Vernier Logger Pro Modeling, Fitting, and Linearization - Manual (Page 2)

Logger Pro Modeling, Fitting, and Linearization
©Vernier Software & Technology 2
people in an amusement park ride. After thinking about the physical model, they call up the Model feature
in the Logger Pro Analyze menu and choose the most appropriate general equation. This should be
y = At 2
 Bt  C
(3)
They use the initial position y
0
to set C and the estimated initial velocity v
y0
to set B. The theoretical
acceleration a
y
= –9.8 m/s 2 that can be calculated from Newton’s gravitational constant and the radius of
the earth is used to set the coefficient A = 0.5 a
y
. They compare a graph of height vs. time from the model
to an overlay graph of height vs. time from the experimental data. The agreement is good, and by making
small changes in the estimates of y
0
and v
y0
they can improve it.
A quick, easy substitute method of determining the coefficients is Curve Fitting. With the advent of
powerful computers in the classroom, curve fitting has become a widely used technique even though it
has strong pedagogical disadvantages. In curve fitting, a standard least-squares algorithm is used to find
the coefficients automatically. The main disadvantage of this method is that students do not have to think
about the physical significance of the coefficients, so the method does not help them learn what the
coefficients mean. Also, they can be deceived about the quality of the model. This is illustrated in the
example of the next section.
Another substitute method is linearization. This method was very useful before computers were
ubiquitous, or even today whenever people plot graphs on paper. The idea is to find a way to plot
functions of the variables so the data points lie along a straight line, then determine the line’s slope and
offset by curve fitting. For example, the impedance of an RC circuit is
Z
 R
2
1 
2 C 2 (4)
where Z and

are the variables. The data can be linearized by plotting Z 2
versus 1

2 , since
Z 2
 1
C 2
1

2
 R
2
 m 1

2
 b .
(5)
where m and b are the slope and offset. The coefficients C and R are determined from m and b. The use of
log paper or log-log paper is another example of linearization.
However, not all interesting functions can be linearized. If B is not zero, Equation 3 cannot be put in the
form
g(y)
 mf (t)  b
(6)
where g(y) and f(t) are functions that do not depend on the coefficients. Thus linearization cannot be used
for free-fall motion unless v
y0
= 0. It also cannot be used for trigonometric functions.
The three methods mentioned above, analytic mathematical modeling, curve fitting and linearization, are
methods of comparing a physical model to experimental data by determining the coefficients in an
analytic mathematical function. Another way of comparing a physical model to data is to skip the step of
finding an analytical mathematical function (especially if it is not possible to find one). Instead, the
differential equation of the model is solved numerically and its numerical solution is compared directly to
the data. This method is called dynamical modeling. It can be carried out by transferring experimental
data from Logger Pro into a spreadsheet or other software that can be set up for numerical integration and
graphing.