Let's start with a quick calculation. You might need your calculator. What is π squared?
Does that number look familiar? Does it look like the local gravitational field on the surface of the Earth, g? Well, no - it doesn't because it doesn't have any units. But the numerical value is similar to the accepted field value of:
Other than the unit issue - it isn't trivial to compare π2 and g. As you move around the Earth, the magnitude of g changes because:
- It changes with altitude.
- It changes with the latitude above the equator.
- It changes due to different densities of the local Earth.
In spite of this, a value of 9.81-ish N/kg is pretty reasonable. And yes, 9.81 N/kg has the exact same units as 9.81 m/s2. However, I like the units of N/kg because it shows the connection between field, mass and force. Please don't call it the 'acceleration due to gravity' - that just brings up a whole bunch of conceptual problems.
What if you use different units for g? In that case, it looks like it doesn't work. Older textbooks will list the gravitational field with a value of 32 ft/s^2. That clearly isn't pi squared.
Seconds Pendulum
Why does this g-π relationship exist? It has to do with the definition of the meter. Before that, let's look at the Seconds pendulum. This is a pendulum that takes exactly 1 second to go from one side of its motion to the other (or a 2 second period). You have probably seen such an example - like this.
Ok, that is a grandfather clock and not actually a seconds pendulum. If you measure the length of the swinging arm, it will be close to 1 meter long. It isn't a simple pendulum, so it doesn't have to be a meter long. A simple pendulum of length L has all of the mass concentrated in a tiny bob at the end of the length. This isn't true for the above pendulum.
Go ahead and try it. Get a small mass like a nut or metal ball. Metal works well since it's weight will likely be significantly larger than the air drag force so that you can ignore it. Now make the distance from the center of the mass to the pivot point 1 meter and let it oscillate with a small angle (maybe about 10°). If you like, you can make a video or just use a stopwatch. Either way, it should take about 1 second to go from one side to the other. Here is a quick example of a seconds pendulum I put together.
I'm not going to derive it, it isn't too difficult to show that for a pendulum with a small angle the period of oscillation is:
What if I want a period of 2 seconds?
That is the length of your seconds pendulum. Suppose we want to call this 1 meter? In that case, I have to have g = π2. That's why these values are related.
Definition of a Meter
The seconds pendulum was one of the ways to define the length of one meter. Of course, there are other ways to define this length. I'm not sure how good of an idea this was, but one definition of the meter was that 10 million meters would be the distance from the North pole to the Equator passing through Paris. It just doesn't seem like this would be easy to measure. But what do I know?
Well, why not use the seconds pendulum? It almost seems like a perfect way to define a standard. Anyone can make one with some very simple tools. However, it is not really reproducible. As you move around the Earth, the value of g changes (as I stated above).
Then how do you define a meter? For a time, the idea was to a particular bar of a certain length and at a certain temperature. Now we define the meter as the distance light travels in a vacuum in a certain amount of time.
But What About Pi?
Yes. This is a Pi Day post, so I should say something more about Pi. Why is Pi in the period expression for a pendulum? That's a great question. Is it because the pendulum moves in a path that follows a circle? No. The equation of motion for a oscillating mass on a spring (simple harmonic motion) has the same form as the small angle pendulum and it isn't moving in a circle. Then why? I guess the best answer is that solution to simple harmonic motion is a sine or cosine function. I don't know what else to say other than that gives us a solution. Since we have a sine function for the answer, the period would have to have a pi in it.
I feel like that is an insufficient answer - but it's the truth. It almost makes Pi magical. It just appears in places you wouldn't expect.
Before I leave you with some more Pi Day links, let me suggest one Pi Day activity based on this seconds pendulum. Get a meter stick. Use it to measure the local gravitational field (which would be the same as the vertical acceleration for a free falling object). Next measure the period of oscillation for a pendulum (well, I would change the length and make a function of period vs. length). From this period and the measured gravitational field, solve for pi. Actually, I think I might do this as homework.
- Can You Determine Pi from the Digits of Pi?
- How Do You Determine Pi Without a Circle?
- On the 8th Day, God Made Pi
- What Is the Best Fractional Representation of Pi?
That should keep you busy for a while.
