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This exercise will help you learn how to use Kaleidagraph or Origin to plot and analyze data.

In 1798 Henry Cavendish conducted an experiment to "weigh the earth." He
did
this by measuring the shift in the equilibrium position of a small
dumbbell
suspended from a torsion fiber caused by two cannonballs. The balls were
placed near the dumbbell so as to make it twist. From the spring constant
of the fiber, the geometry and masses of the dumbbell and cannonballs,
and the magnitude of the angular shift the cannonballs cause, it is
possible to measure the value of *G*, the universal constant of
gravitation. Combined with knowledge of the radius of the earth and the
local value of (little) * g*, the local acceleration due to
gravity,
one can deduce the mass of the earth.

Data have been collected for the angular position of the dumbbell as a function of time, and are available by clicking here. You may retype them or select them in your browser for pasting into the data analysis program.

The motion is described by a damped sinusoid. Its functional form is

where the quantities in the equation are defined by

a |
(angular) amplitude of oscillation |

T |
period of oscillation |

φ | initial phase of the motion |

τ | exponential decay time of the motion |

θ_{o} |
shift in the equilibrium position of the dumbbell |

There are six steps to this exercise. Each task has a help link for Kaleidagraph and Origin, which you may obtain by clicking the link.

- Load the data (
KG Help,
Origin 3.5 Help,
Origin 4.0 Help
).
- Plot the data (
KG Help,
Origin 3.5 Help,
Origin 4.0 Help
).
- Add error bars (
KG Help,
Origin 3.5 Help,
Origin 4.0 Help
)
to the plot.
Do the data look like they could represent the motion of a twisting pendulum slowed down by air resistance? Save the plot.

- Define a fitting function. (
KG Help, Origin 3.5 Help,
Origin 4.0 Help
)
that calculates a damped sinusoid.
In Kaleidagraph, be careful to check whether you wish the trigonmetric
function to be evaluted in radians or degrees. Also be sure to check the
**Weight Data**box in the fit function dialog box.Fit your data.

**Be very careful to choose good guesses for your parameters.**If you don't, you will very likely get an error like**Singular Matrix Error**or some such*"useful"*diagnostic information. In Origin, you can press the Update Plot button to see how you are doing. In Kaleidagraph, there is no such facility. - When you have finished the fitting procedure, you should have a graph
showing the data with error bars, the fitted curve, and some fit
information
(
KG Help,
Origin 3.5 Help,
Origin 4.0 Help
).
When you perform a proper χ
^{2}fit, the error associated with each fit parameter gives you a sensible estimate of the accuracy of the parameter, as determined by the fitting operation. That's pretty nice. Have a look at your graph and see how well you could eyeball the angular offset. It's tough!If you're curious for more details about the fitting operation, click here.

To make sure you've done the fit properly, fill out and submit the form below.

- Lastly, print out a copy of the graph ( KG Help, Origin 3.5 Help, Origin 4.0 Help ). Make the plot area about 4 inches wide by 3 inches high. This is a nice size for taping into a lab notebook.

The function χ^{2} is defined by

which looks more frightening than it is. This function adds up the
differences between each data point and the fitted curve, symbolized here
by *f*. In this equation, the *x _{i}* are the values of
the independent variable. In this example, the independent variable is
time. The

There are two subtleties. First, the differences are squared, to make them all add up positively. If they weren't, the average would be zero! Second, each difference is divided by the uncertainty of the data point. This means that you get penalized a lot when the curve is far from a "good" data point (that has small uncertainty), and not much at all when the curve is far from a "lousy" data point (that has a large uncertainty).

Since χ^{2} is the sum of nonnegative
quantities, the smallest value it can have is zero. This happens when the
curve goes through each and every data point. You might think that this is
the best you can do, but you'd be wrong. *It means either that you
cheat or that you're really sloppy!* (Newton cheated with his data and
got away with it, because nobody was very sophisticated about this sort of
thing when Newton was working out the foundations of mechanics.) Real data
have random noise at some level, and this means that they will virtually
never lie right on the curve. The uncertainty of a data point is an
estimate of the likely discrepancy between the point and its true value.
Roughly speaking, a typical data point should be within about one
uncertainty ("one error bar") from its true value. This means that each
data point should contribute a value of about one to the sum. So the sum
should be about *N*.

It is often more convenient to compute the "per point" value of χ^{2}.
This is called the reduced χ^{2}
or χ^{2} per degree of freedom. It turns out that the best procedure is
to divide χ^{2} not by the number of
data points, *N*, but by *N - m*, where *m* is the number
of fitting parameters. The number you get should be in the ballpark of
unity for a good fit. If it's much smaller than you probably overestimated
your errors. If it's much bigger, you may have underestimated your errors
or your model may not describe the data very well.

Origin reports a value of `chisq`. Don't be fooled. **It is the
reduced χ ^{2}.**

Kaleidagraph reports a value of `chi^2`. It is χ^{2}. To get the
reduced χ^{2}, divide by the number of
degrees of freedom (*N - m*).

Copyright © 2001 Harvey Mudd College Physics Department http://www.physics.hmc.edu/ WebMaster@Physics.hmc.edu This page was last modified on Fri, Jun 29, 2001. |