The simplest facts would be the size and mass of the earth, from these we can calculate the average density of the earth and try to guess what it might be made out of to give that density.

CHAPTER 1. Size of Earth.

Eratosthenes of Cyrene (276 BC - 194 BC) was the first to figure out the size of the Earth. His results have been confirmed by numerous other techniques beginning with Columbus finding out that there was a whole other continent sitting in the Atlantic and Pacific oceans between him and China.

here is an explanation with diagrams of how he did it:

http://www.eg.bucknell.edu/physics/astronomy/astr101/specials/eratosthenes.html

more info on Eratosthenes:

note:

his result is: 250,000 stadia. From various records we think his stadia are somewhere around 160m each+/-20m. Let's say 160m. 250,000 stadia X ~160m = 4,000,000 meters in circumference/3.14= 1,300,000 meters, so the radius is 6.5X10^6meters. We'll use that value, it's not bad.

(this also comes out to: .. X 1mile/1600meters =25,000 miles in circumference or 8000 miles in diameter.)

so now we can find its volume=(4/3)pi*r^3

=4*(6.5X10^6)^3

=4X275X10^18 cubic meters

1100X10^18

10^21 cubic meters

that's big!

CHAPTER 2. Mass of the Earth (Or How Much Does the Earth 'Weigh'?)

Well, by watching things fall and observing planets make ellipses around the sun, Isaac Newton summarized that the gravitational force between two bodies is equal to G*M*m/d^2, where M,m are masses of the bodies, d is their distance and G is some constant that he didn't know very accurately. The point of his formula is that even without knowing the constant, the formula gives the right SHAPES for the orbits of the planets. It even gives all three of Keplers laws:

[link to summary of Kepler's laws?]

He also worked out that this force is also = the mass of either of the bodies times its acceleration.

so GMm/d^2 = ma

notice that m cancels out! the mass of the falling body doesn't affect the acceleration of its fall, something Galileo first showed.

GM/d^2=a

so if we drop a rock here near the Earth's surface, we can measure the acceleration: a; it's 9.8m/s^2, so

GM/d^2 =9.8m/s^2

d=distance from our rock near the surface of the earth to center of mass of the Earth=the radius of the Earth

so

G*M/(6.5X10^6)^2 =9.8m/s^2

look: we've got our equation down to only two variables! If we can measure G, we can figure out M the mass of the earth. Weighing the earth! Not bad! How can we figure out G?

Let's go back to f=GMm/d^2. If we can somehow measure the force between two known masses, say bowling balls, and measure their distance, we could do it! The problem is that we hardly find any force at all when we try this, gravity is actually a pretty weak force, it only SEEMS so strong to us, because the earth is SO MASSIVE.

But Henry Cavendish managed to measure this small force in 1783! Very clever man. he hung something like a dumbbell from a quartz fiber with a mirror attached to it. He then shined a beam of light at the mirror and watched its reflection move across the opposite wall. This setup amplified the subtle twisting motions of the apparatus. He then put two weights near the dumbbell to attract it, and observed the oscillations as the apparatus settled down. By measuring the oscillations he could determine the gravitational force between the masses that twisted the fiber.

here's a long description of how careful you have to be in setting one up in your basement:

Here is a description of the calculation of G from the angle of deflection of the dumbbell:

INTERLUDE OF CONFUSION TO BE CLEARED UP

( I am still unclear at exactly what parameters of the oscillation is being measured, and the final formula giving the torque on the fiber. torque=-k*theta

and

torque =f*moment arm=GMmL/r^2

to his

G=r^2*pi^2*L*theta/MT^2

where T is the period of the oscillator.

But what theta is he measuring? As the device settles, doesn't theta get smaller?

i think i need to use the harmonic oscillator equation:

a=-omega^2*Asin(sqrt(k/m)t)) and some working out. I’ll have to do that later.)

END INTERLUDE

so G turns out to be a ridiculously small number:

~6 x 10^-11 N m2/kg2

we are ready to weigh the earth!

G*M/(6.5X10^6)^2 =9.8m/s^2

M=gr^2/G ok

M= 9.8m/s^2 X 42 X 10^12m^2 /( 6X10^-11 Nm^2/kg^2)

9.8m/s^2 X 42X10^12 m^2 / (6X10^-11 kgm^3/kg^2s^2)

9.8m/s^2 X 42 X 10^12 m^2 / (6X10^-11 m^3/kg/s^2)

= 10X42/6 X 10^12 /10^-11 m^3 kg s^2 / (s^2Xm^3)

=70X10^23kg (WHEW! the units work, no mistakes)

7X10^24kg!

The currently accepted value is 5.9742 × 10^24 kilograms not bad using these simple experiments!

now we are ready to calculate the density of the Earth:

CHAPTER 3. Calculating the Density of The Earth and Guessing at What it is Made of.

d=mass/volume

ok

d=7X10^24kg/10^21 m^3

=7X10^3kg /m^3.

lets convert that to grams/cc

7X10^3kg/m^3X1000g/kgXm^3/10^6cm

=7g/cc! now we know something about the Earth! That is a very reasonable number for density, We are on the right track!

water has density 1g/cc, and here is the clincher: most rocks we find at the surface of the earth and even down a few miles by drilling, have a density of between 2 and 3g/cc

so the earth is full of something even more dense than these minerals at depths deeper than we can drill. that is our FIRST clue as to what's inside.

to bring the overall density of the earth to 7g/cc the minerals below 2 miles or so have to have average density MORE than 7g/cc.

what minerals are like that? we can guess iron because it's the most common dense element in crustal rocks, all others are very rare in comparison.

density of iron is: 7.8g/cc hmm.. maybe there are denser metals down there? or is it almost all iron? or does the density increase under pressure?

we could use some geometry to figure out how large a proportion of the earth's center needs to be iron to bring the average density up to 7 from the 2.5 of the density at the crust.

The next chapters could be: (4) We can find more clues by listening to the whole Earth ring like a bell when it receives a shock from an earthquake or nuclear explosion. We can even listen more carefully to three DIFFERENT types of sound waves that travel through the Earth after earthquakes. (5) We can do experiments on various minerals to see how they behave under different pressures. (6) We can explore the electromagnetic properties of the Earth which seem to be telling us something about processes in the core. (7) There is also the convenient fact that corals and algae have been patiently recording the length of the day, and the length of the lunar month over the past 2 billion years! I think this data can be used to get some ideas of the density profile vs. depth of the earth using principles of moment of inertia.

In sum, it is pretty amazing that from some observations of shadows in a few different cities, plotting the courses of the planets in the night sky, fitting them to a mathematical formula, measuring the acceleration of falling bodies, measuring the twist of a fiber given by balls attracted to each other, and measuring the density of minerals, we can get some guesses at what kinds of minerals there are 4000 miles down in the center of the Earth.

On the other hand the amount of work put in by Copernicus, Tycho Brahe, Kepler, Galileo, and Newton was immense. And the experiment of Cavendish, genius and delicate. This is the story of science.

## Sunday, February 15, 2009

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