**This question is the third in a series, after:**

1. Improved Typing as a result of slight movement

2. Neural Processes of Inducing Flow

**Background:**

Pseudo-random, 'swaying', motion appears to help induce a flow state, in that it 'captures' the movements of an activity and entrains them to an underlying rhythm of activity. Time between activity-related movements is reduced, and error-recovery time is also reduced. (again, this is from first-hand observation, not published results).

*The following is the relevant explanatory section of the 2nd question's answer:*

Seen from this perspective, cyclical movements are the norm for almost any animal, whereas short-duration, "single-use" movements like typing or playing the piano are rather unusual. It could be the case that if the motor cortex (or even the basic movements encoded in the spinal cord) is inherently tuned to modeling cyclic movements, then adding some continuous motion could help the motor cortex capture the intended typing or playing movements as part of the larger, continuous movement.

The last question I have is if the frequency of the periodic movements, which in general follow a 'figure-8' track, could be related to the frequency of other cycles in the body? For example, could the frequency of motion reflect the active frequency of the brain, e.g. EEG waves? Or could it reflect a different neural/physiological cycle or state?

Naturally this is a tricky question: on one hand, the activity itself serves to entrain the neural system to a certain periodicity. On the other hand, if the rhythmic movement activity is allowed to 'float' in frequency organically, does its characteristics reflect anything about the initial state of the neural system?

**Final question(s):Does the frequency of movement give clues to other periodic cycles, such as EEG waves?Does said movement give any other insight into the brain/body state?**

## 7 Answers 7

Sinusoidal waves don't have harmonics because it's exactly sine waves which combined can construct other waveforms. The fundamental wave is a sine, so you don't need to add anything to make it the sinusoidal signal.

About the oscilloscope. Many signals have a large number of harmonics, some, like a square wave, in theory infinite.

This is a partial construction of a square wave. The blue sine which shows 1 period is the fundamental. Then there's the third harmonic (square waves don't have even harmonics), the purple one. Its amplitude is 1/3 of the fundamental, and you can see it's three times the fundamental's frequency, because it shows 3 periods. Same for the fifth harmonic (brown). Amplitude is 1/5 of the fundamental and it shows 5 periods. Adding these gives the green curve. This is not yet a good square wave, but you already see the steep edges, and the wavy horizontal line will ultimately become completely horizontal if we add more harmonics. So this is how you will see a square wave on the scope if only up to the fifth harmonic are shown. This is really the minimum, for a better reconstruction you'll need more harmonics.

Like every non-sinusoidal signal the AM modulated signal will create harmonics. Fourier proved that *every* repeating signal can be deconstructed into a fundamental (same frequency as the wave form), and harmonics which have frequencies that are multiples of the fundamental. It even applies to non-repeating waveforms. So even if you don't readily see what they would look like, the analysis is always possible.

This is a basic AM signal, and the modulated signal is the product of the carrier and the baseband signal. Now

So you can see that even a product of sines can be expressed as the sum of sines, that's both cosines (the harmonics can have their phase shifted, in this case by 90°). The frequencies $(f_C - f_M)$ and $(f_C + f_M)$ are the *sidebands* left and right of the carrier frequency $f_C$.

Even if your baseband signal is a more complex looking signal you can break the modulated signal apart in separate sines.

Pentium100's answer is quite complete, but I'd like to give a much simpler (though less accurate) explanation.

The reason because sinewaves have (ideally) only one harmonic is because the sine is the "smoothest" periodic signal that you can have, and it's therefore the "best" in term of continuity, derivability and so. For this reason is convenient to express waveforms in terms of sinewaves (you can do it with other waves as well, as well as they are $C^

Just an example: why in the water you usually see curved waves? (for this sake, ignore the effect of the beach or wind) Again, it's because it's the shape that requires less energy to form, since all the ramps and edges are smooth.

In some cases, like the Hammond organ, sinewaves are actually used to *compose* the signal, because with decomposition is possible to synthesize a lot of (virtually all) sounds.

There is a beautiful animation by LucasVB explaining the Fourier decomposition of a square wave:

These images explain better the square wave decomposition in harmonics:

You can decompose any waveform into an infinite series of sine waves added together. This is called Fourier analysis (if the original waveform is repeating) or Fourier transform (for any waveform).

In case of a repeating waveform (like a square wave), when you do Fourier analysis you find that all the sines that compose the waveform have frequencies that are an integer multiple of the frequency of the original waveform. These are called "harmonics".

A sine wave will only have one harmonic - the fundamental (well, it already is sine, so it is made up of one sine). Square wave will have an infinite series of odd harmonics (that is, to make a square wave out of sines you need to add sines of every odd multiple of the fundamental frequency).

The harmonics are generated by distorting the sine wave (though you can generate them separately).

- You can make a sine wave out of any wave of a fixed frequency, as long as you have a filter that passes the fundamental frequency, but blocks 2x frequency (as you would be leaving only one harmonic in place).
- Actually, you can make a sine wave that has different frequency than the orginal - just use a bandpass filter to pass the harmonic you want. You can use this to get a sine wave of a frequency that is a multiple of the frequency of another sine - just distort the original sine and pick out the harmonic you want.
- RF systems have to put out waveforms that do not contain harmonics outside the allowed frequency range. This is how a PWM power supply (operating frequency

The derivative - rate of change - of a sinusoid is another sinusoid at the same frequency, but phase-shifted. Real components - wires, antennas, capacitors - can follow the changes (of voltage, current, field-strength, etc.) of the derivatives as well as they can follow the original signal. The rates of change of the signal, of the rate-of-change of the signal, of the rate-of-change of the rate-of-change of the signal, etc., all exist and are finite.

The harmonics of a square wave exist because the rate of change (first derivative) of a square wave consists of very high, sudden peaks infinitely high spikes, in the limit-case of a so-called perfect square wave. Real physical systems can't follow such high rates, so the signals get distorted. Capacitance and inductance simply limit their ability to respond rapidly, so they ring.

Just as a bell can neither be displace nor distorted at the speed with which it is struck, and so stores and releases energy (by vibrating) at slower rates, so a circuit doesn't respond at the rate with which it is struck by the spikes which are the edges of the square wave. It too rings or oscillates as the energy is dissipated.

One conceptual block may come from the concept of the harmonics being higher in frequency than the fundamental. What we call the frequency of the square wave is the number of transitions it makes per unit time. But go back to those derivatives - the rates of change the signal makes are huge compared to the rates of change in a sinusoid at that same frequency. Here is where we encounter the higher component frequencies: those high rates of change *have the attributes of higher frequency sine waves*. The high frequencies are implied by the high rates of change in the square (or other non-sinusoid) signal.

The fast rising edge is not typical of a sinusoid at frequency *f*, but of a much higher frequency sinusoid. The physical system follows it the best it can but being rate limited, responds much more to the lower frequency components than to the higher ones. So we slow humans see the larger amplitude, lower frequency responses and call that *f*!

In practical terms, the reason harmonics "appear" is that linear filtering circuits (as well as many non-linear filtering circuits) which are designed to detect certain frequencies will perceive certain lower-frequency waveforms as being the frequencies they're interested in. To understand why, imagine a large spring with a very heavy weight which is attached to a handle via fairly loose spring. Pulling on the handle will not directly move the heavy weight very much, but the large spring and weight will have a certain resonant frequency, and if one moves the handle back and forth at that frequency, one can add energy to the large weight and spring, increasing the amplitude of oscillation until it's much larger than could be produced "directly" by pulling on the loose spring.

The most efficient way to transfer energy into the large spring is to pull in a smooth pattern corresponding to a sine wave--the same movement pattern as the large spring. Other movement patterns will work, however. If one moves the handle in other patterns, some of the energy that gets put into the spring-weight assembly during parts of the cycle will be taken out during others. As a simple example, suppose one simply jams the handle to the extreme ends of travel at a rate corresponding to the resonant frequency (equivalent to a square wave). Moving the handle from one end to the other just as the weight reaches end of travel will require a lot more work than would waiting for the weight to move back some first, but if one doesn't move the handle at that moment, the spring on the handle will be fighting the weight's attempt to return to center. Nonetheless, clearly moving the handle from one extreme position to the other would nonetheless work.

Suppose the weight takes one second to swing from left to right and another second to swing back. Now consider what happens if one moves the handle from one extreme of motion to the other has before, but lingers for three seconds on each side instead of one second. Each time one moves the handle from one extreme to the other, the weight and spring will have essentially the same position and velocity as they had two seconds earlier. Consequently, they will have about as much energy added to them as they would have two seconds before. On the other hand, the such additions of energy will only be happening a third as often as they would have when the "linger time" was only one second. Thus, moving the handle back and forth at 1/6Hz will add a third as much energy per minute (power) to the weight as would moving it back and forth at 1/2Hz. A similar thing happens if one moves the handle back and forth at 1/10Hz, but since the motions will be 1/5 as often as at 1/2Hz, the power will be 1/5.

Now suppose that instead of having the linger time be an odd-numbered multiple, one makes it an even-numbered multiple (e.g. two seconds). In that scenario, the position of the weight and spring for each left-to-right move will be the same as its position on the next right-to-left move. Consequently, if the handle adds any energy to the spring in the former, such energy will be essentially cancelled out by the latter. Consequently, the spring won't move.

If, instead of doing extreme motions with the handle, one moves it more smoothly, then at lower frequencies of handle motion there are apt to be more times when one is fighting the motion of the weight/spring combo. If one moves the handle in a sine-wave pattern, but at a frequency substantially different from the resonant frequency of the system, the energy that one transfers into the system when pushing the "right" way will be pretty well balanced by the energy taken out of the system pushing the "wrong" way. Other motion patterns which aren't as extreme as the square wave will, at at least some frequencies, transfer more energy into the system than is taken out.

## Measurement of Time

**Time**

Like length and mass, time is also a fundamental physical quantity. Any event which repeats itself after an equal interval of time can be used for measurement of time. In nature, there are many such events which are repetitive. Examples are (i) motion of earth round the sun, (ii) motion of earth on its own axis, (iii) motion of wall clock or pendulum, and many more.

In fact, rotation of earth on its own axis has been taken to fix a unit of time, called second.

**Second**

It is basic unit of time. It is defined as l/86400th part of a mean solar day. A mean solar day is defined as the interval between two consecutive overhead passages of the sun at one place on the earth, as the earth rotates on its axis from west to east. The value of solar day varies over the year, hence its mean over one year is taken as mean solar day.

A second is also defined as the time interval in which 9,192,631,770 vibrations in an atom of cesium-133 take place during transition between the two hyper fine levels of ground state.

**Time measuring devices**

The various time measuring devices are :

**Clocks.**These are mechanical devices in which potential energy given to its spring during its binding, is converted into kinetic energy of its moving parts called hands. The ends of its two hands move along a circular path graduated to measure hours and minutes.**Watches.**These are also mechanical devices much smaller in size but more fine in time measurement. Watches have three hands and measure hours, minutes and as well as seconds.**Electric clocks.**These are run by a fixed frequency alternating current which passes through its synchronous motor. Currents from high frequency electronic oscillators work electronic watches.**Quartz-crystal clocks.**A quartz crystal controls an electronic oscillator by its Own natural frequency. The current from quartz controlled oscillator works the quartz crystal clocks which have higher accuracy and precision.**Atomic clocks.**Vibration of atoms has been utilised for making atomic clocks. These atomic vibrations are so steady that the clocks controlled by them hardly show a difference of a fraction of a second over thousands of years.

**Measurment of time in laboratories**

In laboratories, we have to measure time intervals ranging approximately from 10-2 seconds to 103 seconds. For the purpose of measuring time in this range, we use following time measuring devices :

**1. Stopwatch**

It is used for more accurate measurement of duration of an event or time interval between two events.

In appearance it also looks like an ordinary watch (Fig. 4.01).

