So it makes sense to define ω in the following way:
and then
From this equation we see that as t goes from 0 to T, the argument of sin goes from 0° to 360°, just as we would like. Physicists often use normalization techniques to make mathematics more understandable. We could call this definition of ω an experimental or "practical" definition.
We should also develop a theoretical expression for ω. We see immediately that ω is proportional to the inverse of T.
Going back to the physics that first predicted harmonic motion, F=ma and F=-kx, we should consider the units of each of the terms. (This technique is called "dimensional analysis".)
F has units of kg m/s2, i.e., kilogram meter per second squared
m has units of kg, i.e., kilogram
a has units of m/s2, i.e., meter per second squared
x has units of m, i.e., meter
k has units of kg/s2, i.e., kilogram per second squared
T and t have units of s, i.e., second
ω has units of 1/s, i.e., 1/second
Angles are dimensionless so things like ωt and δ have no units.
Again going back to the equations that predicted harmonic motion, what quantities can we change that will change the period T? What do we have to do to those quantities to make the period longer or shorter? Feel free to run quick experiments to validate your conjectures. Finally, using dimensional analysis, define a more theoretical expression for ω. Test the prediction with experiment.
The students should discover that ω is proportional to the square root of k/m, or
Both the instructor and the students should be aware that any expression created by dimensional analysis is not a rigorous mathematical derivation but a "scientific" guess. In this particular case, we are still missing a constant of proportionality.
In this mathematically oriented lesson, the students will examine simple harmonic motion in terms of energy rather than force.
Using the old reliable demonstration of a mass bouncing on the end of a spring, encourage the students to examine simple harmonic motion in terms of energy. Remind them that they should think in terms of an ideal harmonic oscillator, that is, one where the spring is massless and there is no energy dissipation. Point out that, in terms of the scientific method, we are looking at our hypothesis and trying to find predictions and extensions to the hypothesis that will lead to more experiments. Let the students wallow in the mathematics up to the point of frustration but not far beyond.
The students should arrive at some of the concepts listed below.
Encourage the students to create graphs for the potential and kinetic energy of the system. The students can choose to graph in terms of position or time or both. Ask them to draw as many conclusions as possible from the graphs. The students can discover some of these results.
Beware! In order to determine the potential energy, the students will have to find the potential energy for a system with a force that varies with position. In all previous classroom situations, the force has been constant so the students could simply say U = Fx. But now
The non-calculus students should arrive at this point and be stumped or get the wrong answer. The instructor can indicate that the proper value may be found graphically using a graph of Hook's law, similar to the one shown below. The area under the line is exactly the value of the integral above.
Suggested homework: Ask the students to think about a simple pendulum. Ask them to attempt to describe a simple pendulum in terms of Hook's law. If they can, what are the physical parameters that will give the period of the pendulum? If they cannot, what keeps the pendulum from behaving like a simple harmonic oscillator? Either answer can be correct if the student can make good supporting arguments.
The students will see examples of both damped and driven harmonic motion. They will explore the limits of damping and driving.
Students will participate in two "bodily" demonstrations. The first will be a demonstration of damped harmonic motion with students swinging in a playground swing and dragging their feet through water in a children's pool just below the swing. The second will be a demonstration of driven harmonic motion where the students again swing in a playground swing while their classmates gently drive them higher or lower with foam pool noodles. The "experiments" will be more dramatic and more helpful if two students of very nearly the same weight are swinging in parallel. By comparing the position and amplitude of the swinging students, our budding physicists will be able to tell when the amplitude and frequency of the "damped" swinger changes.
The instructor should first attempt to get the students back into a reasonable state of mind after the splashing, shoving, demonstration. Ask the students to apply the scientific method to the "experiments" they have just performed. There are two or three major forks the students may take. The first is a theoretical, mathematical approach where Newton's second law and Hook's law are combined with some damping or driving term. This approach will lead to a differential equation which will be difficult to solve but very entertaining to create. The second is a more pragmatic approach. In this approach, the student will probably start with the basic description of harmonic motion, x = A sin(ωt + δ), and convert A to some function of t. The third approach the student might take is to consider the energy description and harmonic and put in a term or two describing the change in energy from damping or driving.
