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 Mass

12 1

6 2

4 3

3 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.