The Electromagnetic Spectrum

Recall that waves can be categorized into two major divisions:

Mechanical waves, which require a medium. These include sound, water and waves on a (guitar, etc.) string

Electromagnetic waves, which travel best where there is NO medium (vacuum), though they can typically travel through a medium as well. All electromagnetic waves can be represented on a chart, usually going from low frequency (radio waves) to high frequency (gamma rays). This translates to: long wavelength to short wavelength.

All of these EM waves travel at the same speed in a vacuum: the speed of light (c). Thus, the standard wave velocity equation becomes:

c = f l

where c is the speed of light (3 x 10^8 m/s), f is frequency (in Hz) and l (which should be lambda) is wavelength (in m).

Recent notes

The Doppler Effect

http://www.lon-capa.org/~mmp/applist/doppler/d.htm

http://falstad.com/mathphysics.html

Run the Ripple tank applet -

http://falstad.com/ripple/

The key in the Doppler effect is that motion makes the "detected" or "perceived" frequencies higher or lower.

If the source is moving toward you, you detect/measure a higher frequency - this is called a BLUE SHIFT.

If the source is moving away from you, you detect/measure a lower frequency - this is called a RED SHIFT. Distant galaxies in the universe are moving away from us, as determined by their red shifts. This indicates that the universe is indeed expanding (first shown by E. Hubble). The 2011 Nobel Prize in Physics went to local physicist Adam Riess (and 2 others) for the discovery of the accelerating expansion of the universe. Awesome stuff!

http://www.nobelprize.org/nobel_prizes/physics/laureates/2011/

It's worth noting that the effect also works in reverse. If you (the detector) move toward a sound-emitter, you'll detect a higher frequency. If you move away from a detector move away from a sound-emitter, you'll detect a lower frequency.

Mind you, these Doppler effects only happen WHILE there is relative motion between source and detector (you).

And they also work for light. In fact, the terms red shift and blue shift refer mainly to light (or other electromagnetic) phenomena.

wave practice

1. Consider a 2-m long string fixed at both ends. The speed of wave travel is 200 m/s in this string. Find the following:

a. wavelengths of the first 3 harmonics
b. frequencies of the first 3 harmonics

(4, 2, 4/3 meters; 50, 100, 150 Hz)

2. Now imagine an organ pipe, 1-m long. The speed of sound (which is the same as the speed of wave travel in an organ pipe) is 340 m/s. Find the same things as above.

(2, 1, 2/3 meters; 170, 340, 510 Hz)

3. What is the effect of capping one end of a tube?

4. Consider a C note, vibrating at 262 Hz. Find the following frequencies:

a. the next C, one octave above this one
b. a C, two octaves above 262 Hz
c. the note C#, one piano key ("semi-tone") above C

Organ pipes - sound in tubes

We will be talking about waves in tubes/pipes (like brass instruments, woodwinds, etc.).

http://www.physics.smu.edu/~olness/www/05fall1320/applet/pipe-waves.html

http://ralphmuehleisen.com/animations.html


Ultimately, I want you to see that waves in a tube are the same (mathematically) as waves on a string - the biggest differences:

the waves are longitudinal/compressional, NOT transverse
the waves have antinodes at each end, NOT nodes

When an organ pipe/tube is open on both ends, you have antinodes (in a longitudinal wave) on both ends. This becomes (mathematically) the same as a vibrating string (though the string has nodes on both ends). The math looks like the same, however:

lambda (l) = 2L/n

The lowest harmonic (f1, where n = 1) is still found by using v = f l, and dividing v by l. Successive harmonics are 2f1, 3f1, 4f1, ....

So, a tube has a lowest possible resonant tone, but if air is pushed through it harder, higher harmonics can be heard. Think about the recorder - you may have learned how to play one in your younger years. Keeping your fingers in the same positions, but blowing a bit harder, gives higher tones.

Some things to try:

Work your way through lessons on PhysicsClassroom.com

http://www.physicsclassroom.com/Class/sound/

PARTICULARLY LESSONS 5c and 2a.