Every path the students take in this lesson will lead them to unfamiliar mathematical concepts and techniques. The instructor needs to be prepared to fill in the concepts where appropriate or indicate that the concept is too advanced to teach in the short time available.
The following list provides some guidance for instructors.
The students will use slinkies to investigate transverse and longitudinal waves. With some luck, they will discover standing waves.
Each group will use a slinky in a non-directed way to discover various properties of waves. Every 15 minutes or so, each group should report any new concept it has discovered. By the end of classtime, it would be good if theclass had discovered the following properties of waves:
Using a tin can telephone and a water wave machine, the instructor will encourage the students to develop the mathematics necessary to describe waves in both space and time.
The water wave machine will be the most helpful experiment for the students because the waves move slow enough that students will be able to see harmonic motion if they look in only one place and a sinusoid if the mentally "freeze" the wave motion in time and look at the whole water tank. The trick will be for the students to recognize that the sinusoid they used in harmonic motion can be simply extended to describe the wave traveling in space. All the have to do is add the spatial term to the temporal term in the argument of the cosine function.
The instructor may want to introduce the concept "radian" for angular measure. Some high school students may not familiar with radian measure. Gently guide the students to the "normalization" concept used earlier for the temporal argument. That is, the distance should be normalized to the wavelength with the x/λ term. Then the normalized distance should be multiplied by the angle of one full circle (360° or 2π radians). We show the radian form of the traveling wave expression above and the degree form below.
The instructor may want to invent some problems to reinforce the indeas just devloped. For instance, the students could consider two wave trains of the same phase velocity and frequency moving in opposite direction. From this starting point, the students could consider waves of different amplitude, direction, and phase velocity. Eventually, the students must consider how circular waves interact but that will be done in a few lessons so no need to consider it here.
Ask the students to consider the tin can telephone and develop a hypothesis on its inner workings. The students should pay special attention to the way the "sound" is transmitted from one can to another.
Using "rubber duckies" or obstacles of various sizes in a wave tank, the students will see waves reflected from obstacles. They will also see wave pass "through" obstacles. The students should accurately describe what they see and formulate a general verbal description of waves interacting with obstacles.
The instructor can divide this lesson into two parts. In the first part, the students will observe how obstructions (rubber duckies) interact with waves. Choose some rubber duckies that are larger than the wavelength, some that are about the same size and some that are much smaller. Encourage the students to write non-mathematical descriptions of how the waves and duckies interact.
In the second part of the lesson, introduce Christiaan Huygens' ideas about the interaction of waves with matter. That is, introduce the Hugyens-Fresnel principal. Recall that Huygens proposed that every point to which a wavelike disturbance reaches becomes a source of a spherical wave, and the sum of these secondary waves determines the form of the wave at any subsequent time. He assumed that the secondary waves traveled only in the "forward" direction.
Ask the students to use Hugyens' principal (in drawings made with a compass) to show how wave interact with objects smaller, about the same size and larger than the wavelength with which the objects interact. They should compare their predictions with what they observed in the tank. HINT: Always use rectangular obstructions aligned perpendicular to the wave motion because the the geometry becomes much simpler.
Also remind the students that Huygens principal is not a theory but a mathematical "technique" for finding answers about wave behavior. One can use more sophisticated wave theory including calculus to show that Huygens principal is a natural consequence of wave theory. However, this fact is of no use to the students at the point.
Using a wave machine and a wave blocker with single and multiple slits, the students can discover one of the most far-reaching properties of waves: interference.
Using a wave machine with appropriate slits, show the students the general technique for generating interference patterns in the surface of the water. Encourage them to see what happens with the frequency changes and the spacing of the slits change. The students should have the mathematical tools (from lesson 5.7) to develop a good hypotheses about this system.