It has an arrangement for the ‘start’ and ‘stop’ of the watch. There is a lever at the top of the body. When this lever is pressed for the first time, the watch starts when the same lever is pressed for the second time, the watch stops. A third press on the lever brings the watch to zero reading.

It has two hands, a short and a long hand.

Hands start moving when the lever is pressed down. The tip of the long hand moves over a bigger circular scale covering the whole face of the watch. This tip moves one small division of circular scale in 0.1 sec or 0.2 sec and completes one complete-rotation in 30 sec or 1 minute. The tip of short hand moves over a small circular scale at the top of face of the watch. It moves one division on the small circular scale when long hand makes one full round on big scale.

Hence one division of small scale correspond to 30 sec or 1 minute. Moving hands are stopped by pressing the lever for the second time. When the lever is pressed for the third time, both hands fly back to. zero positions.

With both hands at zero positions, the watch is started when an event starts and stopped when the event ends. The position of the two hands gives the duration of the event in fraction of a second. The least count of such stop watches is 0.1 sec or 0.2 sec.

**2. Simple pendulum**

A simple pendulum is the simplest device for measuring time.

(i) Construction. An ideal simple pendulum consists of a heavy point mass (called bob) tied to one end of a perfectly inextensible, flexible and weightless string. In practice, we make it by tying a metallic spherical bob to a fine cotton stitching thread (Fig. 4.02).

(ii) Working. The free end of the string is tied to a point of suspension S in a rigid support. When left free, the string becomes vertical and the bob stays in mean position A.

When displaced to one extreme position B (right extreme) and left free, the bob returns to mean position A, goes to extreme left position C and again returns to mean position A. This completes one vibration and the motion is repeated. The motion of the bob is a simple harmonic motion (S.H.M.). The time period, T of the simple pendulum depends upon its length l which is the distance between point of suspension S and C.G. of the bob, which is also C.G. of the pendulum.

(iii) Theory. Let a simple pendulum of length l have a bob of mass m. Let the bob be displaced from its vertical mean position A to position B by an angle θ and distance (Fig. 4.03).

The component mg cos θ of its weight balances tension in string. Component mg sin θ acts on the bob as restoring force and produces acceleration in it.

This time period is independent of amplitude of the motion of the bob, provided it remains small (so that θ = sin θ).

Knowing l and g, T can be known.

Hence, a simple pendulum becomes a good time-indicator and as such can be used as a time measuring device.

(iv) Damping. Under ideal conditions of motion (vibration), the amplitude of vibration of the pendulum bob must remain constant. This will be the case when the point of suspension is free from friction and there is no air surrounding the bob. But actually both the factors are present.

The presence of air causes damping and the amplitude of vibration goes on decreasing continually as shown in Fig. 4.04. But even then the time period is not much effected. Amplitude decreases by a constant ratio.

(v) Seconds pendulum. A simple pendulum, whose time period is two seconds (i.e. whose hob takes one second to move from one extreme to other

extreme), is called seconds pendulum. Its length is 4-04. Damped oscillations. about 99.4 cm.

**3. Ticker-Tape timer (Not in CBSE Syllabus)**

It is another device used for measuring time in the laboratory. It can measure time intervals of the order of 1/50 th or 1/100 th of a second.

(i) Construction. Its simple type consists of an electromagnet whose coils are fed with alternating current of known steady frequency. A soft iron strip fixed at one end and free at the other end, runs through the core of the electromagnet. The free end of the strip has a small pointed hammer attached to it which strikes a disc made of carbon paper. A paper tape passes from under the disc.

(ii) Working. When alternating current is passed through the electromagnet, the soft iron strip gets magnetised and demagnetised twice during one cycle of current. It makes soft iron strip vibrate inside the core of the electromagnet with a frequency double of the frequency of the alternating current. As the strip vibrates, the pointed hammer strikes the carbon disc which leaves a dot on the paper tape below it.

As the paper tape is pulled from under the disc, dots are marked on it at a constant interval of time. This interval is half the time period of the alternating electromagnet current and can be known. With a 50 cycles per sec alternating current in laboratory, this time interval will be 1/100 sec. This time interval is called a tick and can be used as a standard unit of time.

**Some important definitions connected with S.H.M.**

- Periodic Motion. A motion that repeats itself in equal intervals of time, is called a periodic motion.
- Oscillatory Motion. The periodic motion in which a body moves to and fro (back and forth) over the same path, is called an oscillatory motion.
- Simple Harmonic Motion (S.H.M.). An oscillatory motion which can be expressed in terms of single harmonic function (e.g., sine and cosine functions), is called a simple harmonic motion (S.H.M.).
- Displacement. At any moment, the distance of the body (particle) from mean position, is called the displacement (it is a vector quantity). Its symbol is y.
- Amplitude. Maximum displacement on either side of mean position is called amplitude. Its symbol is A.
- Vibration. Motion from mean position to one extreme, then to other extreme and then back to mean position from same side, is called one vibration.
- Time Period. The time taken by the body to complete one vibration, is called the time period of the body. Its symbol is T.
- Frequency. The number of vibrations made by the body in one second, is called the frequency of the body. Its symbol is v.

Hence, v = 1/T or vT=1.

**Question. 1. Define a second.**

**Answer.** It is a basic unit of time. It is defined as l/86400th part of a mean solar day. [Read Art. 4.02 for recent definition.]

**Question. 2. Define a mean solar day.**

**Answer.** A mean solar day is defined as the interval between two consecutive overhead passages of the sun at one place on the earth, as the earth rotates on its axis from west to east.

**Question. 3. Is solar day constant ?**

**Answer.** No, it varies in value over the year.

**Question. 4. What is the difference between a clock and a watch ?**

**Answer.** A clock has a big size. Its least count is 1 sec. A watch has a smaller size. Its least count is 0.1 sec or 0.2 sec.

In the Music Theory AP course, students learn to recognize, understand, and describe the basic materials and processes of music that are heard or presented in a musical score. You can expect to practice and develop musical skills while building your understanding of music composition and theory. You will also develop your music vocabulary through class discussions and written analyses of listening selections. Although there are no prerequisite classes for enrollment in the Music Theory course, you do need to be able to read and write musical notation. It is also strongly recommended that you have at least basic performance skills in voice or on an instrument.

You’re expected to possess four core skills at the end of the AP Music Theory course. These skills form the basis of many tasks you’re asked to perform on the AP Music Theory exam. The core skills are:

Skill | Description | Percentage of Exam Score (Multiple-Choice Section) |

Analyze Performed Music | Take musical terms, concepts, and relationships and apply them to performed music (aural). | 48% |

Analyze Notated Music | Use musical terms, concepts, and relationships and apply them to notated music (written). | 44% |

Convert Between Performed and Notated Music | Convert music between aural and written forms using conventions of musical notation and performance. | 8% |

Complete Based on Cues | Following 18th-century stylistic norms complete music based on cues. | Not assessed in multiple-choice questions |

Through the four core skills you’ll explore four big ideas that serve as the foundation of the AP Music Theory course. Using these big ideas, you’ll create connections between the concepts and skills learned and deepen your understanding of musical theory. The College Board describes the four big ideas of AP Music Theory as:

**1. Pitch:** Specific frequencies of sound, known as pitches, are basic units of music. Pitches that are deliberately sequenced through time create melodies, and groups of pitches presented successively or simultaneously form chords. Within an established musical style, chords relate to one another in the context of harmony.

**2. Rhythm:** Music exists in the dimension of time, where long and short sounds and silences can be combined in myriad ways. This temporal aspect, called rhythm, is often governed by a layered structure of interrelated pulses known as meter. Rhythms are typically grouped into distinctive rhythmic patterns, which help define the specific identity of a musical passage. Musicians use established rhythmic devices to expand expressive possibilities, often achieving their effect by challenging the regularity of the meter or transforming rhythmic patterns.

**3. Form:** Music exhibits a structural aspect known as form, in which a musical composition is organized in a hierarchy of constituent parts. The specific ways these parts are related, contrasted, or developed produce the unique profile of an individual composition. Specific formal types and functions may be identified when parts of a composition follow established melodic-harmonic patterns, or fulfill established roles within the overall hierarchical structure.

**4. Musical Design:** Texture, timbre, and expression contribute to the overall design and character of a piece of music or musical performance. The texture of a musical passage arises from the way its layers are produced and distributed, and how they interact to form the totality of sound. Timbre refers to the distinct sounds of specific instruments and voices, arising from the physical manner in which those sounds are produced. Expressive elements are related to musical interpretation and include dynamics, articulation, and tempo.

**AP Music Theory Course Content**

The AP Music Theory Course is divided into eight units. Below is a suggested sequence of the units from the College Board, along with a short description of what’s covered in each one.

Unit | Content Taught |

Music Fundamentals 1: Pitch, Major Scales and Key Signatures, Rhythm, Meter, and Expressive Elements | How pitch and rhythm work together to become melody and meter—building musical compositions. |

Music Fundamentals 2: Minor Scales and Key Signatures, Melody, Timbre, and Texture | Use knowledge of pitch patterns and relationships in major scales and apply it to minor scales. |

Music Fundamentals 3: Triads and Seventh Chords | The fundamentals of harmony. |

Harmony and Voice Leading I: Chord Function, Cadence, and Phrase | Apply knowledge of harmonic materials and processes using it to explore the procedures of 18th-century style voice leading. |

Harmony and Voice Leading II: Chord Progressions and Predominant Function | Describe, analyze, and create complex harmonic progressions in the form of four-part (SATB: soprano, alto, tenor, and bass) voice leading. |

Harmony and Voice Leading III: Embellishments, Motives, and Melodic Devices | Exploration of the skills and concepts of harmony and voice leading. |

Harmony and Voice Leading IV: Secondary Function | Building on knowledge of harmonic relationships and procedures to deepen your understanding of keys, scale degrees, and chords. |

Modes and Form | The conventions that influence the character of music such as modes, phrase relationships, and forms. |

## 14 - The Performance of Music

Music performance is a large subject that can be approached in many different ways. This chapter focuses on empirical research of music performance and related matters. Most of this research is concerned with Western tonal music and mainly art music. Excellence in music performance involves two major components like a genuine understanding of what the music is about, its structure and meaning, and a complete mastery of the instrumental technique. Evaluation of performance included many studies which are reviewed earlier. Evaluation occurs in the everyday activity of music critics, music teachers, and musicians. An overall evaluation is considered as a weighted function of the evaluations in the specific aspects. In order to maintain the tempo and to achieve perceived synchrony, musicians should therefore play a small amount ahead of the beat they hear. With sharp attacks the delay is less, and instruments with sharp attacks may therefore serve as “beat-definers” for the rest of an ensemble. In addition, some attempts are made to predict evaluation of music performances from the physical characteristics of the performances.

## 4 Which of the following is not an example of simple harmonic motion?

SHM is a periodic motion where the restoring force acting on the moving object is directly proportional to the magnitude of displacement and acts towards the equilibrium position. It's typified by a pendulum or mass on a spring subject to the linear elastic restoring force obeying Hooke's law.

All four answer options seem to meet the definition, however, It's that 'system' qualifier in the first option 'A' which is most important:

All of the options given result in periodic motion. However, the first option for the car bouncing on it's suspension 'system', implies the incorporation of an oscillation damper applied to four wheels.

i.e. The car suspension system applied to four wheels produces complex interactions amongst them and cannot be considered 'simple '.

It's a trampoline which is not considered SHM. Think about the forces acting and about how (for what period of time) they act. Remember that the definition of SHM comes from the restoring force.

(The Q is from Edexcel P5, June 2012)

(Original post by **phys981**)

It's a trampoline which is not considered SHM. Think about the forces acting and about how (for what period of time) they act. Remember that the definition of SHM comes from the restoring force.

