introduction to linearization ( no units, no uncertainties, just the core idea )
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Mr. Klapholz Shaker Heights High School. Introduction to Linearization ( No units, no uncertainties, just the core idea ). The purpose of linearization is to get the equation that describes real data. A scientist varies the mass, and measures the acceleration. Force is kept constant. - PowerPoint PPT PresentationTRANSCRIPT
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Introduction to Linearization(No units, no uncertainties, just the core idea)
The purpose of linearization is to get the equation that describes real data.
Mr. KlapholzShaker Heights
High School
![Page 2: Introduction to Linearization ( No units, no uncertainties, just the core idea )](https://reader036.vdocuments.net/reader036/viewer/2022081520/5681628c550346895dd2f8fb/html5/thumbnails/2.jpg)
A scientist varies the mass, and measures the acceleration. Force is kept constant.
Acceleration Mass12 16 24 33 4
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Acceleration
Mass
What shape will we seewhen we graph it?
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Acceleration
Mass
The greater the Mass, the less the
acceleration
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Mass
It is tough to know the equation of this function.
a = ?
Acceleration
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Mass
So we linearize it.
Acceleration
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We guess that Acceleration = k / Mass
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We guess that acceleration = k / Mass
Acceleration Mass 1 ÷ M
12 1 1.00
6 2 ?4 ? 0.33
? 4 0.25
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We guess that acceleration = k / Mass
Acceleration Mass 1 ÷ M
12 1 1.00
6 2 0.50
4 3 0.33
3 4 0.25
![Page 10: Introduction to Linearization ( No units, no uncertainties, just the core idea )](https://reader036.vdocuments.net/reader036/viewer/2022081520/5681628c550346895dd2f8fb/html5/thumbnails/10.jpg)
Acceleration
1 / Mass
What shape will we seewhen we graph it?
![Page 11: Introduction to Linearization ( No units, no uncertainties, just the core idea )](https://reader036.vdocuments.net/reader036/viewer/2022081520/5681628c550346895dd2f8fb/html5/thumbnails/11.jpg)
Acceleration
1 / Mass
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Acceleration
1 / Mass
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Acceleration
y = mx +b
1 / Mass
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Acceleration
a = (slope)×(1/Mass) + b
1 / Mass
y = mx +b
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Find the slope and the intercept.
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Slope
Slope = Rise / RunSlope = Da / D(1/M)
Slope = ( 12 – 3 ) / (1 – 0.75)Slope = 9 / 0.75
Slope = 12
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Intercept
Since the graph goes through the origin, the intercept is 0.
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So, what is the equation?
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a = ?
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a = 12 (1/M)• Notice that we were able to write down the
conclusion to the lab only because we had linearized the data.
• The function could be said to be “linear in 1/M”.
• But what we really wanted was the function, and we have it: a = 12 / M.
• FYI: Newton’s second law says, in part, that acceleration = Force / mass.
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Our last example…
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A researcher changes the distance that a spring is compressed, and measures the
energy in the spring.Energy Distance
2 1
8 2
18 3
32 4
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Energy
Distance
What shape will we seewhen we graph it? Energy = ?
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Energy
Distance
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The greater the Distance, the greater
the acceleration.
Energy
Distance
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Energy
Distance
It is tough to know the equation of this function.
E = ?
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Energy
Distance
Let’s linearize it.
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We guess that E = k × D2
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We guess that E = k × D2
Energy Distance D2
2 1
8 2
18 3
32 4
![Page 30: Introduction to Linearization ( No units, no uncertainties, just the core idea )](https://reader036.vdocuments.net/reader036/viewer/2022081520/5681628c550346895dd2f8fb/html5/thumbnails/30.jpg)
We guess that E = k × D2
Energy Distance D2
2 1 1
8 2 4
18 3 9
32 4 16
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What shape will we seewhen we graph it?
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Energy
Distance
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Energy
Distance
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y = mx +b
Energy
Distance
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a = (slope)×(1/Mass) + b
Energy
Distance
y = mx +b
![Page 36: Introduction to Linearization ( No units, no uncertainties, just the core idea )](https://reader036.vdocuments.net/reader036/viewer/2022081520/5681628c550346895dd2f8fb/html5/thumbnails/36.jpg)
Find the slope and the intercept.
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Slope
Slope = Rise / RunSlope = DE / D(D2)
Slope = ( 32 – 2 ) / ( 16 – 1 )Slope = 30 / 15
Slope = 2
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Intercept
Since the graph goes through the origin, the intercept is 0.
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So, what is the equation?
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E = ?
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E = 2 D2
• The data indicate that the energy stored in a spring is proportional to the square of the compression distance.