It might be worthwhile for the instructor to indicate the importance of the idea of interference. It can be used to show that light behaves as if it is a wave. Even more intriguing is that it shows one of the most mind-boggling properties of quantum mechanics: quantum action at a distance.
The instructor should try to play a tune on a yardstick or ruler. To perform this feat, the instructor should hold one end of the ruler firmly on a desk and allow the other end to protrude over the edge of the desk several inches. Then the instructor should gently "pluck" the end of the ruler and it should vibrate. By changing the length of the protruding portion of the ruler, the instructor can change the pitch. Provide the students with their own rules and create a jazz ensemble.
Show the students that when the free end of the ruler is long, the vibrations in the plucked ruler are clearly visible. That is, the ruler is performing periodic motion of a relatively low frequency. When the free end is short, the vibrations are too fast to see. The ruler is performing periodic motion of a higher frequency. Challenge the students to measure the frequency of the vibrating ruler.
Introduce the idea of pitch and relate it to frequency. That is, a higher frequency corresponds to a higher pitch.
Ask the students to relate the vibrating ruler to sound and develop a hypothesis on the way the vibrations in the ruler are perceived as sound.
In the lesson, the students will attempt to measure the speed of sound.
Permit the students to discovery ways to measure the speed of sound. Help them think about techniques but let them do most of the work. It would be good if the students performed a quick experiment to get a ballpark figure for the speed of sound. Such an approximation will help them make better decisions when devising a technique for finding a more accurate value.
If the instructor wants to keep this lesson simple, consider the use of echos. A student might repeatedly clap his hands at such a rate that an echo arrives back at the student's ears just as the student claps his hands. Determine the clapping period and the total distance the sound traveled. The speed immediately falls out of these two numbers.
If an echo producing site is not immediately available, perhaps one student could move far away and start clapping at a period such that the sound arrives back at the origin just as the distant student claps again. (A second student could use a metonome to help the first student maintain a constant period.) When the distant students achieve the correct period, the sound of the first clap will arrive at the origin just as the second clap occurs and it will produce the illusion that the sound is arriving instantaneously. Cell telephones could be used by the students to help adjust the clapping period to just the right value. Again, determine the clapping period accurately and the distance between the origin and the far students to find the speed of sound.
Hopefully, at least one of the students in class is a string player. Ask the string players to bring their instruments to class. If there are no string players, ask the orchestra instructor or some other string player to be your guest for the day and bring a bass to class.
Just for fun, attempt to play a simple duet where the instructor uses the ruler (as in the previous lesson) and the class or guest plays the stringed instrument. The students should immediately understand that instruments are designed to produce sound and rulers are not.
Use a slinky, as in lesson 5.6, to review the concept of
"standing" waves. Attempt to show at least three standing waves. Define
the term "resonance" for the students.
By analogy, ask the string players to play the fundamental tone on their "lowest" string, that is, the string that produces the lowest frequency. Then ask the students to play the string again but attempt to produce an overtone. From the "Violin" entry of Wikipedia, "Lightly touching the string with a fingertip at a node creates harmonics. Instead of the normal tone, a higher pitched note sounds. Each node is at an integer division of the string, for example half-way or one-third along the length of the string."
Ask the students to develop a theoretical and experimental method to determine the frequency ratio between the first harmonic, f1 and the nth harmonic, fn.
This lesson presents some of the basic theory of the tone system used in most western music. This lesson is not strictly physics but many physics students are also musicians so this lesson can be fun for them. Because the lesson is not physics but more about the definition of tone and harmony, it must be presented in a lecture format. The instructor must be sure to include the students in the musical formulation.
First discuss harmony as a combination of tones where the tones are, in a sense, overtones of the fundamental tone. We know that the frequency of overtones are integer multiples of the fundamental tone. We also know that when the integer multiple is a power of two, the fundamental and overtone have a special relationship. We call these multiples "octaves". The table below emphasizes this relationship.