(The Q is from Edexcel P5, June 2012)

I'm thinking it's a faulty question unless there's diagrams or something - if you watch a video of someone bungee jumping and pay attention it's pretty clear they get 'airtime' and the rope goes slack at the top of the first bounce (or two) after the first drop.

e.g. https://www.youtube.com/watch?v=1RzsHbBIJ14 2:32

if you watch the shots from ground level (same camera position - different levels of zoom) she bounces back till she looks roughly level with the tree tops (2.41) - but on the way down that's a long way above the point at which the rope starts to go taut on the first drop. She passes the same point on the first drop about 2.35

risk of penalising a smart, observant pupil - bad

(Original post by **Joinedup**)

I'm thinking it's a faulty question unless there's diagrams or something - if you watch a video of someone bungee jumping and pay attention it's pretty clear they get 'airtime' and the rope goes slack at the top of the first bounce (or two) after the first drop.

e.g. https://www.youtube.com/watch?v=1RzsHbBIJ14 2:32

if you watch the shots from ground level (same camera position - different levels of zoom) she bounces back till she looks roughly level with the tree tops (2.41) - but on the way down that's a long way above the point at which the rope starts to go taut on the first drop. She passes the same point on the first drop about 2.35

risk of penalising a smart, observant pupil - bad

The reasoning is that on a trampoline, the acceleration of the child is g for most of the time, other than when in contact with the trampoline, and thus cannot possibly be proportional to displacement.

As for the bungee cord, it says bouncing rather than jumping, means bouncing with the cord extended, not during the jump.

While some may misinterpret that, the trampoline is very *definitely* not SHM - it's worth remembering in case you ever see it again.

The advice, when answering multiple choice questions, is to always look for the BEST answer.

## Teaching harmonic motion in trigonometry: Inductive inquiry supported by physics simulations.

In this paper, we would like to present a lesson whose goal is to utilise a scientific environment to immerse a trigonometry student in the process of mathematical modelling. The scientific environment utilised during this activity is a physics simulation called Wave on a String created by the PhET Interactive Simulations Project at Colorado University at Boulder and available free on the Internet. The outline of the activity, situated in inductive inquiry, is written in a format that is adaptable to various classroom settings students can work independently in front of a computer or in groups. If a computer lab is not available, the simulation can be projected on a screen in a regular math classroom. In all of these settings, the teacher takes the role of a facilitator. Although, the lesson was developed following trigonometry curriculum in the US, its cognitive learning objectives fit well into the scope of the proposed Australian mathematics curriculum (ACARA, 2010) that also emphasises the development of the skills of mathematical modelling, data collection, and analysis. The activity, presenting applications of periodic functions in a non-geometric setting, can be conducted in Australian Upper Secondary or Lower Tertiary Trigonometry courses. With some extensions, including dumped oscillation, its content will fit into Queensland Mathematics C syllabus (QSA, 2008), in particular the section of Advanced Periodic and Exponential Functions.

Inductive reasoning is a thought process whose ultimate goal is knowledge acquisition. Inductive reasoning encompasses a range of instructional methods, including inquiry learning, problem-based learning, project-based learning, case-based teaching, discovery learning, and just-in-time teaching (Prince & Felder, 2006). Inductive reasoning is commonly applied in science, where data is gathered and mathematical models are formulated to predict future behaviour of highlighted quantities. Literature (Felder & Brent 2004) shows that challenges provided by inductive methods serve as precursors to students' intellectual development.

Learning methods involving inductive reasoning are characterised as constructivist methods. They build on the widely accepted principle that students construct their own versions of reality rather than simply absorb versions presented by their instructors. In cognitive constructivism, which originated primarily in the work of Piaget (1972) an individual's reactions to experiences lead to learning.

Although inductive reasoning produces multiple learning outcomes and it is extensively used in sciences (Thacker, Eunsook, Trefz, & Lea, 1994), this inquiry method is rarely used in trigonometry. We argue that applying trigonometric functions to model harmonic motion can provide a rich scientific context to exercise mathematical modelling through inductive inquiry in trigonometry classes as well.

Inquiry learning, one of the simplest forms of the inductive reasoning was selected to construct this activity. Staver and Bay (1987) identified three stages of inquiry learning: structured inquiry, where students are given a problem and an outline for how to solve it, guided inquiry where in addition, students are supposed to figure out the solution method, and open inquiry where students must formulate the problem and find the solution.

We situated this activity in a guided inquiry stage. By modifying some of its auxiliary elements, it can be changed to either structured or open inquiry depending on students' responses.

The framework of the guided inquiry follows Joice's (2009) four layers of inductively organised learning environment: focus, conceptual control, inference, and confirmation. In order to parallel this inquiry with its scientific counterpart, problem statement was included as a catalyst of the process. The following descriptions of the layers served as theoretical foundations of the activity. Problem statement is a form of a question that students answer because of conducting an experiment. Focus is building (collecting) data and asking students to analyse the attributes of the data and formulate the hypothesis. Conceptual control (analysis) is classifying the facts and identifying patterns of regularity. Inference is a generalisation (formulation of a pattern or law) about the relations between the collected facts that leads to acquiring a general formula or mathematical function. Confirmation is a verification of the derived model in new (physical) circumstances conducted through testing inference and further observations.

An effective implementation of the inquiry method provides also the students with practicing a conduct of scientific experiment of how to identify and collect appropriate evidence, analyse and interpret results, formulate conclusions, and evaluate the conclusions (Lee, 2004). Since understanding physics concepts requires formal reasoning, familiarity with the process of mathematisation of generated data, and extracting general principles from specific cases (Bellomonte, Guastella, & Sperandeo-Mineo, 2005), exposing trigonometry students to a simulated physics phenomena might produce multiple results. It not only develops their modelling skills but also helps them understand the laws of physics. Furthermore, while conducting this activity, students will be placed in the roles of scientists actively constructing new knowledge. By referring to a scientific environment, practitioners will learn to select information based on scientific validity, a cognitive skill that they can apply in other subject areas as well as in their work places.

Physics simulations selected for this project are provided free online by PhET Interactive Simulations Project at Colorado University. Although their primary purpose is enhancing the teaching of physics, we argued that they could be integrated into the process of teaching and learning of mathematics. While working on the virtual labs, students can state hypotheses, observe scientific processes, take measurements, construct mathematical models, and validate them. They can modify the variables of the experiments, as well as predict and verify the respective outputs. Inquiry conducted by Lima (2010) shows that when prediction is used effectively students are likely to progress from passive listeners to active thinkers, simultaneously expanding and deepening their mathematical knowledge. With the aid of graphing technology, derived mathematical models can be further verified. Research conducted by PhET (Finkelstein, Perkins, Adams, Kohl, & Podolefsky, 2004) showed that these simulations can be substituted effectively for real laboratory equipment in physics courses. Findings of research (Sokolowski & Walters, 2010, p. 110) conducted in a South-Central Texas high school proved that mathematics students not only learned more and scored higher on the district and state standardised test items related to analysis and synthesis but that they also enjoyed and appreciated the new learning environment.

The structure of the activity

The activity uploaded also at http://phet.colorado.edu/en/contributions/ view/3340 evolves within the five stages of guided inquiry. The purpose of the commentaries is to help walk students through the inquiry process.

Introduction of the concept and demonstration of the simulation

The teacher opens the simulation at http://phet.colorado.edu/en/ simulation/wave-on-a-string and demonstrates its features, focusing the students' attention on the shape of the string while it transmits energy. The oscillations can be generated manually, or they can be produced periodically by checking the button Oscillate located on the left side of the simulation.

The instructor might begin the process with generating the wave manually and mention to students some obstacles that they would face to mathematically describe this irregular movement. The oscillator, a wheel rotating periodically due assigned frequency, produces a regular wave on the string. It is important that the No End mode located on the bottom right corner of the simulation is checked. The instructor might also want to demonstrate the effect of the different damping factors and string tensions on the motion of the energy. At last, the instructor directs students' attention to frequency and amplitude, as these two physical factors of the periodic motion will dictate the formulation of respective periodic function. Since mathematical modelling refers to successive approximation, yet more regularity in the movement would help construct the model. This can be achieved by reducing the damping factor to zero. Under these circumstances, the energy transmitted by string is not dissipated to the environment, and therefore the amplitude of the wave indicating the amount of energy remains constant. It might be interesting to students to note that the damping factor does not affect the frequency of the wave. The frequency of the wave depends only on the frequency of the source producing the wave.

What mathematical function can be applied to model the path of motion of the energy generated by the wheel? What will the independent and dependent variables of the function be? Students might be given some time to discuss their answers in groups.

The instructor elicits from the students a periodic nature of the motion that suggests sinusoidal function to be applied. There is one important element that the instructor might want to address to the students at this point. Since the path of the motion is two- dimensional (the energy oscillates up and down and moves forward) there can be two different ways undertaken to model the movement:

A. Applying parametric representation to model independently vertical y = f(t) and horizontal position x = f(t) of the front of the wave. In this case, the vertical movement would be modelled by a sinusoidal function and horizontal by a linear (energy moves with a constant forward velocity). Both functions would be expressed in terms of time. Although, this model depicting a dynamics of the system is physically rich, students who did not study properties of parametric equations might find it difficult to apply. Therefore, another (B), a simplified version of this representation is suggested to be used.

B. Expressing vertical position of the wave in terms of horizontal position, y = f(x)

This representation is easily conveyable to an average trigonometry student and it is suggested to be adopted to model the sinusoidal path. If an opportunity exists, the instructor might want to prove mathematically that the parametric forms discussed in A can be converted into a singular representation discussed in B.

Focus/gathering information/stating hypothesis

In this part, the teacher discusses, in detail, the critical components of the sinusoidal function such as amplitude, period, horizontal, and vertical shift, and how these quantities can be identified and measured in the experiment.

Due to chosen singular model, the period of anticipated function will be expressed in the units of metres (here millimetres). In physics, the length of one wave, expressed in metres, represents the wavelength of the wave denoted by [lambda]. Although students who took physics course will correlate the distance to wavelength, for the purpose of this activity, the distance will be labelled [DELTA]x.

The instructor might want to demonstrate some measuring devices embedded in the simulation, such as a ruler and a stopwatch that help quantify the highlighted quantities.

Establishing a frame of reference (x- and y-axes) is also important. Due to its location, function vertical and horizontal shifts will be referenced. It is suggested that to develop the general model, the x-axis is established at the equilibrium line of the string and the y-axis is aligned with the centre of the oscillator (initial position of the propagated energy).

Once a tentative model has been concluded, students focus on measuring necessary quantities that will constitute the form of the sinusoidal function. In order to further generalise the model, only one full cycle can be shown on the screen for the analysis. Students can be given rulers and be asked to measure necessary quantities using screenshots of the simulation copied in their lab outlines. This approach has some advantages it makes the activity more tangible. Students prefer this approach to just using the numbers embedded in the scenario. The lengths can be expressed in millimetres or centimetres.

Generalisation of the analysis/inference

During this stage, students transfer measured quantities into a mathematical model. In order to review essential elements of a periodic function, they can work on multiple-choice questions. Samples of such items are provided below.

A general form of a periodic function is given by y = A sin (2[pi]/T)1.

1. Select the quantity that represents the measured average time of one full cycle.

D. General time variable (t)

2. Select the quantity that represents the dependent variable in this function.

B. Vertical position of wave (y)

C. General time variable (t)

3. Suppose that the length of one full cycle of a wave is denoted by Ax and expressed in the units of metres. Which expression can be used to model the wave?

A. y = A sin (2[pi]/[DELTA]x) [DELTA]x

Following the review, the teacher discusses the answers and then lets the students construct their functions. Students substitute the values of these quantified components to the general form of

and then verify the function equation. Building on Bateman (1990), inductive instruction should be spirally organised students should be directed to continually revisit critical concepts and improve their cognitive models, thus verification, the next stage of the inquiry follows.

