# New Particles

#### December 16, 2015

Physicists in Europe Find Tantalizing Hints of a Mysterious New Particle: This new particle, if confirmed to exist (the data is not conclusive) seems to go beyond the Standard Model of physics that we know and love.

The last sub-atomic particle discovered was the Higgs boson, which is shown in the graph below.

Finding the Higgs Boson “The strongest evidence for this new particle comes from analysis of events containing two photons. The smooth dotted line traces the measured background from known processes. The solid line traces a statistical fit to the signal plus background. The new particle appears as the excess around 126.5 GeV. The full analysis concludes that the probability of such a peak is three chances in a million. ” from ATLAS.

Citing this post: Urbano, L., 2015. New Particles, Retrieved April 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Note Cards for Equations

#### November 16, 2015

Part of physical science for the middle school is to start going beyond the conceptual, and making the connection between equations in science and algebra. So, we’ve started making note cards for the numerous laws we’ve encountered so far.

Citing this post: Urbano, L., 2015. Note Cards for Equations, Retrieved April 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Black Hole Consuming a Star

#### October 25, 2015

For the student who asked how do we know about black holes if we can’t see them. From NASA. Based on the paper: http://www.nasa.gov/mission_pages/chandra/destroyed-star-rains-onto-black-hole-winds-blow-it-back.html

Citing this post: Urbano, L., 2015. Black Hole Consuming a Star, Retrieved April 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Synthetic Muscle Fibers from Fishing Line

#### August 16, 2015

I need some students to try this at school. Muscle fibers that contract on heating sounds like a great way to open and close vents for air circulation (in the chicken coop to start with).

Citing this post: Urbano, L., 2015. Synthetic Muscle Fibers from Fishing Line, Retrieved April 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Relativity in a Canoe

#### September 29, 2014

The world moves around the canoe.

Perhaps not surprisingly, my middle school students have a difficult time wrapping their heads around the idea of multiple frames of reference. We were in a canoe on the Current River and I asked the student paddling in the rear of the boat to look at me and answer the question, “Are we in the canoe moving, or are we steady in one spot and everything around us moving?”

This resulted in some serious cognitive processing. And she still has not gotten back to me with an answer.

Another student, faced with the same question, thought it over overnight and concluded that it was a riddle. He figured the correct answer was that the canoe was moving and the land was still. I asked him to think about it a little more (because he was only half right).

Interestingly enough, I’ll be teaching my Advanced Physics class this block, and the first chapter has a neat little section on coordinate systems. I’m curious to see if the 11th and 12th graders have an easier time with the concept.

The canoe moves.

Citing this post: Urbano, L., 2014. Relativity in a Canoe, Retrieved April 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Arduino for Beginners

#### July 22, 2014

Arduino UNO connected to a breadboard from the starter kit.

I’ve been avoiding working with the Arduino microcontrollers because I’d prefer to be able to program in Python with the Raspberry Pi (for example). However, since the 3d printer we just built this summer uses an Arduino for a brain, I broke down and picked up the Arduino Starter Kit (via Adafruit).

The Arduino Projects Book is an excellent resource for the beginner.

What I liked most about the Starter Kit most is the Arduino Projects Book that comes with it. It’s a wonderful introduction to circuits, electronics, circuit diagrams, and microcontrollers at the beginners level. If I offer an Arduino elective, I’ll use it as a textbook. Indeed, I’ll probably use bits of it as a reference when I teach circuits in middle school and Advanced Physics.

As for the programming, the basics, at least, are pretty straightforward. I got a blinking LED controlled by a switch input up an running pretty quickly. The code requires two loops, one to set up the inputs and the output, and a loop for the program to follow. The code below has a blinking light that’s controlled via pin 4, but changes to a solid light when the switch is pressed (the input for the switch is pin 2). The wiring for the circuit is shown in the picture at the top of the page.

int switchOn = 0;

void setup(){
pinMode(2, INPUT);
pinMode(4, OUTPUT);
}

void loop(){

if (switchOn == HIGH) {
digitalWrite(4, HIGH);
} else {
digitalWrite(4, LOW);
delay(500);
digitalWrite(4, HIGH);
delay(200);
}

}

Citing this post: Urbano, L., 2014. Arduino for Beginners, Retrieved April 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Necks, Fretboards, and Scale Length

#### June 11, 2014

Pluck a string on a guitar and the sound you hear depends on how fast it vibrates. The frequency is how many times it vibrates back and forth in each second. An A4 note has a frequency of 440 vibrations per second (one vibration per second is one Hertz).

The vibration frequency of a guitar string depends on three things:

• the mass of the string
• the tension on the string (how tight it’s pulled)
• and, the length of the string.

Guitar string sets come with wires of different masses. The guitar has little knobs on the end for adjusting the tension. For building the guitar, you have the most control over the last last parameter, the length of the string, which is called the scale length. Since the guitar string masses are pretty much set, and the strings can only hold so much tension, there are limits to the scale length you can choose for your guitar.