Frequency Multiplier | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Overtone | Fundamental | First | Second | Third | Fourth | Fifth | Sixth | Seventh | Eightth |
Octave | Fundamental | One octave above fundamental | Two octaves above fundamental | Three octaves above fundamental |
We know that tones lower in frequency than overtones are used to produce harmony. These lower frequency tones are related to overtones by octave displacements. Take the second overtone for an example. It has a frequency three times higher than the fundamental. However, if we lower its pitch by an octave (divide the frequency by two), its frequency is 3/2 times the fundamental. Dividing in this way make the "reduced" overtone have a frequency between the fundamental and the first octave above the fundamental. Similarly, the fourth overtone can be transformed into a pitch whose frequency is 5/4 the fundamental. Our goal is to produce a pitch (frequency) that lies between the fundamental and its first overtone. Equivalently we can say we want to produce a multiplier for the fundamental frequency that lies between 1 and 2. We can use any appropriate power of two (octave) as our divisor to produce a fraction in the required range. If we continue this process, we can create a table like the one below.
Frequency Multiplier | 3 | 5 | 6 | 7 | 9 | 10 | 11 | 12 | 13 |
"Reduced" Pitch Multiplier | 3/2 | 5/4 | 3/2 | 7/4 | 9/8 | 5/4 | 11/8 | 3/2 | 13/8 |
440 Hz Fundamental Tone | 660 Hz | 550 Hz | 660 Hz | 770 Hz | 495 Hz | 550 Hz | 605 Hz | 660 Hz | 715 Hz |
Table 5.13.1. An overtone table.
Let's take a different approach to thinking about pitch. Recall that the higher tone in an octave interval is characterized by a frequency that is exactly double the lower tone. That is, the higher tone is the first overtone of the lower tone. We can express this frequency relationship with the equation
fo = 2 ff
where fo is the frequency of the first octave and ff is the frequency of the fundamental. We can generalize the equation to describe the frequency of several octaves both above and below the fundamental with the equation
fon = 2nff
where n is an integer (positive or negative), and fon is the frequency of the nth octave from ff. Theoreticians have found that this equation is even more useful if it is expressed as a logarithm. They write
Musicians noted that, within the logrithmic expression above, they could identify each octave with a separate integer n. They thought that they could identify each note within an octave by a constant value also. Since they liked the twelve note octave, they created a table for the tones in the first octave similar to the one below. Since n is the octave index in the equations above, we need to change the index to another symbol, m, to indicate the tone index within an octave in the table below. Then we can fill in the table using the equation above, replacing the n with the values of m below.
m | 0/12 | 1/12 | 2/12 | 3/12 | 4/12 | 5/12 | 6/12 | 7/12 | 8/12 | 9/12 | 10/12 | 11/12 |
2m | 1.00000 | 1.05964 | 1.12246 | 1.18921 | 1.25992 | 1.33484 | 1.41421 | 1.49831 | 1.58740 | 1.68179 | 1.78180 | 1.88775 |
440 Hz Fundamental Tone | 440.0 | 466.2 | 493.9 | 523.3 | 554.4 | 587.3 | 622.3 | 659.3 | 698.5 | 740.0 | 784.0 | 830.6 |
Table 5.13.2 A scale table
The students should see that at least two of the interior tones in the scale table nearly match the overtone table. That is, the m=4/12 nearly matches the fifth overtone and the 7/12 tone nearly matches the sixth overtone. One can find higher overtones that will nearly match every note in the scale, but the lower overtones (those 17 and below) are not matched very well.
We know that lower overtones sound good when played with the fundamental. They make harmony. We know that a scale will divide an octave into a reasonable set of tones. However, the two are not well matched except in a few places. This mismatch has been a source of entertainment for musicians for more than 300 years. Students can find lots of sources describing how musicians have dealt with this conflict over the years.