Verification and confirmation of the derived model

This stage is very significant in the process of the guided inquiry. When working on typical paper-and-pencil problems, this stage is often omitted because the physical representation of a wave is usually not provided and the movement is not observable. The availability of the simulation presents a great opportunity for contrasting observed wave with its mathematical representation. Students can use a graphing calculator or any technological tool (sketchpad) that converts algebraic function into a graph. They might be asked to determine the dimensions of the window of a graphing calculator so that their functions resemble the screen shot as closely as possible.

If graphing technology is not available, students can verify the model the old fashioned way using a table of values. They could calculate outputs for selected inputs and compare the values with the model. Further verification can refer to modification of the components of the parent function. Sample questions in regards to this mode of verification are presented below. For each modification, the students are supposed to use the simulation to observe the change, write new function, and use graphing technology to check if derived function corresponds to observable wave.

1. Suppose that the x-axis--the draggable reference (dotted) line shown on the simulation--is moved 30 mm below the string. Which component of the derived function should be changed to reflect this transformation?

B. Vertical transformation

D. Horizontal transformation

2. Due to a modified frequency, there are twice as many waves observed on the string. Which component of the sinusoidal function should be changed to reflect this transformation?

B. Vertical transformation

3. Suppose that the maximum height of the wave as measured from the equilibrium line increased by 10 cm. Which parameter changed?

B. Vertical transformation

Exchanging thoughts about the experiment, and suggesting ways of improving this learning environment concludes the activity.

What impact do the simulations have on student learning process?

Students highly praise the new learning environment and find the lessons utilising simulations very attractive. As we mentioned before, this environment affects positively their test scores. Following this positive feedback, more simulations were adopted to enhance the modelling process in other sections of mathematics such as polynomial or transcendental functions. The simulations were also utilised in calculus to enhance the teaching of limits, derivatives, and integrals including the First Fundamental Theorem of Calculus. We argue that the virtual physical world and the inquiry processes that the students were immersed in to model the world helps them to understand the modelling process and prepares them for engineering classes and college. We foresee a need for a more systematic research study of the influence of the simulations on mathematical knowledge acquisition by mathematics students. We hope that the readers will find the journey interesting enough to also get involved in similar research.

ACARA. (2010). Australian Curriculum. Retrieved 20 December 2010, from www.australiancurriculum.edu.au

Bateman, W. (1990). Open to question: The art of teaching and learning by inquiry. San Francisco: Jossey-Bass.

Bellomonte, L., Guastella I., & Sperandeo-Mineo, R. M. (2005). Mechanical models of amplitude and frequency modulation. European Journal of Physics 26, 409-422.

Felder, R. M., & Brent R. (2004). The intellectual development of science and engineering students. Journal of Engineering Education, 93(4), 269-77.

Finkelstein, N.D., Perkins, K.K., Adams, W., Kohl, P., & Podolefsky, N. (2004). Can computer simulations replace real equipment in undergraduate laboratories? PERC Proceedings. Colorado.

Joyce, B., Weil, M., Calhoun, E. (2009). Models of teaching (8th ed.). Boston, MA: Pearson.

Lee, V.S. (2004). Teaching and learning through inquiry. Sterling, VA: Stylus Publishing.

Lima, K., Buendi'ab, G., Kimc, K., Corderod, F., & Kasmere, L. (2010). The role of prediction in the teaching and learning of mathematics. International Journal of Mathematical Education in Science and Technology, 41(5), 595-608.

Piaget, J. (1972). The psychology of the child. New York: Basic Books.

Physics Simulations. Retrieved 10 December 2010 from http://phet.colorado.edu.

Prince M. J., & Felder R. (2006). Inductive teaching and learning methods: Definitions, comparisons, and research bases. Journal of Engineering Education, 95(2), 123-138.

Queensland Studies Authority, QSA. (2008). Mathematics C syllabus. Available: http://www.qsa.qld.edu.au/downloads/senior/snr_maths_c_08_syll.pdf

Staver, J. R., & Bay, M. (1987). Analysis of the project synthesis goal cluster orientation and inquiry emphasis of elementary science textbooks. Research in Science Teaching, 24, 629-643.

Sokolowski, A., &Walters, L. (2010). Mathematical modeling in trigonometry enhanced by Physics simulations. In Proceedings of International Technology, Education and Development (INTED) Conference, Valencia, Spain. (pp. 106-112). [CD]

Thacker, B., Eunsook, K., Trefz, & S. Lea (1994). Comparing problem solving performance of physics students in inquiry-based and traditional introductory physics courses. American Journal of Physics, 62(7), 627-633.

## Hooke’s Law and Simple Harmonic Motion

To determine the spring constant of a spring by measuring its stretch versus applied force, to determine the spring constant of a spring by measuring the period of oscillation for different masses, and also to investigate the dependence of period of oscillation on the value of mass and amplitude of motion.

### Hypothesis

If the applied force (mass) is to remain the same while the vertical displacement is increased, there period will remain the same. However, if the vertical displacement is held constant while the applied force in increased, the period will increase. This is in accordance with the derived equation T = 2π (M / k) 0.5 .

### Labeled Diagrams

A (m) | ∆t_{1} (s) | ∆t_{2} (s) | ∆t_{3} (s) | ∆t avg. (s) | α_{t} (s) | T (s) |

0.0200 | 7.17 | 7.38 | 7.40 | 7.32 | 0.127 | 0.0736 |

0.0400 | 7.40 | 7.42 | 7.33 | 7.38 | 0.0473 | 0.0273 |

0.0600 | 7.33 | 7.38 | 7.44 | 7.38 | 0.0551 | 0.0318 |

0.0800 | 7.33 | 7.31 | 7.29 | 7.31 | 0.0200 | 0.0115 |

0.100 | 7.10 | 7.14 | 7.08 | 7.11 | 0.0306 | 0.0176 |

M (kg) | ∆t_{1} (s) | ∆t_{2} (s) | ∆t_{3} (s) | ∆t avg. (s) | α_{t} (s) | T (s) | T 2 (s 2 ) |

0.0500 | 7.55 | 7.63 | 7.55 | 7.58 | 0.0462 | 0.0267 | 0.758 |

0.0600 | 8.10 | 8.15 | 8.13 | 8.13 | 0.0252 | 0.0145 | 0.813 |

0.0700 | 8.75 | 8.85 | 8.73 | 8.78 | 0.0643 | 0.0371 | 0.878 |

0.0800 | 9.37 | 9.34 | 9.36 | 9.36 | 0.0153 | 0.0088 | 0.936 |

0.0900 | 9.95 | 9.94 | 10.00 | 9.96 | 0.0321 | 0.0186 | 0.996 |

0.100 | 10.50 | 10.48 | 10.46 | 10.48 | 0.0200 | 0.0115 | 1.05 |

### Graphs

### Questions

**1. Do the data from Part 1 verify Hooke’s Law? State clearly the evidence for your answer.**

The data correlate close to Hooke’s Law, but not quite. The law states that F = -ky, where F is in this case Mg and y equals the negative displacement. After graphing forces versus displacement, a value of 3.53 N/m was determined as the spring constant. However, when applying this value to the equation and using recorded displacement values, the calculated force come up less than the actual for used. For example, in the first trial y = -0.055 m. Multiplying that value by the extrapolated spring constant gives a theoretical force of 0.194 N, but the actual force used was 0.244 N. All other trials yield a similar lowball theoretical force.

**2. How is the period T expected to depend upon the amplitude A? Do your data confirm this expectation?**

Period is not expected to depend upon amplitude, as suggested by the equation T = 2π (M / k) 0.5 , where amplitude is absent as a variable. The data confirms this expectation, as the period was nearly the same for each trial.

**3. Consider the value you obtained for C. If you were to express C as a whole number fraction, which of the following would best fit your data (1/2, 1/3, 1/4, 1/5)?**

The obtained value of C is 0.694, which is closest to 1/3.

**4. Calculate T predicted by the equation T = 2π (M / k) 0.5 for M = 0.050 kg. Calculate T predicted by T = 2π ( [M + Cm _{s}] / k ) 0.5 for M = 0.050 kg and your value of C. What is the percent difference between them? Repeat for a value for M of 1.000 kg. Is there a difference in the percent differences? If so, which is greater and why?**

T predicted using the first equation and M = 0.050 kg is 0.726 s. T predicted using the second equation and M = 0.050 kg is 0.771 s. This is a percent difference of 6.01%.

T predicted using the first equation and M = 1.000 kg is 3.24 s. T predicted using the second equation and M = 1.000 kg is 3.26 s. This is a percent difference of only 0.615%. The greater percent difference occurs at the lower weight because the weight of the spring is almost insignificant at higher weight. The proportion of the mass to the spring is so great that it has almost no effect on the calculation.

### Conclusion

During part one of the experiment, the vertical displacement of a spring was measured as a function of force applied to it. The starting position of the spring was recorded using a stretch indicator. Mass was added to the spring, and the displacement was recorded. This was repeated with various amounts of mass. From these data, a graph of force versus displacement was plotted, and a linear fit slope revealed the spring constant. In this endeavor, the spring constant was valued at 3.53 N/m.

However, when applying this spring constant to the recorded displacements in the Hooke’s Law equation, the calculated forces are lower than the recorded forces. In similar manner, when rearranging Hooke’s Law to solve for displacement, the calculated displacements are larger than the actual recorded displacements. This means there was human error, most likely in terms of not being precise with the displacement readings because the recordings for the masses used were accurate. Because such small masses were used, any error in displacement readings was augmented. The spring used may also not have been perfect.

During part two of the experiment, the period of the spring was measured as amplitude changed while mass remained constant. The period remained nearly the same throughout every trial, which was to be expected. Any differences in period may be accounted to inadequate stopwatch usage and inaccurate starting displacements throughout the trials. It should be noted that at amplitude of 0.100 m, the hook lost contact with the spring for a split second at the apex of oscillation, which accounts for its oddity in period. This was something that could not be avoided at that amplitude the spring pulled the hook up too quickly which caused the loss of contact. The resulted graph of period versus amplitude yielded a linear fit slope of close to 0 (-0.247 s/m), which was predicted.

During part three of the experiment, the period of the spring was measured as mass was varied while amplitude remained constant. As the mass was increased, the period also increased. This was not surprising considering the given equations. The square of the period versus mass for each trial was plotted and a linear fit was taken. The slope and y-intercept of this line was then used to determine the spring constant and C, the fraction of the spring’s mass that should be taken into account for the equation T = 2π ( [M + Cm_{s}] / k ) 0.5 . The calculated value of k was 3.75 N/m, which is only 6.04% different from the value determined earlier of 3.53 N/m. The value of C was determined to be 0.694, which is closest to the whole number fraction of 1/3. Any error during parts two and three can be attributed to inaccurate stopwatch recordings and slight variance in displacement and release of the masses at each amplitude.

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## Harmonic Links Cycles, Harmonics & the Universe

Cosmology -The supreme task of the physicist is to arrive at those **universal elementary laws** from which the **cosmos** can be built by pure **deduction (Albert Einstein, 1954)**. The **Wolff-Haselhurst Cosmology** explains a **Perpetual Finite Spherical Universe** within an **Infinite Eternal Space**. The Spherical Standing waves determines the size of our finite spherical universe within an infinite Space (Matter is large not small, we only 'see' the Wave-Center / 'particle' effect which has greatly confused physics). Huygens' Principle explains how other matter's out waves combine to form our matter's spherical In-Waves, which then deduces both Mach's Principle and the redshift with distance (without assuming Doppler effect due to an expanding universe / Big Bang). This also explains how matter interacts with all other matter in the universe (why we can see stars) as matter is the size of the universe, but we only 'see' the high wave amplitude wave-centers / 'particles'.

Physics and Metaphysics - The Metaphysics of Space and Motion Sensibly Unites **Albert Einstein's Relativity, Quantum Theory, and Cosmology.** This 40 page Treatise (written over five years) will be published in 'What is the Electron' (Apeiron, 2005).

