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
A scientist varies the mass, and measures the acceleration. Force is kept constant.
Acceleration Mass12 16 24 33 4
Acceleration
Mass
What shape will we seewhen we graph it?
Acceleration
Mass
The greater the Mass, the less the
acceleration
Mass
It is tough to know the equation of this function.
a = ?
Acceleration
Mass
So we linearize it.
Acceleration
We guess that Acceleration = k / Mass
We guess that acceleration = k / Mass
Acceleration Mass 1 ÷ M
12 1 1.00
6 2 ?4 ? 0.33
? 4 0.25
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
Acceleration
1 / Mass
What shape will we seewhen we graph it?
Acceleration
1 / Mass
Acceleration
1 / Mass
Acceleration
y = mx +b
1 / Mass
Acceleration
a = (slope)×(1/Mass) + b
1 / Mass
y = mx +b
Find the slope and the intercept.
Slope
Slope = Rise / RunSlope = Da / D(1/M)
Slope = ( 12 – 3 ) / (1 – 0.75)Slope = 9 / 0.75
Slope = 12
Intercept
Since the graph goes through the origin, the intercept is 0.
So, what is the equation?
a = ?
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.
Our last example…
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
Energy
Distance
What shape will we seewhen we graph it? Energy = ?
Energy
Distance
The greater the Distance, the greater
the acceleration.
Energy
Distance
Energy
Distance
It is tough to know the equation of this function.
E = ?
Energy
Distance
Let’s linearize it.
We guess that E = k × D2
We guess that E = k × D2
Energy Distance D2
2 1
8 2
18 3
32 4
We guess that E = k × D2
Energy Distance D2
2 1 1
8 2 4
18 3 9
32 4 16
What shape will we seewhen we graph it?
Energy
Distance
Energy
Distance
y = mx +b
Energy
Distance
a = (slope)×(1/Mass) + b
Energy
Distance
y = mx +b
Find the slope and the intercept.
Slope
Slope = Rise / RunSlope = DE / D(D2)
Slope = ( 32 – 2 ) / ( 16 – 1 )Slope = 30 / 15
Slope = 2
Intercept
Since the graph goes through the origin, the intercept is 0.
So, what is the equation?
E = ?
E = 2 D2
• The data indicate that the energy stored in a spring is proportional to the square of the compression distance.