Here is a problem to try:

1. Consider a tube that is 0.8-m long. The speed of sound is 345 m/s. Find the following:

a. the wavelengths of the first 4 harmonics
b. the frequencies of the first 4 harmonics
c. the wave shapes of the first 5 harmonics - see the applet, as well as my note sheet below

Note that the mathematics in this problem are IDENTICAL to those of the standing waves in the string - the speed, however, is the speed of sound.



http://www.physics.smu.edu/~olness/www/05fall1320/applet/pipe-waves.html

Music!

Music can be thought of (rather unartistically) as collections of frequencies. Here are the ways that we get notes.

In Western music, the standard is (generally) that a "Concert A" is tuned to 440 Hz. All other notes can be tuned with respect to that. Some orchestras tune to different standards, but we won't worry about those now.

An octave "above" a note is defined as twice the frequency of the note. Similarly, an octave "below" is taking a note and dividing it by two.

The tougher concept is - how do we get from one note on a piano (or any instrument) to the next note (and beyond)? The answer is wrapped up in the "equal tempered scale", a scale such that the ratio of one note to the previous note is always the same.

In short - to get from one note to its octave, multiply by two. But since there are 12 "jumps" or "semi-tones" or "half-steps" from one note to its octave, we ask ourselves (in an equal-tempered scale), what number multiplied by itself 12 times gives us two? The answer?

The twelfth root of 2, or 2 to the 1/12 power --- around 1.0594

This number to the 12th power is 2.

But wait, what did I mean by 12 "jumps"?

A A# B C C# D D# E F F# G G# A

That's 13 notes, but 12 "jumps" or piano keys from A to the next A.

To go from A to A#, multiply the frequency of A by 1.0594.

If you wanted to get to B instead, multiply A's frequency by 1.0594 twice (or by 1.0594^2). To get to C, multiply A by 1.0594^3.



In gory detail:

http://en.wikipedia.org/wiki/Equal_temperament

Wave problems.

Wave problems - try all

Some review questions (and a couple new ideas).

1. Differentiate between mechanical and electromagnetic waves. Give examples.

2. Differentiate between longitudinal and transverse waves. Give examples.

3. Draw a wave and identify (or just define) the following parts: crest, trough, amplitude, frequency, period.

4. What is the frequency of a wave that travels at 25 m/s, if 3 complete waves can fit in a 10-m space?

5. Draw the first 3 harmonics for a string that is 3-m in length. Also, find the first 3 frequencies and wavelengths, if the wave speed is 100 m/s.

6. The speed of sound (in air) is approximately 345 m/s. If you stand far from a mountainside and yell at it, the echo returns to your ear in 1.8 seconds. How far is the mountain from you?

7. Approximately how much greater is the speed of light than the speed of sound in air?

8. Discuss the physics of the Chladni plate.

9. Find the wavelength of a 89.7 MHz radio wave.

Harmonics on a String

Much like "pumping a swing," a guitar string (or any vibrating string) will vibrate will certain "modes of vibration". That is, it will vibrate in certain specific configurations that satisfy the geometry of the string - fixed at both ends.

Since the string is fixed at both ends, it must have nodes (points of NO disturbance) at both ends. However, other possible modes of vibration can satisfy this condition. The wavelength must be satisfied by:

wavelength (lambda) = 2L / n

where n is the so-called "harmonic number," or if you prefer, the number of HALF-waves.

So, the wavelengths can be given by:

n wavelength
1 2L
2 L
3 2L/3
4 2L/4, or L/2
5 2L/5
6 2L/6, or L/3

The frequencies that correspond to these "harmonics" are given by:

v = f * lambda

or...

f = v/lambda

It is also important to note that the frequencies increase LINEARLY. That is, the frequency for n=2 is twice that of n=1. The frequency for n=3 is three times that of n=1. Got it?

This turns out to have musical significance.

Doubling a frequency generates something called and OCTAVE. For those of you who do NOT speak music, an octave is the difference between DO and DO, if you sing:

DO RE MI FA SO LA TI DO

Note that there are 8 notes here, thus the term octave.

