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. Klapholz Shaker Heights High School

A scientist varies the mass, and measures the acceleration. Force is kept constant. AccelerationMass

Acceleration Mass What shape will we see when 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 Mass1 ÷ M ? 4 ? 0.33 ? 40.25

We guess that acceleration = k / Mass Acceleration Mass1 ÷ M

Acceleration 1 / Mass What shape will we see when 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 / Run Slope =  a /  (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. EnergyDistance

Energy Distance What shape will we see when 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 × D 2

EnergyDistanceD2D

We guess that E = k × D 2 EnergyDistanceD2D

What shape will we see when 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 / Run Slope =  E /  D 2 ) 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 D 2 The data indicate that the energy stored in a spring is proportional to the square of the compression distance.