Why Is There a Connection Between Pi, the Gravitational Field, and the Meter?

Photo: Rhett Allain. A mass on a string.

Here’s a fun one to try. Take out your calculator. Now use it to take the square root of 9.8 (remember that g = 9.8 Newtons per kilogram). What do you get? SPOILER ALERT — it’s 3.13 something. Yes, that’s pretty close to the value of pi. Is it a coincidence? NOPE.

But why? Why is the square root of g approximately pi? Let’s start with g. Although it’s often called “the acceleration due to gravity”, it’s actually the magnitude of the local gravitational field. It has a value of 9.8 Newtons per kilogram, but this is the same as 9.8 meters per second². The important thing in this unit is the meter. Yes, it’s the meter’s fault.

The Meter and the Seconds Pendulum

Humans need to measure distance — how else could you build stuff? In the very early days, different groups of humans had different standards for measuring length. That’s how we get silly units like “feet” and “inches” — which are sadly still used today.

However, in 1660 the Royal Society wanted to define the length of a meter using a seconds pendulum. The idea is to make a pendulum (a mass on the end of a swinging string) such that it swings from one side to the other in 1 second or back and forth in 2 seconds (so it has a period of oscillation of 2 seconds)…