Tripling the frequency also gives something of musical significance - it is 3/2 times greater than n=2. In music, a frequency multiple of 3/2 is defined as a "fifth", so named since it is the difference between DO and SO (5 notes).

But don't worry about that business, please - I mention it for your interest.

Play with this:

http://zonalandeducation.com/mstm/physics/waves/standingWaves/understandingSWDia1/UnderstandingSWDia1.html

This one is ok, but the pictures are a little misleading:

http://zonalandeducation.com/mstm/physics/waves/standingWaves/standingWaveDiagrams1/StandingWaveDiagrams1.html

A word on terminology:

The lowest harmonic (n=1) is called the fundamental. Any other n-value above 1 is called an overtone. So, the second harmonic (n=2) is called the first overtone.

Got it?

Next up.... how to translate these ideas to sounds in tubes/pipes, and how to construct musical scales.

Energy and the "blocks story"

I stole my energy story from the famous American physicist Richard Feynman. Here is a version adapted from his original energy story. He used the character, "Dennis the Menace." The story below is paraphrased from the original Feynman lecture on physics (in the early 1960s).

Dennis the Menace

Adapted from Richard Feynman

Imagine Dennis has 28 blocks, which are all the same. They are absolutely indestructible and cannot be divided into pieces.

His mother puts him and his 28 blocks into a room at the beginning of the day. At the end of each day, being curious, she counts them and discovers a phenomenal law. No matter what he does with the blocks, there are always 28 remaining.

This continues for some time until one day she only counts 27, but with a little searching she discovers one under a rug. She realises she must be careful to look everywhere.

One day later she can only find 26. She looks everywhere in the room, but cannot find them. Then she realises the window is open and two blocks are found outside in the garden.

Another day, she discovers 30 blocks. This causes considerable dismay until she realises that Bruce has visited that day, and left a few of his own blocks behind.

Dennis' mother removes the extra blocks, gives the remaining ones back to Bruce, and all returns to normal.

We can think about energy in this way (except there are no blocks!). We can use this idea to track energy transfers during changes. We need to be careful to look everywhere to ensure that we can account for all of the energy.

Some ideas about energy

• Energy is stored in fuels (chemicals).
• Energy can be stored by lifting objects (potential energy).
• Moving objects carry energy (kinetic energy).
• Electric current carries energy.
• Light (and other forms of radiation) carries energy.
• Heat carries energy.
• Sound carries energy.

Wave properties

A wave is essentially a traveling disturbance - motion of "energy", NOT the motion of stuff. Fundamentally, there are two varieties of these - one that requires a medium (mechanical) and one that does not require a medium (electromagnetic). An electromagnetic wave can often travel through a medium, but it always travels fastest (at the speed of light, c) where nothing gets in the way.

Properties of a wave:

Wavelength - the distance between 2 successive crests, or 2 successive troughs, or any 2 points "in phase" with each other. Wavelength is represented by the Greek letter lambda (which unfortunately I can't show here). The unit for wavelength is generally the meter, but it could be any unit of length.

Frequency is the number of waves/oscillations per second. It is represented by the letter f. The unit is the "cycle per second", usually called the hertz (Hz).

The period (T) is the amount of time for one oscillation. It is the inverse of the frequency. That is, if you have 2 oscillations/waves per second, the time for each is 1/2 second. In equation form:

T = 1/f or f = 1/T

Amplitude - the distance from equilibrium (the horizontal line) to the peak/crest of the wave, or to the trough/valley of the wave. The amplitude is usually a representation of the volume/loudness (if it's a sound wave) or intensity/brightness (if a light wave).


The velocity/speed (v) of a wave is the rate at which the energy travels. Simply, v is given by:

v = d/T

Since the wavelength is the distance in question, and T = 1/f, the equation can be written more conveniently for a wave:

v = d/T = wavelength * 1/T = wavelength * f

So.....

v = wavelength * frequency

We can imagine waves that travel outward from the origin - maybe in one direction (like sending a pulse on a spring) or in 3 dimensions (like a sound wave).

However, often waves interfere with other waves - to produce NEW waves. Sometimes waves interfere with themselves. This is the case with "standing waves", or waves on a string. It will also be the case with music in tubes or organ pipes.

