why is pi squared approximately equal to g

Let's go back to school for a moment and recall some math and physics lessons. Do you remember what the number π is? And what is the square of π? This is also a strange question. Of course, it is 9.87. And do you remember the value of the acceleration of gravity g? Of course, this number is so thoroughly hammered into our memory that it is impossible to forget it: 9.81 m/s². Naturally, it can change, but to solve elementary school problems, we usually used this value.

Mysterious equality

Now the next question: how did it happen that π² is approximately equal to g? You might say that such questions are not asked in polite society. First, they are not exactly equal. The difference is already in the second decimal place. Second, π is a dimensionless number, and g is a physical quantity with its own units of measurement.

And yet, no matter how you look at it, this cannot be a simple coincidence.

Not everything is as simple as it seems

First, let's take a closer look at the right-hand side. The value 9.81 is expressed in m/s². But these are far from the only units of measurement. If you express this value in any other units, the magic immediately disappears. So this is no coincidence – let's delve into meters and seconds.

What is a “meter,” and how does it relate to π? At first glance, not at all. According to Wikipedia, “a meter is the distance light travels in a vacuum during a time interval of 1/299,792,458 of a second.” Great, now we’re dealing with seconds — good! But there’s still nothing about π.

Wait a minute, why 1/299,792,458? Why not, say, 1/300? Where did this number even come from? It seems that to better understand this, we need to delve into the history of the unit of length itself.

A standard for every honest trader

In the past, people didn't really care about standards: they were only interested in what was convenient to measure. For example, why not measure length in human cubits? It might not be accurate, but it was cheap, reliable, and practical. And what about the fact that everyone's cubits were different lengths? Sometimes that was even useful. If you needed to buy more fabric, you'd call the tallest man in the village and ask him to measure the fabric in his cubits.

Later, of course, people started thinking about standardization. They started creating different standards. But this turned out to be inconvenient and cumbersome: it was impossible to constantly refer to a single standard for measurement. So copies of standards began to appear. And then copies of copies…

Serious people decided that such chaos was an obstacle to serious work, and set a goal: to come up with a definition of a unit of length that would not depend on any arbitrary standards. It should depend only on natural constants, so that any person with basic tools could reproduce and measure it.

Bright dreams of standardization and insidious gravity

A definition of a meter without standards was proposed back in the 17th century. The Dutch mechanic, physicist, mathematician, astronomer and inventor Christiaan Huygens proposed using a simple pendulum for this purpose. Take a small object and hang it on a thread. The length of the thread should be such that the pendulum completes a full swing (returns to its original position) in exactly two seconds. This length of thread was called the “universal measure” or “Catholic meter”. This length differed from the modern meter by about half a centimeter.

The proposal was well received and accepted. However, problems soon arose. First, Huygens was dealing with what he called a “mathematical pendulum.” This is “a material point suspended by a weightless, inextensible string.” A material point and a weightless string are hardly simple tools that every tradesman would have on hand.

Secondly, it quickly became clear that the length of the pendulum thread was not the same in different parts of the Earth. Gravity cleverly decreased as it approached the equator and did not contribute to humanity's bright dream of standardization.

Amazing equation

But let's return to our mysterious equation. To find the period of small oscillations of a mathematical pendulum depending on the length of the suspension, the following formula is used:

And here it is – our π! Let's substitute the parameters of the Huygens pendulum into this formula. The length of the thread l in the Huygens pendulum is equal to 1. T – oscillations is equal to 2. Substituting these values ​​into the formula, we get π²=g.

So, have we found the answer to our question? Well, not quite. We have already seen that this equality is only approximate. It is not quite correct to equate 9.87 and 9.81. Does this mean that the meter has changed since then?

With revolutionary greetings from France

Yes, it really did change! It happened during the reform of units of measurement initiated by the French Academy of Sciences in 1791. Smart people proposed to keep the definition of the meter by the pendulum, but with the clarification that it should be a French pendulum – at latitude 45° north (roughly between Bordeaux and Grenoble).

However, this did not satisfy the commission responsible for the reform. The problem was that the head of the commission, Jean-Charles de Borda, was an ardent supporter of the transition to a new (revolutionary) system of measuring angles – using grads (a grad is one hundredth of a right angle). Each grad was divided into 100 minutes, and each minute – into 100 seconds. The method of the second pendulum did not fit into this harmonious concept.

The true and final meter

In the end, they managed to get rid of the seconds and define the metre as one forty-millionth of the Paris meridian. Or, in other words, as one ten-millionth of the distance from the North Pole to the equator along the surface of the earth's ellipsoid at the longitude of Paris. This measurement was somewhat different from the “pendulum” metre. The commission, without false modesty, called the resulting value “the true and final metre”.

The idea of ​​a universal standard, accessible to everyone, waved goodbye and sank into oblivion. Need an accurate standard for a meter? No problem! Just measure the length of the meridian and divide it by several million. By the way, the French actually did this – they physically measured part of the Paris meridian, an arc from Dunkirk to Barcelona. They laid a chain of 115 triangles across all of France and part of Spain. Based on these measurements, they created a brass standard. By the way, they made a mistake – they did not take into account the flattening of the Earth from the polar side.

Conclusion

Let's return to our equation once again. Now we know where the inaccuracy comes from: π² and g differ by about 0.06. If it weren't for another attempt to reform and improve everything, we would now have a slightly different value of the meter and the elegant equation π² = g. Later, scientists nevertheless returned to defining the meter through unchanging and reproducible natural constants, but the standard of the meter was no longer the same.