(Pi)^2 = Gravity on earth?

1. Nov 25, 2011

bmcgann

So while i was doing some homework i noticed that when you square pie you end up with 9.869 which is very close to the gravity on earth.

Just wondering if there is any relationship between the two, i don't see how there can be but it is a bit strange, is it coincidence?

2. Nov 25, 2011

DaveC426913

Coincidence. Get enough numbers, perform enough arbitrary operations on them, with a large enough margin of error, and it would be a miracle if you didn't find coinciding numbers.

In case there's any doubt, note that the number you use to define gravity on Earth is based on metres and seconds. In feet, g is closer to 30. In furlongs per fortnight^2 you'll get a different number again. Pi is unitless (it works in feet and metres and furlongs).

Last edited: Nov 25, 2011
3. Nov 25, 2011

clearwater304

Gravity is calculated by g=(gravitational constant* mass)/radius^2.

If you were correct, the gravitational constant would be equal to (pi^2*(radius)^2)/(mass)

The actuall equation for calculating the gravitational constant is (pi^2*(r)^2*d*s)/(M*T*L)
(page 6 from link provided at bottom)

G is the gravitational constant;
r is the distance between the center of the small sphere and its nearest large sphere;
d = 0.05 m, is the half the distance between the centers of the small spheres;
M is the mass of the large spheres;
T is the undamped period of oscillation;
L is the perpendicular distance from the mirror to the wall; and
S is the distance on the wall between positions of the spots of light associated with the equilibrium positions of the small spheres-rod-mirror assembly if the large spheres are fully clockwise to the equilibrium position, and if the large spheres are fully counterclockwise position.

4. Nov 25, 2011

You've got that backwards. It is 32.2 ft/s^2 or 9.81 m/s^2

5. Nov 25, 2011

DaveC426913

Good catch. S'what I get for doing too much editing and not enough proofing. Fixed.

6. Nov 25, 2011

clearwater304

7. Nov 25, 2011

Probably also related to the time of the post. I find I make a lot more errors of logic and typos when I post closer to when I wake up in the morning. If I don't wait at least an hour to really wake up, there is no telling what I might post.

8. Nov 25, 2011

D H

Staff Emeritus
Not nearly as coincidental as you think. It is coincidental that g defined in terms of time as 1/86400 of a day and in terms of the predecessor of the meter, the length of the king's arm / length of the king's stride (which varied widely from place to place), happens to be close to $\pi^2$.

This happy coincidence led many to propose a definition of the meter that would have made g have a numeric value that was exactly equal to π2 at 45o latitude. This was, for a time, the leading contender for the definition of the meter. It was the definition favored by most French scientists. (At least amongst those who had not lost their heads to the guillotine. Those who had lost their heads no longer had a say in the matter.)

However, the French revolutionaries also happened to be the very first aficionados of political correctness. Defining the meter to be the length of a seconds pendulum at 45o latitude was just too placist. So they settled instead on an alternate definition, one ten millionth of the distance from the equator to a pole measured along a meridian, and commissioned a scientific survey to measure this length precisely. In the interim, they built a prototype meter bar based on a preliminary estimate of this length. After the expedition showed this prototype meter bar was off by a bit, the committee chose to make the prototype meter bar the standard for length. It took quite some time to find something that was consistent with this prototype but that could be defined in terms of observable physics.

So yes, you are correct that this is a happy circumstance, but only thanks to human stupidity.

9. Nov 25, 2011

Staff Emeritus
A blog is not an acceptable reference, but in any event, the length of the seconds pendulum was not adopted for the definition of the meter. Which takes all the wind out of the sails of that argument.

10. Nov 25, 2011

D H

Staff Emeritus
I didn't look past DaveC's post. Honest!

There was an immense amount of play in the standards of length that preceded the meter. By sheer dumb luck, those definitions coupled with the already well-defined value for the second (well-defined by the standards of the late 1700s) happened to more or less be such that g had a numeric value of approximately π2. Another happy circumstance is that the length of a human arm is approximately one ten millionth of the distance from the equator to the north pole as measured along a meridian at sea level.