Consider this applet, where the fundamental (lowest) possible resonant frequency is 25 Hz. The resonant frequency is that frequency that generates the largest possible amplitude for the energy investment. Think of it as the "just right" rate at which you'd need to "pump" a swing to get it higher and higher. Too little and you go nowhere. Too much and you also go nowhere. There is a "just right" amount of frequency - that's the resonant frequency.

http://ngsir.netfirms.com/englishhtm/StatWave.htm

Note what happens when you move the frequency to multiples of the resonant frequency: 50 Hz, 75 Hz, etc. This same sort of thing happens routinely on stringed instruments, as we shall see in class.

Harmonic motion

There is a peculiar type of motion that is worth discussion: Simple Harmonic Motion (SHM).

SHM refers to a regular oscillation, such as you might see with a pendulum or mass bobbing up and down (or back and forth) on a spring.

http://www.walter-fendt.de/ph14e/pendulum.htm

Note in this applet that while the "bob" swings back and forth, the displacement (elongation) changes SINUSOIDALLY, as shown on the graph. The velocity and acceleration of the bob also changes in a similar fashion.

Note the similar behavior for a mass on a spring:

http://www.walter-fendt.de/ph14e/springpendulum.htm

This type of motion is called "Simple Harmonic Motion," and it assumes the following reasonable conditions:

- the initial amount of pull (as long as it is "small") does not matter
- the mass of the string (for a pendulum) is small compared to the bob
- there are no significant frictional losses or effects

If these conditions are not met, the oscillator would likely be a complex harmonic oscillator, and there are different rules for those!

I introduce the idea of SHM, as it leads us nicely into the concept of a wave. If you imagine a mass bobbing up and down on a spring, and can imagine a pen attached to the mass, it could draw out the shape of the graph above. You would have to have the pen hit a piece of paper that is being moved at a constant speed horizontally.

In this case, a sine wave would be generated. You can see a little of this here:

http://phet.colorado.edu/sims/wave-on-a-string/wave-on-a-string_en.html

What should I know for the test?

Basics of SI units - what do the units mean, etc.

How to use the formulas (which will be given):
v = d/t
a = (vf - vi)/t
vf = vi + at
d = 0.5(vi + vf) t
d = vi t + 0.5at^2
F = ma
W = mg
F = G m1 m2 / d^2
g = G M / d^2
a^3 = T^2

the meaning of velocity, acceleration

gravitational acceleration, g

center of gravity

universal gravitation

Newton's laws

Kepler's laws

basic science history (Copernicus, Kepler, Newton)

epicycles and retrograde motion

speed of light (meaning, etc.)

inverse square law

weightlessness

Expect a test that is multiple-choice, with somewhere in the neighborhood of 20% numerical problems - a calculator is permitted but not really needed.

Also expect that there will be 40-50 questions, though more are possible. You will have the entire class to finish the test.

If you've attended class, reviewed the notes, understood the demonstrations and worked the problems, you should be fine.

Center of Mass / Center of Gravity

The Center of Gravity (CG) of an object can be thought of as several things:

- the point about which an object best rotates

- the point on which an object best balances

- the point where all "Torques" are balanced*

- a fictitious point, not even necessarily ON the object (consider a donut)

* Torque - the produce of weight and distance; W d

It is easy to visualize the CG concept with a simple see-saw:

W1 W2

------------o--------------

d1 d2

Weights are W and distances are d.

Two people (W1 and W2) are balanced on a see-saw; however, for balance/equilibrium, they don't have to have the same weight. The product of their weight and distance (from the fulcrum) must be equal. That is, the torque on the left (W1 d1) must equal the torque on the right (W2 d2).

So, the CG of the see-saw above is the point where the torques are balanced - NOT the weights. In other words, the weight on both sides are NOT equal; the torques are.

For something to be stable, the CG must be supported - either by a base below or above the CG. Consider the demonstrations from class.

- spoon / fork

- balancing guy

- Ernest the balancing bear

- the balancing meter stick lever (like a see-saw)

- balancing a meter stick on your finger

- the CG of a man vs. a woman