### g = \pi^2

16Sep07

Yesterday I stumbled upon a fantastic blog by Isabel, a fellow grad student blogger at God Place Dice. She has some very well thought-out and fascinating posts (certainly more so than my blog) and GPD has quickly become one of my  favourite blogs.

I wanted to share a particularly excellent post about why the gravitational acceleration at the surface of the earth is approximately $g \approx \pi^2$ in SI units. No, it’s not a coincidence; the meter was originally defined to be the length ($L$) of a pendulum with a two second period ($\tau$). And, using the small angle approximation, we know from high school that:

$\tau \approx 2\pi \sqrt{\frac{L}{g}}$

Solving this, we get $g = \pi^2$ meters per (second)2. This is a very cute result with a nice touch of history, see the original post.

Anyway, thanks to Isabel for enlightening my day. 🙂

#### 7 Responses to “g = \pi^2”

1. Thanks for the compliment!

It’s a nice counterpoint to the large majority of the people who commented when this was posted at reddit (which is where I’m guessing you stumbled in from, because that’s where something like thirteen thousand people stumbled in from) who claimed that I was some sort of crackpot because “of course g would be different if we used different units”. (I suspect many of them didn’t actually read the post.)

2. Isabel — the random clueless comments were half of the fun. 🙂

3. hmmm, do you particle physicists take g=1 as well then …

🙂

Evan

4. My undergraduate adviser once said that pi is almost dimensionful. And you’re familiar with a Cambridge cosmologist who set pi = 1, concluding that pi^100 is also approximately 1. 🙂

5. Looks like cosmologists really can get away with everything… 😉

6. Indeed pi is dimensionful; it has units of diameter 😀 After all, it stands for “perimeter”.
But, thinking in this way, all trigonometric functions are dimensionful, in units of radius. Even the tangent, and here the paradox: on one side is sin/cos, so units should cancel. On other hand, it is the length of the tangent line, in the triangle composed by radius, secant and tangent lines, when the radius is the unit (hint: solve the paradox by recovering R, instead of the unit, in all the trigonometric formulae).

I am a bit puzzled and surprised by Isabel’s history, and the original research involved. I have always though that the goal in the meter was to be near of the yard.

7. \pi^{100}\approx1 -> that is so funny! it will never get old! he’s like that magic mathematician on the simpsons who made remainders disappear!

🙂

Evan