Propagating Uncertainties through the Logarithms

In this section, we will be converting our (L, T) data points with uncertainties to (ln(L), ln(T/2π)) data points with uncertainties. We do this the only way we know how to:  by Monte Carlo simulations.  Our “Lengths” sheet has our (L, T) data points with uncertainties. We add two new sheets to our workbook, named ln(L) MC and ln(T) MC respectively. ln(L) MC will have all our converted lengths, and ln(T) MC will have all our converted periods (divided by 2π).

At this point, you know how to do Monte Carlo simulations as you’ve had to do at least one for every lab so far. Thus this section will just be review.

View the picture below, where the first L is converted to the first ln(L). We see that the only “numbers” here are the first length, and the standard deviation on the first length. Additionally, these values are referenced, not copy-pasted. For Length# 2, copy columns A and B, and paste them on columns D and E. All you need to update is the Length#, L [m], and sigma L [m]. After updating these values ( L [m] and sigma L [m] by reference), ln(L) and sigma ln(L) for Length #2 pop right out. Do this for all Length #.

Then you can set up your ln(T) MC sheet almost exactly the same, so it makes sense to duplicate sheet ln(L) MC, then all you need to do is switch all “L”s to “T”s, “[m]”s to “[s]”s, divide by 2π, and update the references so that they reference the periods not the lengths.