23)+Simple+Harmonic+Motion+based+on+Given+Data

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__**Simple Harmonic Motion Based on a Set of Given Data**__ by, Ethan Ossolinski
 * Reflection:** Personally this section was easy for me to imagine in my head. I'm able to picture what is going on in the problem without having to actually draw it. Because of this I was able to determine very quickly what each part of the equation is in relation to the problem. To me this type of problem is very intuitive and logical as I can compare it to the real world, and that is how I function the best.

There are many examples of Simple Harmonic motion in nature. Some include: Average monthly temperature, a pendulum, and even the electrons in a wire carrying alternating current.

__**What is Simple Harmonic Motion:**__ Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement.

In other words for every amount that the graph goes up it goes equally down, which forms a type of sinusoidal graph.

__ **Cosine** __ : Y = a cos (b (t - c)+v
 * __Sine__ ** : Y = a sin (b (t - c))+v

__**Its Parts**__

__Amplitude (a):__ Half the height of the entire graph. It is obtained by finding the average range between the highest and lowest y-values and then dividing the average by two.

__Frequency (k):__ This is 1/period

__Period:__ This is the average distance between two consecutive relative maximums or two consecutive relative minimums. It is found by finding the average distance between two high points in a set of data. So in other words, when talking about temperature and you take for granted that July is the hottest month you find the difference in the x values from two consecutive years. If you are using years it will be twelve months of the amount of time it takes to get from one July to the next.

__b-Value:__ To find the b-value divide 360 or 2pi by the period.

__Phase Shift (c):__ This is the horizontal shift of the graph. When talking in terms of months it will be shifted to the right one (-1) to take into account for the fact that the first month isn't month 0 it's month one.

__Vertical Shift (b):__ This is the height that the graph is shifted up or down, or the amplitude plus the distance between a relative minimum and the x-axis.

__**Example Problems:**__ 1.)
 * || Avg.Lows || Avg.Highs ||
 * January || 27 || 38 ||
 * February || 29 || 42 ||
 * March || 35 || 50 ||
 * April || 45 || 61 ||
 * May || 54 || 71 ||
 * June || 64 || 79 ||
 * July || 69 || 84 ||
 * August || 68 || 83 ||
 * September || 61 || 75 ||
 * October || 50 || 64 ||
 * November || 42 || 54 ||
 * December || 32 || 43 ||

Based on the following data given which represents the average low and high temperatures in New York City give the equation for a sinusoidal graph that will plot the average temperatures year round (find the average of the averages).

__**Period:**__ We know that every year in a perfect world that the temperature will repeat again so we will use **12** as the period

__**b-value:**__ We also know that the b-value is 360 divided by the period so the b-value will be **360/12 or 30**

__**Amplitude:**__ This is simple enough. First find the average for the month of January (the coldest) and July (the warmest) 27+38=68 68/2=32.5 then July 69+84=153 153/2=76.5 Then find the difference between the two averages: 76.5-32.5=44 There is on average a 44 degree difference in temperature. Therefore there is on average a difference of 44 degrees between the coldest part of the year and the warmest. To find the amplitude simply divide 44 by 2 to get **22,** this is the amplitude.

__**Vertical Shift:**__ Both of the shifts require a little bit of common sense and thinking. First of all we must pick whether we will use sine or cosine as our parent function. In this instance we will use cosine. Vertically we must move it up the amplitude plus the lowest point. So simply add 22 to 32.5 which we found to be the average temperature in January. **+54.5** is the vertical shift.

__**Horizontal Shift:**__ The horizontal shift is a little bit tricky. We want the first peak of the cosine wave to be at July on the x-axis. July is the hottest month and is the seventh month of the year. So we will shift it to the **right 7**. Note that there is no 0 month of the year so be careful when you shift the equation horizontally.