Physics: Cosmology: Problems of the Big Bang Theory - Three Famous 'Dissident Scientists' on the many errors of the Big Bang Theory - **Eric Lerner, Halton Arp, Bill Mitchell**. Information and links to problems of the **Big Bang Theory**.

Physics: Cosmology: Edwin Hubble Space Telescope - Deducing the **Hubble Redshift** with distance from **Huygens' Principle** rather than **Doppler Shift** / expanding Universe. (Includes some nice pictures from Hubble Space Telescope).

Physics: Cosmology: CMBR - Cold Hydrogen in Space (not Big Bang) as the Cause of the **Cosmic Microwave Background Radiation?**

Physics: Cosmos Harmony Music Universe - Cosmic Harmony and the **Musical Universe** - The Wave Structure of Matter in Space explains how we live in a Musical Universe (on a Planet in need of better Harmony!).

https://en.wikipedia.org/wiki/Cycles - Excellent summary of various types of cycles and comprehensive links.

https://www.cyclespage.com/ - Interesting website for women which helps track your monthly menstrual cycle. This easy-to-use service lets you: Keep an online record of your past menstrual cycles. View predictions of your upcoming menstruation and ovulation dates. Know when you are most likely to be fertile.

akwu/stckpre.html - On the relationship between planetary cycles and business / stock market cycles, has some understanding of the Wave Structure of Matter. Human beings tend to respond to these invisible electromagnetic waves in the spherical resonating cavity surrounding the earth in such a way as to be modulated indirectly by the motion and position of the planets.

icouzin/Boi%20et%20al,%201999.pdf - ' We investigated the phenomenon of activity cycles in ants, taking into account the spatial structure of colonies. In our study species, Leptothorax acervorum, there are two spatially segregated groups in the nest. We developed a model that considers the two groups as coupled oscillators which can produce synchronized activity. By investigating the effects of noise on the model system we predicted how the return of foragers affects activity cycles in ant colonies.' (School of Mathematical Sciences and Department of Biology and Biochemistry, Centre for Mathematical Biology, University of Bath, Claverton Down, Bath BA2 7AY, UK)

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## 4 Which of the following is not an example of simple harmonic motion?

SHM is a periodic motion where the restoring force acting on the moving object is directly proportional to the magnitude of displacement and acts towards the equilibrium position. It's typified by a pendulum or mass on a spring subject to the linear elastic restoring force obeying Hooke's law.

All four answer options seem to meet the definition, however, It's that 'system' qualifier in the first option 'A' which is most important:

All of the options given result in periodic motion. However, the first option for the car bouncing on it's suspension 'system', implies the incorporation of an oscillation damper applied to four wheels.

i.e. The car suspension system applied to four wheels produces complex interactions amongst them and cannot be considered 'simple '.

It's a trampoline which is not considered SHM. Think about the forces acting and about how (for what period of time) they act. Remember that the definition of SHM comes from the restoring force.

(The Q is from Edexcel P5, June 2012)

(Original post by **phys981**)

It's a trampoline which is not considered SHM. Think about the forces acting and about how (for what period of time) they act. Remember that the definition of SHM comes from the restoring force.

(The Q is from Edexcel P5, June 2012)

I'm thinking it's a faulty question unless there's diagrams or something - if you watch a video of someone bungee jumping and pay attention it's pretty clear they get 'airtime' and the rope goes slack at the top of the first bounce (or two) after the first drop.

e.g. https://www.youtube.com/watch?v=1RzsHbBIJ14 2:32

if you watch the shots from ground level (same camera position - different levels of zoom) she bounces back till she looks roughly level with the tree tops (2.41) - but on the way down that's a long way above the point at which the rope starts to go taut on the first drop. She passes the same point on the first drop about 2.35

risk of penalising a smart, observant pupil - bad

(Original post by **Joinedup**)

I'm thinking it's a faulty question unless there's diagrams or something - if you watch a video of someone bungee jumping and pay attention it's pretty clear they get 'airtime' and the rope goes slack at the top of the first bounce (or two) after the first drop.

if you watch the shots from ground level (same camera position - different levels of zoom) she bounces back till she looks roughly level with the tree tops (2.41) - but on the way down that's a long way above the point at which the rope starts to go taut on the first drop. She passes the same point on the first drop about 2.35

risk of penalising a smart, observant pupil - bad

The reasoning is that on a trampoline, the acceleration of the child is g for most of the time, other than when in contact with the trampoline, and thus cannot possibly be proportional to displacement.

As for the bungee cord, it says bouncing rather than jumping, means bouncing with the cord extended, not during the jump.

While some may misinterpret that, the trampoline is very *definitely* not SHM - it's worth remembering in case you ever see it again.

The advice, when answering multiple choice questions, is to always look for the BEST answer.

## Measurement of Time

**Time**

Like length and mass, time is also a fundamental physical quantity. Any event which repeats itself after an equal interval of time can be used for measurement of time. In nature, there are many such events which are repetitive. Examples are (i) motion of earth round the sun, (ii) motion of earth on its own axis, (iii) motion of wall clock or pendulum, and many more.

In fact, rotation of earth on its own axis has been taken to fix a unit of time, called second.

**Second**

It is basic unit of time. It is defined as l/86400th part of a mean solar day. A mean solar day is defined as the interval between two consecutive overhead passages of the sun at one place on the earth, as the earth rotates on its axis from west to east. The value of solar day varies over the year, hence its mean over one year is taken as mean solar day.

A second is also defined as the time interval in which 9,192,631,770 vibrations in an atom of cesium-133 take place during transition between the two hyper fine levels of ground state.

**Time measuring devices**

The various time measuring devices are :

**Clocks.**These are mechanical devices in which potential energy given to its spring during its binding, is converted into kinetic energy of its moving parts called hands. The ends of its two hands move along a circular path graduated to measure hours and minutes.**Watches.**These are also mechanical devices much smaller in size but more fine in time measurement. Watches have three hands and measure hours, minutes and as well as seconds.**Electric clocks.**These are run by a fixed frequency alternating current which passes through its synchronous motor. Currents from high frequency electronic oscillators work electronic watches.**Quartz-crystal clocks.**A quartz crystal controls an electronic oscillator by its Own natural frequency. The current from quartz controlled oscillator works the quartz crystal clocks which have higher accuracy and precision.**Atomic clocks.**Vibration of atoms has been utilised for making atomic clocks. These atomic vibrations are so steady that the clocks controlled by them hardly show a difference of a fraction of a second over thousands of years.

**Measurment of time in laboratories**

In laboratories, we have to measure time intervals ranging approximately from 10-2 seconds to 103 seconds. For the purpose of measuring time in this range, we use following time measuring devices :

**1. Stopwatch**

It is used for more accurate measurement of duration of an event or time interval between two events.

In appearance it also looks like an ordinary watch (Fig. 4.01).

It has an arrangement for the ‘start’ and ‘stop’ of the watch. There is a lever at the top of the body. When this lever is pressed for the first time, the watch starts when the same lever is pressed for the second time, the watch stops. A third press on the lever brings the watch to zero reading.

It has two hands, a short and a long hand.

Hands start moving when the lever is pressed down. The tip of the long hand moves over a bigger circular scale covering the whole face of the watch. This tip moves one small division of circular scale in 0.1 sec or 0.2 sec and completes one complete-rotation in 30 sec or 1 minute. The tip of short hand moves over a small circular scale at the top of face of the watch. It moves one division on the small circular scale when long hand makes one full round on big scale.

Hence one division of small scale correspond to 30 sec or 1 minute. Moving hands are stopped by pressing the lever for the second time. When the lever is pressed for the third time, both hands fly back to. zero positions.

With both hands at zero positions, the watch is started when an event starts and stopped when the event ends. The position of the two hands gives the duration of the event in fraction of a second. The least count of such stop watches is 0.1 sec or 0.2 sec.

**2. Simple pendulum**

A simple pendulum is the simplest device for measuring time.

(i) Construction. An ideal simple pendulum consists of a heavy point mass (called bob) tied to one end of a perfectly inextensible, flexible and weightless string. In practice, we make it by tying a metallic spherical bob to a fine cotton stitching thread (Fig. 4.02).

(ii) Working. The free end of the string is tied to a point of suspension S in a rigid support. When left free, the string becomes vertical and the bob stays in mean position A.

When displaced to one extreme position B (right extreme) and left free, the bob returns to mean position A, goes to extreme left position C and again returns to mean position A. This completes one vibration and the motion is repeated. The motion of the bob is a simple harmonic motion (S.H.M.). The time period, T of the simple pendulum depends upon its length l which is the distance between point of suspension S and C.G. of the bob, which is also C.G. of the pendulum.

(iii) Theory. Let a simple pendulum of length l have a bob of mass m. Let the bob be displaced from its vertical mean position A to position B by an angle θ and distance (Fig. 4.03).

The component mg cos θ of its weight balances tension in string. Component mg sin θ acts on the bob as restoring force and produces acceleration in it.

This time period is independent of amplitude of the motion of the bob, provided it remains small (so that θ = sin θ).

Knowing l and g, T can be known.

Hence, a simple pendulum becomes a good time-indicator and as such can be used as a time measuring device.

(iv) Damping. Under ideal conditions of motion (vibration), the amplitude of vibration of the pendulum bob must remain constant. This will be the case when the point of suspension is free from friction and there is no air surrounding the bob. But actually both the factors are present.

The presence of air causes damping and the amplitude of vibration goes on decreasing continually as shown in Fig. 4.04. But even then the time period is not much effected. Amplitude decreases by a constant ratio.

(v) Seconds pendulum. A simple pendulum, whose time period is two seconds (i.e. whose hob takes one second to move from one extreme to other

extreme), is called seconds pendulum. Its length is 4-04. Damped oscillations. about 99.4 cm.

**3. Ticker-Tape timer (Not in CBSE Syllabus)**

It is another device used for measuring time in the laboratory. It can measure time intervals of the order of 1/50 th or 1/100 th of a second.

(i) Construction. Its simple type consists of an electromagnet whose coils are fed with alternating current of known steady frequency. A soft iron strip fixed at one end and free at the other end, runs through the core of the electromagnet. The free end of the strip has a small pointed hammer attached to it which strikes a disc made of carbon paper. A paper tape passes from under the disc.

(ii) Working. When alternating current is passed through the electromagnet, the soft iron strip gets magnetised and demagnetised twice during one cycle of current. It makes soft iron strip vibrate inside the core of the electromagnet with a frequency double of the frequency of the alternating current. As the strip vibrates, the pointed hammer strikes the carbon disc which leaves a dot on the paper tape below it.

As the paper tape is pulled from under the disc, dots are marked on it at a constant interval of time. This interval is half the time period of the alternating electromagnet current and can be known. With a 50 cycles per sec alternating current in laboratory, this time interval will be 1/100 sec. This time interval is called a tick and can be used as a standard unit of time.

**Some important definitions connected with S.H.M.**

- Periodic Motion. A motion that repeats itself in equal intervals of time, is called a periodic motion.
- Oscillatory Motion. The periodic motion in which a body moves to and fro (back and forth) over the same path, is called an oscillatory motion.
- Simple Harmonic Motion (S.H.M.). An oscillatory motion which can be expressed in terms of single harmonic function (e.g., sine and cosine functions), is called a simple harmonic motion (S.H.M.).
- Displacement. At any moment, the distance of the body (particle) from mean position, is called the displacement (it is a vector quantity). Its symbol is y.
- Amplitude. Maximum displacement on either side of mean position is called amplitude. Its symbol is A.
- Vibration. Motion from mean position to one extreme, then to other extreme and then back to mean position from same side, is called one vibration.
- Time Period. The time taken by the body to complete one vibration, is called the time period of the body. Its symbol is T.
- Frequency. The number of vibrations made by the body in one second, is called the frequency of the body. Its symbol is v.

Hence, v = 1/T or vT=1.

**Question. 1. Define a second.**

**Answer.** It is a basic unit of time. It is defined as l/86400th part of a mean solar day. [Read Art. 4.02 for recent definition.]