In a guitar, the scale length only refers to the length of the string that’s actually vibrating when you pluck the string, so it’s the distance between the nut and the bridge. For many guitars this turns out to be about 24.75 inches.

For a guitar, the scale length is the length of the strings that are free to vibrate.

# Frets

To play different notes, you shorten the vibrating length of the string by using your finger to hold down the string somewhere along the neck of the instrument. The fret board (which is attached to the neck) has a set of marks to help locate the fingering for the different notes. How do you determine where the fret marks are located?

Well, the music of math post showed how the frequency of different notes are related by a common ratio (r). With:

$r = \sqrt[12]{2}$

So given the notes:

Note Number (n) Note
0 C
1 C#
2 D
3 D#
4 E
5 F
6 F#
7 G
8 G#
9 A
10 A#
11 B
12 C

Since the equation for the frequency of a note is:

$f_n = f_0 \; r^n$

we can find the length the string needs to be to play each note if we know the relationship between the frequency of the string (f) and the length of the string (l).

It turns out that the length is inversely proportional to the frequency.

$l = \frac{1}{f}$

So we can calculate the length of string for each note (ln) as a fraction of the scale length (Ls).

$l_n = \frac{1}{f_n}$

substituting for fn gives:

$l_n = \frac{1}{f_0 \; r^n}$

but since we know the length for f0 is the scale length (Ls) (that inverse relationship again):

$l_n = \frac{1}{\frac{1}{L_s} \; r^n}$

giving:

$l_n = \frac{L_s}{r^n}$

When we play the different notes on the guitar, we move our fingers along the neck to shorten the vibrating parts of the string, so the base of the string stays at the same place–at the bridge. So, to mark where we need to place our fingers for each note, we put in marks at the right distance from the bridge. These marks are called frets, and we’ll call the distance from the bridge to each mark the fret distance (D_n). So we reformulate our formula to subtract the length of the vibrating string from the scale length of the guitar:

$D_n = L_s - \frac{L_s}{r^n}$

Showing the fret distance.

The fret marks are cut into a fret board that was supplied by the guitarbuilding team, which we glued onto the necks of our guitars. We did, however, have to add our own fret wire.

Placing the fret wire into the fret cuts. The wire still needs to be fully pressed in.

The team also has an activity for students to use a formula (a different one that’s recursive) to calculate the fret distance, but the Excel spreadsheet fret-spacing.xls can be used for reference (though it’s a good exercise for students to make their own).

Citing this post: Urbano, L., 2014. Necks, Fretboards, and Scale Length, Retrieved April 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# The Math of Music

#### June 10, 2014

Mark French has an excellent YouTube channel on Mechanical Engineering, including the above video on Math and Music. The video describes the mathematical relationships between musical notes.

Given the sequence of notes: C, C#, D, D#, E, F, F#, G, G#, A, A#, B, C.

Let the frequency of the C note be f0, the frequency of C# be f1 etc.

The ratio of any two successive frequencies is constant (r). For example:

$\frac{f_1}{f_0} = r$

so:

$\frac{f_1}{f_0} = \frac{f_2}{f_1} = \frac{f_4}{f_3} = \frac{f_{12}}{f_{11}} = r$

We can find the ratio of the first and third notes by combining the first two ratios. First solve for f1 in the first equation:

$\frac{f_1}{f_0} = r$

solving for f1,

$f_1 = f_0 \; r$

now take the second ratio:

$\frac{f_2}{f_1} = r$

and substitute for f1,

$\frac{f_2}{f_0 \; r} = r$

which gives:

$\frac{f_2}{f_0} = r^2$

We can now generalize to get the formula:

$\frac{f_n}{f_0} = r^n$

or

$f_n = f_0 \; r^n$

where,

• n – is the number of the note

From this we can see that comparing the ratio of the first and last notes (f12/f0) is:

$\frac{f_{12}}{f_0} = r^{12}$

Now, as we’ve seen before, when we talked about octaves, the frequency of the same note in two different octaves is a factor of two times the lower octave note.

Click the waves to hear the different octaves. The wavelengths of the sounds are shown (in meters).

So, the frequency ratio between the first C (f0) and the second C (f12) is 2:

$\frac{f_{12}}{f_0} = 2$

therefore:

$\frac{f_{12}}{f_0} = 2 = r^{12}$

so we can now find r:

$r^{12} = 2$

$r = \sqrt[12]{2}$

Finally, we can now find the frequency of all the notes if we know that the international standard for the note A4 is 440 Hz.

Mark French has details on the math in his two books: Engineering the Guitar which is algebra based, and Technology of the Guitar, which is calculus based.

Citing this post: Urbano, L., 2014. The Math of Music, Retrieved April 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.