**y=22cos(30(x-7))+54.5**
Suppose you and a friend decide to do something completely unnessecary. You ride a bike in an oval around a section of bushes in your yard starting at the Northern end. Your friend is a mathematical genius and decides to record how far you are away from the center of the bushes as you cross by the North, East, South, and West side of the bushes. He records the data in a table. Write the equation that will model your path around the bushes assuming that you make a full revolution every 15 seconds. Also assume that you ride the same distance away from the bushes at both the North and the South side and the same distance at the East and the West side. Also take into account that at the Easy and West side of the bushes there are branches that stick out quite a ways that you must swerve to avoid.
 * 2.)**

21.67-12.87=8.8 feet which is the average difference between how far you swerved out of the way compared to the side you didn't have to. The amplitude will be half of this or **4.4**.
 * || 1st || 2nd || 3rd ||
 * North/South || 12.5 || 13.2 || 12.9 ||
 * East/West || 21.3 || 22.0 || 21.7 ||
 * Amplitude:** The amplitude will be the difference between the average shortest distance you get away from the bushes and the average longest distance you get away from the bushes. 12.5+13.2+12.9=38.6 38.6/3=12.87 21.3+22.0+21.7=65 65/3=21.67
 * Amplitude:** The amplitude will be the difference between the average shortest distance you get away from the bushes and the average longest distance you get away from the bushes. 12.5+13.2+12.9=38.6 38.6/3=12.87 21.3+22.0+21.7=65 65/3=21.67
 * Amplitude:** The amplitude will be the difference between the average shortest distance you get away from the bushes and the average longest distance you get away from the bushes. 12.5+13.2+12.9=38.6 38.6/3=12.87 21.3+22.0+21.7=65 65/3=21.67


 * Period**: equals **7.5 seconds** because 15 second measures how long it takes to get from North to North. However, in actuality because what you are really modeling is from North to East to South to West 7.5 is the right answer because you are the same distance away from the center when you are at North compared to when you are at the Southern end of the bushes.


 * b-Value:** The b-value is simply **360/7.5**


 * Vertical Shift**: The Vertical Shift will be the amplitude plus the closest that you get to the bush or 4.4+12.87=**17.27**


 * Horizontal Shift:** Assuming that we use cosine again the horizontal shift will need to move the graph so that a relative minimum lies on the y-axis since you starts at North which is as close as you get. Because the period is 7.5 you need to chop this in half or the time it takes you to get from North to the nearest maximum or East. This is simply to the right 7.5/2=**3.75**

y=4.4cos(360/7.5(x-3.75))+17.27
=__Youtube Video Link:__= []

=__**Portfolio Problem:**__= Write the appropriate simple harmonic motion equation for the monthly average temperatures using the average seasonal temperatures from 2009-2011 in Central Park, New York, New York.  2009
 * Month || Temperature (Fahrenheit) ||
 * 1 || 27.9 ||
 * 2 || 36.7 ||
 * 3 || 42.4 ||
 * 4 || 54.5 ||
 * 5 || 62.5 ||
 * 6 || 67.5 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">7 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">72.7 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">8 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">75.7 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">9 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">66.3 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">10 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">55.0 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">11 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">51.2 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">12 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">35.9 ||

<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">2010
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">Month || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">Temperature (Fahrenheit) ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">13 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">32.5 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">14 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">33.1 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">15 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">48.2 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">16 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">57.9 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">17 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">65.3 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">18 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">74.7 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">19 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">81.3 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">20 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">77.4 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">21 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">71.1 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">22 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">58.1 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">23 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">47.9 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">24 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">32.8 ||

<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">2011
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">Month || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">Temperature (Fahrenheit) ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">25 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">29.7 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">26 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">36.0 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">27 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">42.3 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">28 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">54.3 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">29 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">64.5 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">30 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">72.3 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">31 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">80.2 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">32 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">75.3 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">33 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">70.0 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">34 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">57.1 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">35 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">51.9 ||
 * <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">36 || <span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-decoration: none; vertical-align: baseline;">43.3 ||