Appendix B


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Graphical Analysis

Figure 2

When the relationship is suspected to be of a quadratic form L = mT2 + b then plotting L versus T2 should yield a straight line (figure 2). How well the graph follows a linear fit is an indication of how well the data follows a quadratic relationship. General scatter in the plotted points or noticeable trends (such as a curve) in the graph indicated the certainty of the fit.

For a suspected form of L = KTn one first takes logarithms of both sides lnL = nlnT + lnK and then plots lnL versus lnT which should yield a straight line (figure 3). Once the slope n and the intercept lnK have been determined then one must double check the values obtained by plotting L versus Tn which should yield a straight line with slope K. This is because logarithm graphs are very inaccurate.

Similarly for a suspected form of L = KaT , lnL = Tlna + lnK. Comparing this with the straight line equation (y = mx + b) on sees that a plot of lnL versus T should be linear. Again because of logarithms one must check this by plotting L versus aT. A special case of this equation (which does occur often in physics) is L = Kect where e = 2.71828... (the base of natural logarithms). This then on taking logarithms reduces to lnL = ct + lnK.

Note that natural logarithms are generally used in physics since in many cases (see, for instance, error analysis of logs) they simplify the math involved. Common logarithms will also work as well in most cases.

Figure 3

Figure 4