Two coincidences just happened to nearly coincide, and this was just coincidence. Of course the PC (and they were very PC) revolutionary council picked the dumbest but most politically correct of the three alternatives for defining the meter. A lot of French scientists who still had heads on their shoulders were quite POed at the choice. It made Poincare resign from the council.

11. Nov 25, 2011

clearwater304

According to the wikipage, the first timeline definition of a meter was:

1790 May 8 – The French National Assembly decides that the length of the new metre would be equal to the length of a pendulum with a half-period of one second.

http://en.wikipedia.org/wiki/Metre

12. Nov 25, 2011

Staff Emeritus
Yes, and if you keep going and read the rest of the article - or for that matter, the rest of the paragraph - you will find that that proposal was not officially adopted as the definition. Indeed, if you read the title of that section, you should suspect the same thing.

13. Nov 25, 2011

Dickfore

First of all, the acceleration due to gravity has DIMENSION [length]×[time]^(-2). Therefore, the numerical value depends on the choice of the system of units. If you want to compare numbers, you must form a DIMENSIONLESS quantity.

Since the meter was originally defined through the length of the Earth's meridian (such that the distance from the North Pole to the Equator is 10^7 m), and the second was defined through the Earth's rotation period (which is 24 × 60 × 60 = 86400 s). Thus, your hypothesis really is:
$$\pi^2 = \frac{g \cdot 1 \, \mathrm{s}^2}{\mathrm{m}}$$
$$\pi^2 = \frac{g \, \left( \frac{T}{86400} \right)^2}{\frac{\frac{R \, \pi}{2}}{10^7}}$$
Simplifying, we get:
$$\frac{g \, T^2}{R} = \left( \frac{36 \, \pi}{5} \right)^3$$

Last edited: Nov 25, 2011
14. Nov 25, 2011

clearwater304

@Vanadium- It was proposed in 1668 by Wilkins, it was adopted on May 8 1790 by the French National Assembly, it was changed only slightly March 30 1791 becuase the force of gravity varies slightly over the surface of the earth. The equation by dickfore explains why they chose the meter to be equal to one ten-millionth of the length of the distance from the equator to the north pole.

15. Nov 25, 2011

AlephZero

I would never question Wikipedia as an authority [/IRONY] but the early navigators knew very well that the length of a seconds pendulum varies with latitude. Finding an explanation for this was one of the major scientific questions of the time, given the importance of timekeeping for accurate navigation, and various experiments had been done on the subject. (One hypothesis was that gravity was a function of temperature). Newton summarised the experimental data in Principia, and modelled the effect in terms of gravitation and the earth's rotation.

Wren was a colleague of Newton (in fact Newton used Wren's the partly-built St Paul's Cathedral in London for some dynamics experiments on damping) so it seems rather odd that Wren would propose a "standard" that he most likely knew was poorly defined. But the French National Assembly may have had the same respect (i.e. not much) for Newton as Newton did for French scientists in general (again, reference Principia)....

But where Wilkins fits into all this, I don't know.

16. Nov 26, 2011

Dickfore

Ok, then go and make a pendulum with length 1 and measure its period and tell us if it is 2. We eagerly await your answer.

17. Nov 26, 2011

clearwater304

I was just quoting the blog, but the point was that length and time are both arbitrary measurements until you define them and correlate one with the other.

18. Nov 26, 2011

Dickfore

No, that's not the point. The point is that g is a CONSTANT in that equation. Given the length, you cannot have arbitrary periods, but only one corresponding to the result obtained from that equation.

19. Nov 26, 2011

clearwater304

The length from the equator to the north pole they would have measure would have been:

We can calculate that this would have been:

(6378100 meters * pi)/2

Which equals

10018696.05 meters which back in the day, since the meter was equal to the equivalent of 997 millimeter, the length they would have measured would have been 100488425.6 meters which is very close 10000000.

The question is, was the length from the equator to the north pole known before they chose to divide it by 10 million. If so, the relation between g and pi squared would have remained highly correlated. If not, and the division by 10 million was a random guess, then that would have been a coincident.

Last edited: Nov 26, 2011
20. Nov 26, 2011

lol.