**Question. 2. Define a mean solar day.**

**Answer.** A mean solar day is defined as the interval between two consecutive overhead passages of the sun at one place on the earth, as the earth rotates on its axis from west to east.

**Question. 3. Is solar day constant ?**

**Answer.** No, it varies in value over the year.

**Question. 4. What is the difference between a clock and a watch ?**

**Answer.** A clock has a big size. Its least count is 1 sec. A watch has a smaller size. Its least count is 0.1 sec or 0.2 sec.

## Harmonic Links Cycles, Harmonics & the Universe

Cosmology -The supreme task of the physicist is to arrive at those **universal elementary laws** from which the **cosmos** can be built by pure **deduction (Albert Einstein, 1954)**. The **Wolff-Haselhurst Cosmology** explains a **Perpetual Finite Spherical Universe** within an **Infinite Eternal Space**. The Spherical Standing waves determines the size of our finite spherical universe within an infinite Space (Matter is large not small, we only 'see' the Wave-Center / 'particle' effect which has greatly confused physics). Huygens' Principle explains how other matter's out waves combine to form our matter's spherical In-Waves, which then deduces both Mach's Principle and the redshift with distance (without assuming Doppler effect due to an expanding universe / Big Bang). This also explains how matter interacts with all other matter in the universe (why we can see stars) as matter is the size of the universe, but we only 'see' the high wave amplitude wave-centers / 'particles'.

Physics and Metaphysics - The Metaphysics of Space and Motion Sensibly Unites **Albert Einstein's Relativity, Quantum Theory, and Cosmology.** This 40 page Treatise (written over five years) will be published in 'What is the Electron' (Apeiron, 2005).

Physics: Cosmology: Problems of the Big Bang Theory - Three Famous 'Dissident Scientists' on the many errors of the Big Bang Theory - **Eric Lerner, Halton Arp, Bill Mitchell**. Information and links to problems of the **Big Bang Theory**.

Physics: Cosmology: Edwin Hubble Space Telescope - Deducing the **Hubble Redshift** with distance from **Huygens' Principle** rather than **Doppler Shift** / expanding Universe. (Includes some nice pictures from Hubble Space Telescope).

Physics: Cosmology: CMBR - Cold Hydrogen in Space (not Big Bang) as the Cause of the **Cosmic Microwave Background Radiation?**

Physics: Cosmos Harmony Music Universe - Cosmic Harmony and the **Musical Universe** - The Wave Structure of Matter in Space explains how we live in a Musical Universe (on a Planet in need of better Harmony!).

https://en.wikipedia.org/wiki/Cycles - Excellent summary of various types of cycles and comprehensive links.

https://www.cyclespage.com/ - Interesting website for women which helps track your monthly menstrual cycle. This easy-to-use service lets you: Keep an online record of your past menstrual cycles. View predictions of your upcoming menstruation and ovulation dates. Know when you are most likely to be fertile.

akwu/stckpre.html - On the relationship between planetary cycles and business / stock market cycles, has some understanding of the Wave Structure of Matter. Human beings tend to respond to these invisible electromagnetic waves in the spherical resonating cavity surrounding the earth in such a way as to be modulated indirectly by the motion and position of the planets.

icouzin/Boi%20et%20al,%201999.pdf - ' We investigated the phenomenon of activity cycles in ants, taking into account the spatial structure of colonies. In our study species, Leptothorax acervorum, there are two spatially segregated groups in the nest. We developed a model that considers the two groups as coupled oscillators which can produce synchronized activity. By investigating the effects of noise on the model system we predicted how the return of foragers affects activity cycles in ant colonies.' (School of Mathematical Sciences and Department of Biology and Biochemistry, Centre for Mathematical Biology, University of Bath, Claverton Down, Bath BA2 7AY, UK)

## Hooke’s Law and Simple Harmonic Motion

To determine the spring constant of a spring by measuring its stretch versus applied force, to determine the spring constant of a spring by measuring the period of oscillation for different masses, and also to investigate the dependence of period of oscillation on the value of mass and amplitude of motion.

### Hypothesis

If the applied force (mass) is to remain the same while the vertical displacement is increased, there period will remain the same. However, if the vertical displacement is held constant while the applied force in increased, the period will increase. This is in accordance with the derived equation T = 2π (M / k) 0.5 .

### Labeled Diagrams

A (m) | ∆t_{1} (s) | ∆t_{2} (s) | ∆t_{3} (s) | ∆t avg. (s) | α_{t} (s) | T (s) |

0.0200 | 7.17 | 7.38 | 7.40 | 7.32 | 0.127 | 0.0736 |

0.0400 | 7.40 | 7.42 | 7.33 | 7.38 | 0.0473 | 0.0273 |

0.0600 | 7.33 | 7.38 | 7.44 | 7.38 | 0.0551 | 0.0318 |

0.0800 | 7.33 | 7.31 | 7.29 | 7.31 | 0.0200 | 0.0115 |

0.100 | 7.10 | 7.14 | 7.08 | 7.11 | 0.0306 | 0.0176 |

M (kg) | ∆t_{1} (s) | ∆t_{2} (s) | ∆t_{3} (s) | ∆t avg. (s) | α_{t} (s) | T (s) | T 2 (s 2 ) |

0.0500 | 7.55 | 7.63 | 7.55 | 7.58 | 0.0462 | 0.0267 | 0.758 |

0.0600 | 8.10 | 8.15 | 8.13 | 8.13 | 0.0252 | 0.0145 | 0.813 |

0.0700 | 8.75 | 8.85 | 8.73 | 8.78 | 0.0643 | 0.0371 | 0.878 |

0.0800 | 9.37 | 9.34 | 9.36 | 9.36 | 0.0153 | 0.0088 | 0.936 |

0.0900 | 9.95 | 9.94 | 10.00 | 9.96 | 0.0321 | 0.0186 | 0.996 |

0.100 | 10.50 | 10.48 | 10.46 | 10.48 | 0.0200 | 0.0115 | 1.05 |

### Graphs

### Questions

**1. Do the data from Part 1 verify Hooke’s Law? State clearly the evidence for your answer.**

The data correlate close to Hooke’s Law, but not quite. The law states that F = -ky, where F is in this case Mg and y equals the negative displacement. After graphing forces versus displacement, a value of 3.53 N/m was determined as the spring constant. However, when applying this value to the equation and using recorded displacement values, the calculated force come up less than the actual for used. For example, in the first trial y = -0.055 m. Multiplying that value by the extrapolated spring constant gives a theoretical force of 0.194 N, but the actual force used was 0.244 N. All other trials yield a similar lowball theoretical force.

**2. How is the period T expected to depend upon the amplitude A? Do your data confirm this expectation?**

Period is not expected to depend upon amplitude, as suggested by the equation T = 2π (M / k) 0.5 , where amplitude is absent as a variable. The data confirms this expectation, as the period was nearly the same for each trial.

**3. Consider the value you obtained for C. If you were to express C as a whole number fraction, which of the following would best fit your data (1/2, 1/3, 1/4, 1/5)?**

The obtained value of C is 0.694, which is closest to 1/3.

**4. Calculate T predicted by the equation T = 2π (M / k) 0.5 for M = 0.050 kg. Calculate T predicted by T = 2π ( [M + Cm _{s}] / k ) 0.5 for M = 0.050 kg and your value of C. What is the percent difference between them? Repeat for a value for M of 1.000 kg. Is there a difference in the percent differences? If so, which is greater and why?**

T predicted using the first equation and M = 0.050 kg is 0.726 s. T predicted using the second equation and M = 0.050 kg is 0.771 s. This is a percent difference of 6.01%.

T predicted using the first equation and M = 1.000 kg is 3.24 s. T predicted using the second equation and M = 1.000 kg is 3.26 s. This is a percent difference of only 0.615%. The greater percent difference occurs at the lower weight because the weight of the spring is almost insignificant at higher weight. The proportion of the mass to the spring is so great that it has almost no effect on the calculation.

### Conclusion

During part one of the experiment, the vertical displacement of a spring was measured as a function of force applied to it. The starting position of the spring was recorded using a stretch indicator. Mass was added to the spring, and the displacement was recorded. This was repeated with various amounts of mass. From these data, a graph of force versus displacement was plotted, and a linear fit slope revealed the spring constant. In this endeavor, the spring constant was valued at 3.53 N/m.

However, when applying this spring constant to the recorded displacements in the Hooke’s Law equation, the calculated forces are lower than the recorded forces. In similar manner, when rearranging Hooke’s Law to solve for displacement, the calculated displacements are larger than the actual recorded displacements. This means there was human error, most likely in terms of not being precise with the displacement readings because the recordings for the masses used were accurate. Because such small masses were used, any error in displacement readings was augmented. The spring used may also not have been perfect.

During part two of the experiment, the period of the spring was measured as amplitude changed while mass remained constant. The period remained nearly the same throughout every trial, which was to be expected. Any differences in period may be accounted to inadequate stopwatch usage and inaccurate starting displacements throughout the trials. It should be noted that at amplitude of 0.100 m, the hook lost contact with the spring for a split second at the apex of oscillation, which accounts for its oddity in period. This was something that could not be avoided at that amplitude the spring pulled the hook up too quickly which caused the loss of contact. The resulted graph of period versus amplitude yielded a linear fit slope of close to 0 (-0.247 s/m), which was predicted.

During part three of the experiment, the period of the spring was measured as mass was varied while amplitude remained constant. As the mass was increased, the period also increased. This was not surprising considering the given equations. The square of the period versus mass for each trial was plotted and a linear fit was taken. The slope and y-intercept of this line was then used to determine the spring constant and C, the fraction of the spring’s mass that should be taken into account for the equation T = 2π ( [M + Cm_{s}] / k ) 0.5 . The calculated value of k was 3.75 N/m, which is only 6.04% different from the value determined earlier of 3.53 N/m. The value of C was determined to be 0.694, which is closest to the whole number fraction of 1/3. Any error during parts two and three can be attributed to inaccurate stopwatch recordings and slight variance in displacement and release of the masses at each amplitude.

## Teaching harmonic motion in trigonometry: Inductive inquiry supported by physics simulations.

In this paper, we would like to present a lesson whose goal is to utilise a scientific environment to immerse a trigonometry student in the process of mathematical modelling. The scientific environment utilised during this activity is a physics simulation called Wave on a String created by the PhET Interactive Simulations Project at Colorado University at Boulder and available free on the Internet. The outline of the activity, situated in inductive inquiry, is written in a format that is adaptable to various classroom settings students can work independently in front of a computer or in groups. If a computer lab is not available, the simulation can be projected on a screen in a regular math classroom. In all of these settings, the teacher takes the role of a facilitator. Although, the lesson was developed following trigonometry curriculum in the US, its cognitive learning objectives fit well into the scope of the proposed Australian mathematics curriculum (ACARA, 2010) that also emphasises the development of the skills of mathematical modelling, data collection, and analysis. The activity, presenting applications of periodic functions in a non-geometric setting, can be conducted in Australian Upper Secondary or Lower Tertiary Trigonometry courses. With some extensions, including dumped oscillation, its content will fit into Queensland Mathematics C syllabus (QSA, 2008), in particular the section of Advanced Periodic and Exponential Functions.

Inductive reasoning is a thought process whose ultimate goal is knowledge acquisition. Inductive reasoning encompasses a range of instructional methods, including inquiry learning, problem-based learning, project-based learning, case-based teaching, discovery learning, and just-in-time teaching (Prince & Felder, 2006). Inductive reasoning is commonly applied in science, where data is gathered and mathematical models are formulated to predict future behaviour of highlighted quantities. Literature (Felder & Brent 2004) shows that challenges provided by inductive methods serve as precursors to students' intellectual development.

Learning methods involving inductive reasoning are characterised as constructivist methods. They build on the widely accepted principle that students construct their own versions of reality rather than simply absorb versions presented by their instructors. In cognitive constructivism, which originated primarily in the work of Piaget (1972) an individual's reactions to experiences lead to learning.

