Foucault's pendulum

I visited the Panthéon yesterday, and saw the original Foucault's pendulum. Well, sort of. It's the same location, but it got taken down when the Panthéon became a church again for a while, and was then restored, and then the wire broke and they had to replace it. And at some point the original 28kg bob was replaced with one almost twice as heavy. So it's a bit like the story about the axe that's had three new handles and two new heads...

I'd recently seen this fun YouTube video of renowned physics lecturer Walter Lewin's farewell lecture, in which among other things he swings from a pendulum to confirm that changing the mass doesn't change the period. So I thought I'd try timing the Panthéon pendulum's swings and see if I could get a good value for the length of the wire...

I thought I'd time 10 swings, but this took longer than I'd imagined, and I got distracted and nearly lost count. But in the end I measured approximately 2 minutes and 44 seconds. Guessing that my reaction time is probably good to about plus or minus half a second, that gives \(\newcommand{\metre}{\text{m}} \newcommand{\second}{\text{s}}10T\approx 164\pm 0.5\thinspace\second\) and so \(T \approx 16.4 \pm 0.05 \thinspace\second\). The formula for the period \(T\) is

\[ T = 2 \pi \sqrt{\frac L g} \]

Inverting that gives

\[ L = g \left( \frac{T}{2\pi}\right)^2 \]

According to Wikipedia, the gravitational acceleration in Paris is \(g \approx 9.809 \thinspace\metre\second^{-2}\). If that last significant figure is indeed significant, then the relative error in this value is a couple of orders of magnitude less than the error in my time measurement and can safely be ignored. So

\[ L \approx \left( 9.809 \thinspace\metre\second^{-2} \right) \times \left( \frac{16.4 \pm 0.05 \thinspace\second}{2\pi} \right)^2 \approx 66.8 \pm 0.4 \thinspace\metre \]

The information panel in the Panthéon states that the length is 67 metres, so we're within the error bars! Not too bad given that the formula is based on the small-angle approximation and doesn't take any sort of friction or drag into account.

There's a mirror on the floor, it looks amazing.