Vernier Logger Pro Modeling, Fitting, and Linearization - Manual

©Vernier Software & Technology 1
Logger Pro Modeling,
Fitting, and Linearization
Excerpt From: Physics with Video Analysis (Appendix C)
Authors: Priscilla Laws, Robert Teese, Maxine Willis and Patrick Cooney
When physicists compare theory with experiment, they usually consider a physical model of the situation.
The Bohr model or quark model may be the first that come to mind, but in fact nearly every application of
physics in introductory physics courses involves a simple model that is
analogous to the real-world phenomenon. A moving object, for
example, regardless of whether it is an atom, an elephant or a
solar system, is often represented in the first approximation as a
point mass with various forces applied to it. A real spring has
mass, it can move in three dimensions and it is not ideal, but we
model it as a straight-line, massless object that obeys Hooke’s
Law. A real capacitor may have internal resistance, frequency
dependence and polarity, but we treat it as a pure capacitance. A
cow rolling down a hill might be modeled as a sphere on an
inclined plane as shown in Figure 1.
Figure 1: A spherical cow
Before we can compare a model to data, we have to turn the physical model into a mathematical model by
applying the laws of physics. Newton’s Second Law applied to a falling object, modeled as a point mass
in a uniform gravitational field, leads to a simple differential equation. Its solution is the analytic
mathematical function
y
 1
2
a
y
(t
 t
0
) 2  v
y 0
(t
 t
0
)
 y
0
(1)
where a
y
is the acceleration of gravity, v
y0
is the initial velocity, y
0
is the initial height and t
0
is the initial
time. Applying Kirchoff’s Laws to an LC circuit, modeled as a pure capacitance connected across a pure
inductance, leads to a differential equation whose solution is the analytic mathematical function
i(t)
 i
0
cos(

(t
 t
0
)
 
) (2)
where i
0
is the current at t = t
0
,

is the phase angle and
  1 LC
is the angular frequency.
An analytic equation contains variables and coefficients. In Equation 1, for example, the measured
quantities, position and time (y and t), are variables whose functional form in the equation determines the
basic shape of its graph. The numbers a
y
, v
y0
, y
0
and t
0
are the coefficients. They determine the size,
orientation and position of the graph. Once the coefficients are determined, the model may be compared
to the experimental data. This can be done visually by plotting the equation and the data on the same
graph. In Logger Pro the Root Mean Square Error (RSME), a measure of the goodness of fit, can also be
calculated.
There are several methods for determining the coefficients. The method that we consider the most
versatile and educationally useful is to exploit the physical significance of each coefficient and determine
its value from the data or other information about the situation. This method is called Analytic
Mathematical Modeling. The Demon Drop activity has a good example of how it is carried out. In this
activity, students use the Logger Pro video analysis tools to find height vs. time data for a falling cage of