Although inductive reasoning produces multiple learning outcomes and it is extensively used in sciences (Thacker, Eunsook, Trefz, & Lea, 1994), this inquiry method is rarely used in trigonometry. We argue that applying trigonometric functions to model harmonic motion can provide a rich scientific context to exercise mathematical modelling through inductive inquiry in trigonometry classes as well.

Inquiry learning, one of the simplest forms of the inductive reasoning was selected to construct this activity. Staver and Bay (1987) identified three stages of inquiry learning: structured inquiry, where students are given a problem and an outline for how to solve it, guided inquiry where in addition, students are supposed to figure out the solution method, and open inquiry where students must formulate the problem and find the solution.

We situated this activity in a guided inquiry stage. By modifying some of its auxiliary elements, it can be changed to either structured or open inquiry depending on students' responses.

The framework of the guided inquiry follows Joice's (2009) four layers of inductively organised learning environment: focus, conceptual control, inference, and confirmation. In order to parallel this inquiry with its scientific counterpart, problem statement was included as a catalyst of the process. The following descriptions of the layers served as theoretical foundations of the activity. Problem statement is a form of a question that students answer because of conducting an experiment. Focus is building (collecting) data and asking students to analyse the attributes of the data and formulate the hypothesis. Conceptual control (analysis) is classifying the facts and identifying patterns of regularity. Inference is a generalisation (formulation of a pattern or law) about the relations between the collected facts that leads to acquiring a general formula or mathematical function. Confirmation is a verification of the derived model in new (physical) circumstances conducted through testing inference and further observations.

An effective implementation of the inquiry method provides also the students with practicing a conduct of scientific experiment of how to identify and collect appropriate evidence, analyse and interpret results, formulate conclusions, and evaluate the conclusions (Lee, 2004). Since understanding physics concepts requires formal reasoning, familiarity with the process of mathematisation of generated data, and extracting general principles from specific cases (Bellomonte, Guastella, & Sperandeo-Mineo, 2005), exposing trigonometry students to a simulated physics phenomena might produce multiple results. It not only develops their modelling skills but also helps them understand the laws of physics. Furthermore, while conducting this activity, students will be placed in the roles of scientists actively constructing new knowledge. By referring to a scientific environment, practitioners will learn to select information based on scientific validity, a cognitive skill that they can apply in other subject areas as well as in their work places.

Physics simulations selected for this project are provided free online by PhET Interactive Simulations Project at Colorado University. Although their primary purpose is enhancing the teaching of physics, we argued that they could be integrated into the process of teaching and learning of mathematics. While working on the virtual labs, students can state hypotheses, observe scientific processes, take measurements, construct mathematical models, and validate them. They can modify the variables of the experiments, as well as predict and verify the respective outputs. Inquiry conducted by Lima (2010) shows that when prediction is used effectively students are likely to progress from passive listeners to active thinkers, simultaneously expanding and deepening their mathematical knowledge. With the aid of graphing technology, derived mathematical models can be further verified. Research conducted by PhET (Finkelstein, Perkins, Adams, Kohl, & Podolefsky, 2004) showed that these simulations can be substituted effectively for real laboratory equipment in physics courses. Findings of research (Sokolowski & Walters, 2010, p. 110) conducted in a South-Central Texas high school proved that mathematics students not only learned more and scored higher on the district and state standardised test items related to analysis and synthesis but that they also enjoyed and appreciated the new learning environment.

The structure of the activity

The activity uploaded also at http://phet.colorado.edu/en/contributions/ view/3340 evolves within the five stages of guided inquiry. The purpose of the commentaries is to help walk students through the inquiry process.

Introduction of the concept and demonstration of the simulation

The teacher opens the simulation at http://phet.colorado.edu/en/ simulation/wave-on-a-string and demonstrates its features, focusing the students' attention on the shape of the string while it transmits energy. The oscillations can be generated manually, or they can be produced periodically by checking the button Oscillate located on the left side of the simulation.

The instructor might begin the process with generating the wave manually and mention to students some obstacles that they would face to mathematically describe this irregular movement. The oscillator, a wheel rotating periodically due assigned frequency, produces a regular wave on the string. It is important that the No End mode located on the bottom right corner of the simulation is checked. The instructor might also want to demonstrate the effect of the different damping factors and string tensions on the motion of the energy. At last, the instructor directs students' attention to frequency and amplitude, as these two physical factors of the periodic motion will dictate the formulation of respective periodic function. Since mathematical modelling refers to successive approximation, yet more regularity in the movement would help construct the model. This can be achieved by reducing the damping factor to zero. Under these circumstances, the energy transmitted by string is not dissipated to the environment, and therefore the amplitude of the wave indicating the amount of energy remains constant. It might be interesting to students to note that the damping factor does not affect the frequency of the wave. The frequency of the wave depends only on the frequency of the source producing the wave.

What mathematical function can be applied to model the path of motion of the energy generated by the wheel? What will the independent and dependent variables of the function be? Students might be given some time to discuss their answers in groups.

The instructor elicits from the students a periodic nature of the motion that suggests sinusoidal function to be applied. There is one important element that the instructor might want to address to the students at this point. Since the path of the motion is two- dimensional (the energy oscillates up and down and moves forward) there can be two different ways undertaken to model the movement:

A. Applying parametric representation to model independently vertical y = f(t) and horizontal position x = f(t) of the front of the wave. In this case, the vertical movement would be modelled by a sinusoidal function and horizontal by a linear (energy moves with a constant forward velocity). Both functions would be expressed in terms of time. Although, this model depicting a dynamics of the system is physically rich, students who did not study properties of parametric equations might find it difficult to apply. Therefore, another (B), a simplified version of this representation is suggested to be used.

B. Expressing vertical position of the wave in terms of horizontal position, y = f(x)

This representation is easily conveyable to an average trigonometry student and it is suggested to be adopted to model the sinusoidal path. If an opportunity exists, the instructor might want to prove mathematically that the parametric forms discussed in A can be converted into a singular representation discussed in B.

Focus/gathering information/stating hypothesis

In this part, the teacher discusses, in detail, the critical components of the sinusoidal function such as amplitude, period, horizontal, and vertical shift, and how these quantities can be identified and measured in the experiment.

Due to chosen singular model, the period of anticipated function will be expressed in the units of metres (here millimetres). In physics, the length of one wave, expressed in metres, represents the wavelength of the wave denoted by [lambda]. Although students who took physics course will correlate the distance to wavelength, for the purpose of this activity, the distance will be labelled [DELTA]x.

The instructor might want to demonstrate some measuring devices embedded in the simulation, such as a ruler and a stopwatch that help quantify the highlighted quantities.

Establishing a frame of reference (x- and y-axes) is also important. Due to its location, function vertical and horizontal shifts will be referenced. It is suggested that to develop the general model, the x-axis is established at the equilibrium line of the string and the y-axis is aligned with the centre of the oscillator (initial position of the propagated energy).

Once a tentative model has been concluded, students focus on measuring necessary quantities that will constitute the form of the sinusoidal function. In order to further generalise the model, only one full cycle can be shown on the screen for the analysis. Students can be given rulers and be asked to measure necessary quantities using screenshots of the simulation copied in their lab outlines. This approach has some advantages it makes the activity more tangible. Students prefer this approach to just using the numbers embedded in the scenario. The lengths can be expressed in millimetres or centimetres.

Generalisation of the analysis/inference

During this stage, students transfer measured quantities into a mathematical model. In order to review essential elements of a periodic function, they can work on multiple-choice questions. Samples of such items are provided below.

A general form of a periodic function is given by y = A sin (2[pi]/T)1.

1. Select the quantity that represents the measured average time of one full cycle.

D. General time variable (t)

2. Select the quantity that represents the dependent variable in this function.

B. Vertical position of wave (y)

C. General time variable (t)

3. Suppose that the length of one full cycle of a wave is denoted by Ax and expressed in the units of metres. Which expression can be used to model the wave?

A. y = A sin (2[pi]/[DELTA]x) [DELTA]x

Following the review, the teacher discusses the answers and then lets the students construct their functions. Students substitute the values of these quantified components to the general form of

and then verify the function equation. Building on Bateman (1990), inductive instruction should be spirally organised students should be directed to continually revisit critical concepts and improve their cognitive models, thus verification, the next stage of the inquiry follows.

Verification and confirmation of the derived model

This stage is very significant in the process of the guided inquiry. When working on typical paper-and-pencil problems, this stage is often omitted because the physical representation of a wave is usually not provided and the movement is not observable. The availability of the simulation presents a great opportunity for contrasting observed wave with its mathematical representation. Students can use a graphing calculator or any technological tool (sketchpad) that converts algebraic function into a graph. They might be asked to determine the dimensions of the window of a graphing calculator so that their functions resemble the screen shot as closely as possible.

If graphing technology is not available, students can verify the model the old fashioned way using a table of values. They could calculate outputs for selected inputs and compare the values with the model. Further verification can refer to modification of the components of the parent function. Sample questions in regards to this mode of verification are presented below. For each modification, the students are supposed to use the simulation to observe the change, write new function, and use graphing technology to check if derived function corresponds to observable wave.

1. Suppose that the x-axis--the draggable reference (dotted) line shown on the simulation--is moved 30 mm below the string. Which component of the derived function should be changed to reflect this transformation?

B. Vertical transformation

D. Horizontal transformation

2. Due to a modified frequency, there are twice as many waves observed on the string. Which component of the sinusoidal function should be changed to reflect this transformation?

B. Vertical transformation

3. Suppose that the maximum height of the wave as measured from the equilibrium line increased by 10 cm. Which parameter changed?

B. Vertical transformation

Exchanging thoughts about the experiment, and suggesting ways of improving this learning environment concludes the activity.

What impact do the simulations have on student learning process?

Students highly praise the new learning environment and find the lessons utilising simulations very attractive. As we mentioned before, this environment affects positively their test scores. Following this positive feedback, more simulations were adopted to enhance the modelling process in other sections of mathematics such as polynomial or transcendental functions. The simulations were also utilised in calculus to enhance the teaching of limits, derivatives, and integrals including the First Fundamental Theorem of Calculus. We argue that the virtual physical world and the inquiry processes that the students were immersed in to model the world helps them to understand the modelling process and prepares them for engineering classes and college. We foresee a need for a more systematic research study of the influence of the simulations on mathematical knowledge acquisition by mathematics students. We hope that the readers will find the journey interesting enough to also get involved in similar research.

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In the Music Theory AP course, students learn to recognize, understand, and describe the basic materials and processes of music that are heard or presented in a musical score. You can expect to practice and develop musical skills while building your understanding of music composition and theory. You will also develop your music vocabulary through class discussions and written analyses of listening selections. Although there are no prerequisite classes for enrollment in the Music Theory course, you do need to be able to read and write musical notation. It is also strongly recommended that you have at least basic performance skills in voice or on an instrument.

You’re expected to possess four core skills at the end of the AP Music Theory course. These skills form the basis of many tasks you’re asked to perform on the AP Music Theory exam. The core skills are:

Skill | Description | Percentage of Exam Score (Multiple-Choice Section) |

Analyze Performed Music | Take musical terms, concepts, and relationships and apply them to performed music (aural). | 48% |

Analyze Notated Music | Use musical terms, concepts, and relationships and apply them to notated music (written). | 44% |

Convert Between Performed and Notated Music | Convert music between aural and written forms using conventions of musical notation and performance. | 8% |

Complete Based on Cues | Following 18th-century stylistic norms complete music based on cues. | Not assessed in multiple-choice questions |

Through the four core skills you’ll explore four big ideas that serve as the foundation of the AP Music Theory course. Using these big ideas, you’ll create connections between the concepts and skills learned and deepen your understanding of musical theory. The College Board describes the four big ideas of AP Music Theory as:

**1. Pitch:** Specific frequencies of sound, known as pitches, are basic units of music. Pitches that are deliberately sequenced through time create melodies, and groups of pitches presented successively or simultaneously form chords. Within an established musical style, chords relate to one another in the context of harmony.

**2. Rhythm:** Music exists in the dimension of time, where long and short sounds and silences can be combined in myriad ways. This temporal aspect, called rhythm, is often governed by a layered structure of interrelated pulses known as meter. Rhythms are typically grouped into distinctive rhythmic patterns, which help define the specific identity of a musical passage. Musicians use established rhythmic devices to expand expressive possibilities, often achieving their effect by challenging the regularity of the meter or transforming rhythmic patterns.

**3. Form:** Music exhibits a structural aspect known as form, in which a musical composition is organized in a hierarchy of constituent parts. The specific ways these parts are related, contrasted, or developed produce the unique profile of an individual composition. Specific formal types and functions may be identified when parts of a composition follow established melodic-harmonic patterns, or fulfill established roles within the overall hierarchical structure.

**4. Musical Design:** Texture, timbre, and expression contribute to the overall design and character of a piece of music or musical performance. The texture of a musical passage arises from the way its layers are produced and distributed, and how they interact to form the totality of sound. Timbre refers to the distinct sounds of specific instruments and voices, arising from the physical manner in which those sounds are produced. Expressive elements are related to musical interpretation and include dynamics, articulation, and tempo.

**AP Music Theory Course Content**

The AP Music Theory Course is divided into eight units. Below is a suggested sequence of the units from the College Board, along with a short description of what’s covered in each one.

Unit | Content Taught |

Music Fundamentals 1: Pitch, Major Scales and Key Signatures, Rhythm, Meter, and Expressive Elements | How pitch and rhythm work together to become melody and meter—building musical compositions. |

Music Fundamentals 2: Minor Scales and Key Signatures, Melody, Timbre, and Texture | Use knowledge of pitch patterns and relationships in major scales and apply it to minor scales. |

Music Fundamentals 3: Triads and Seventh Chords | The fundamentals of harmony. |

Harmony and Voice Leading I: Chord Function, Cadence, and Phrase | Apply knowledge of harmonic materials and processes using it to explore the procedures of 18th-century style voice leading. |

Harmony and Voice Leading II: Chord Progressions and Predominant Function | Describe, analyze, and create complex harmonic progressions in the form of four-part (SATB: soprano, alto, tenor, and bass) voice leading. |

Harmony and Voice Leading III: Embellishments, Motives, and Melodic Devices | Exploration of the skills and concepts of harmony and voice leading. |

Harmony and Voice Leading IV: Secondary Function | Building on knowledge of harmonic relationships and procedures to deepen your understanding of keys, scale degrees, and chords. |

Modes and Form | The conventions that influence the character of music such as modes, phrase relationships, and forms. |

## 7 Answers 7

Sinusoidal waves don't have harmonics because it's exactly sine waves which combined can construct other waveforms. The fundamental wave is a sine, so you don't need to add anything to make it the sinusoidal signal.

About the oscilloscope. Many signals have a large number of harmonics, some, like a square wave, in theory infinite.

This is a partial construction of a square wave. The blue sine which shows 1 period is the fundamental. Then there's the third harmonic (square waves don't have even harmonics), the purple one. Its amplitude is 1/3 of the fundamental, and you can see it's three times the fundamental's frequency, because it shows 3 periods. Same for the fifth harmonic (brown). Amplitude is 1/5 of the fundamental and it shows 5 periods. Adding these gives the green curve. This is not yet a good square wave, but you already see the steep edges, and the wavy horizontal line will ultimately become completely horizontal if we add more harmonics. So this is how you will see a square wave on the scope if only up to the fifth harmonic are shown. This is really the minimum, for a better reconstruction you'll need more harmonics.

Like every non-sinusoidal signal the AM modulated signal will create harmonics. Fourier proved that *every* repeating signal can be deconstructed into a fundamental (same frequency as the wave form), and harmonics which have frequencies that are multiples of the fundamental. It even applies to non-repeating waveforms. So even if you don't readily see what they would look like, the analysis is always possible.

This is a basic AM signal, and the modulated signal is the product of the carrier and the baseband signal. Now

So you can see that even a product of sines can be expressed as the sum of sines, that's both cosines (the harmonics can have their phase shifted, in this case by 90°). The frequencies $(f_C - f_M)$ and $(f_C + f_M)$ are the *sidebands* left and right of the carrier frequency $f_C$.

Even if your baseband signal is a more complex looking signal you can break the modulated signal apart in separate sines.

Pentium100's answer is quite complete, but I'd like to give a much simpler (though less accurate) explanation.

The reason because sinewaves have (ideally) only one harmonic is because the sine is the "smoothest" periodic signal that you can have, and it's therefore the "best" in term of continuity, derivability and so. For this reason is convenient to express waveforms in terms of sinewaves (you can do it with other waves as well, as well as they are $C^

Just an example: why in the water you usually see curved waves? (for this sake, ignore the effect of the beach or wind) Again, it's because it's the shape that requires less energy to form, since all the ramps and edges are smooth.

In some cases, like the Hammond organ, sinewaves are actually used to *compose* the signal, because with decomposition is possible to synthesize a lot of (virtually all) sounds.

There is a beautiful animation by LucasVB explaining the Fourier decomposition of a square wave:

These images explain better the square wave decomposition in harmonics:

You can decompose any waveform into an infinite series of sine waves added together. This is called Fourier analysis (if the original waveform is repeating) or Fourier transform (for any waveform).

In case of a repeating waveform (like a square wave), when you do Fourier analysis you find that all the sines that compose the waveform have frequencies that are an integer multiple of the frequency of the original waveform. These are called "harmonics".

A sine wave will only have one harmonic - the fundamental (well, it already is sine, so it is made up of one sine). Square wave will have an infinite series of odd harmonics (that is, to make a square wave out of sines you need to add sines of every odd multiple of the fundamental frequency).

The harmonics are generated by distorting the sine wave (though you can generate them separately).

- You can make a sine wave out of any wave of a fixed frequency, as long as you have a filter that passes the fundamental frequency, but blocks 2x frequency (as you would be leaving only one harmonic in place).
- Actually, you can make a sine wave that has different frequency than the orginal - just use a bandpass filter to pass the harmonic you want. You can use this to get a sine wave of a frequency that is a multiple of the frequency of another sine - just distort the original sine and pick out the harmonic you want.
- RF systems have to put out waveforms that do not contain harmonics outside the allowed frequency range. This is how a PWM power supply (operating frequency

The derivative - rate of change - of a sinusoid is another sinusoid at the same frequency, but phase-shifted. Real components - wires, antennas, capacitors - can follow the changes (of voltage, current, field-strength, etc.) of the derivatives as well as they can follow the original signal. The rates of change of the signal, of the rate-of-change of the signal, of the rate-of-change of the rate-of-change of the signal, etc., all exist and are finite.

The harmonics of a square wave exist because the rate of change (first derivative) of a square wave consists of very high, sudden peaks infinitely high spikes, in the limit-case of a so-called perfect square wave. Real physical systems can't follow such high rates, so the signals get distorted. Capacitance and inductance simply limit their ability to respond rapidly, so they ring.

Just as a bell can neither be displace nor distorted at the speed with which it is struck, and so stores and releases energy (by vibrating) at slower rates, so a circuit doesn't respond at the rate with which it is struck by the spikes which are the edges of the square wave. It too rings or oscillates as the energy is dissipated.

One conceptual block may come from the concept of the harmonics being higher in frequency than the fundamental. What we call the frequency of the square wave is the number of transitions it makes per unit time. But go back to those derivatives - the rates of change the signal makes are huge compared to the rates of change in a sinusoid at that same frequency. Here is where we encounter the higher component frequencies: those high rates of change *have the attributes of higher frequency sine waves*. The high frequencies are implied by the high rates of change in the square (or other non-sinusoid) signal.

The fast rising edge is not typical of a sinusoid at frequency *f*, but of a much higher frequency sinusoid. The physical system follows it the best it can but being rate limited, responds much more to the lower frequency components than to the higher ones. So we slow humans see the larger amplitude, lower frequency responses and call that *f*!

In practical terms, the reason harmonics "appear" is that linear filtering circuits (as well as many non-linear filtering circuits) which are designed to detect certain frequencies will perceive certain lower-frequency waveforms as being the frequencies they're interested in. To understand why, imagine a large spring with a very heavy weight which is attached to a handle via fairly loose spring. Pulling on the handle will not directly move the heavy weight very much, but the large spring and weight will have a certain resonant frequency, and if one moves the handle back and forth at that frequency, one can add energy to the large weight and spring, increasing the amplitude of oscillation until it's much larger than could be produced "directly" by pulling on the loose spring.

The most efficient way to transfer energy into the large spring is to pull in a smooth pattern corresponding to a sine wave--the same movement pattern as the large spring. Other movement patterns will work, however. If one moves the handle in other patterns, some of the energy that gets put into the spring-weight assembly during parts of the cycle will be taken out during others. As a simple example, suppose one simply jams the handle to the extreme ends of travel at a rate corresponding to the resonant frequency (equivalent to a square wave). Moving the handle from one end to the other just as the weight reaches end of travel will require a lot more work than would waiting for the weight to move back some first, but if one doesn't move the handle at that moment, the spring on the handle will be fighting the weight's attempt to return to center. Nonetheless, clearly moving the handle from one extreme position to the other would nonetheless work.

Suppose the weight takes one second to swing from left to right and another second to swing back. Now consider what happens if one moves the handle from one extreme of motion to the other has before, but lingers for three seconds on each side instead of one second. Each time one moves the handle from one extreme to the other, the weight and spring will have essentially the same position and velocity as they had two seconds earlier. Consequently, they will have about as much energy added to them as they would have two seconds before. On the other hand, the such additions of energy will only be happening a third as often as they would have when the "linger time" was only one second. Thus, moving the handle back and forth at 1/6Hz will add a third as much energy per minute (power) to the weight as would moving it back and forth at 1/2Hz. A similar thing happens if one moves the handle back and forth at 1/10Hz, but since the motions will be 1/5 as often as at 1/2Hz, the power will be 1/5.

Now suppose that instead of having the linger time be an odd-numbered multiple, one makes it an even-numbered multiple (e.g. two seconds). In that scenario, the position of the weight and spring for each left-to-right move will be the same as its position on the next right-to-left move. Consequently, if the handle adds any energy to the spring in the former, such energy will be essentially cancelled out by the latter. Consequently, the spring won't move.

If, instead of doing extreme motions with the handle, one moves it more smoothly, then at lower frequencies of handle motion there are apt to be more times when one is fighting the motion of the weight/spring combo. If one moves the handle in a sine-wave pattern, but at a frequency substantially different from the resonant frequency of the system, the energy that one transfers into the system when pushing the "right" way will be pretty well balanced by the energy taken out of the system pushing the "wrong" way. Other motion patterns which aren't as extreme as the square wave will, at at least some frequencies, transfer more energy into the system than is taken out.

## 14 - The Performance of Music

Music performance is a large subject that can be approached in many different ways. This chapter focuses on empirical research of music performance and related matters. Most of this research is concerned with Western tonal music and mainly art music. Excellence in music performance involves two major components like a genuine understanding of what the music is about, its structure and meaning, and a complete mastery of the instrumental technique. Evaluation of performance included many studies which are reviewed earlier. Evaluation occurs in the everyday activity of music critics, music teachers, and musicians. An overall evaluation is considered as a weighted function of the evaluations in the specific aspects. In order to maintain the tempo and to achieve perceived synchrony, musicians should therefore play a small amount ahead of the beat they hear. With sharp attacks the delay is less, and instruments with sharp attacks may therefore serve as “beat-definers” for the rest of an ensemble. In addition, some attempts are made to predict evaluation of music performances from the physical characteristics of the performances.