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---------------------------- - 2291 - Archimedes - Density of the Earth
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- The Density of Mother Earth doing it the Ancient’s Way. Density is what something weighs divided by its volume. The Earth is two-thirds water and the density of water is 1,000 kilogram per cubic meter. So if we know the volume of the Earth we could calculate the weight.
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- The density of a typical rock is 3,000 kilograms/m^3. And, the Earth’s average density is 5,520 kilograms per cubic meter. Since the average density is so much higher it leads astronomers to conclude the Earth has a heavy iron core. But how did the ancients calculate the density in their time?
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-------------------------------- Average Density = mass / volume.
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- So, if we know how much the Earth weighs, its mass, and we know the volume we can calculate the average density. Actually the weight of the Earth is zero because weight is the “force” of gravity and Force = mass * acceleration. (F = ma)
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- It is the mass we need to measure , not the force. Archimedes was an ancient Greek astronomer in 225 B.C. He understood the principle of the lever and used it to calculate the volume of a sphere. Eratosthenes was a Greek scholar who calculated the circumference of the Earth. As you will see these ancients were brilliant and profound thinkers. They were able to calculate the density of the Earth.
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- Archimedes could calculate the density of water by putting a 1 kilogram weight on one side of a lever. Equal length from the fulcrum on the other side he would put a barrel and begin filling it with water until the lever was in balance. Balancing like a teeter totter he knew the water now also weighed 1 kilogram.
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- The volume of a cylinder he calculated by multiplying the area of the base, which is a circle, times the height of the water in the barrel. He could determine the circumference of a circle as compared to its diameter by taking a string across the circle to measure its diameter; then, laying the length of string around the edge of the circle end for end.
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- It will always go around 3.14 times for any circle. The diameter times 3.14 = circumference for all perfect circles. 3.14 is a constant that we call pi. (pi = 3.141592654).
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- We can get the formula for the area of a circle by cutting the circle up in wedges like a pizza. Laying the wedges in an alternating pattern side by side we can form a rectangle. The thinner the slices the more perfect the rectangle that gets formed. The length of two sides will be (pi*radius) which is half the circumference. The width of the other two sides of the rectangle will be the radius. The Area is length times width so the area of a circle is pi* radius^2. (Area = pi*r^2).
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- Archimedes’ barrel had a radius of .56 meters and the water was 1 meter high. The volume was pi*r^2, or 3.14 * (.56)^2 = 3.14 * .318 = 1 meter^3. The density of water is 1 kilogram per cubic meter. Which is, by definition, a kilogram, the weight of 1 cubic meter of water.
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- Archimedes said that if he had a long enough lever and a place to stand he could move the Earth. He was also ingenious enough to think of another alternative to calculate the volume of a sphere. The Earth was a sphere and in 225 B.C. no one knew how to calculate the volume of a sphere. His method used to get this formula was pure genius. See if you can visualize his thought process, it‘s amazing:
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- He started with a lever and he put a sphere inside a cone inside a cylinder on one side of the lever. Nothing was on the left side, so obviously the heavier, right side was down. The radius of the sphere is R, and it just touches the top and bottom of each cylinder base. Therefore, the cylinder height was 2*R. The base of the cone was the same as the right side base of the cylinder having a radius of 2*R.
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- This is obviously a thought experiment since each of these is a solid object nested one inside the other. Archimedes thought that he could slice these objects and move parts to the other side of the fulcrum until the lever was in balance. Since he knew how to calculate the volume of a cone and a cylinder, he could use this to determine the volume of the third object, the sphere.
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- Take a thin slice through the center of the cylinder and it will be a circle of radius 2*R and a thickness of delta R. As we make the delta R very small the volume of this slice approaches the area of the circle, pi*(2R)^2.
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- As you slice through the cylinder you also slice through the sphere and the cone. Again, since the slice is very thin the volume of the sphere’s slice becomes pi*R^2 and the volume of the cone’s slice becomes pi*R^2.
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- Archimedes could see that if he slid the slice of the sphere and the slice of the cone across the fulcrum to the 2*R position on the other side he could exactly counterbalance the slice of the cylinder, that remained in place.
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------------------------------ Slice of sphere + slice of cone = slice of cylinder
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------------------------------ 2R*(pi*R^2) + 2R*(pi^2) = R*pi*(2R)^2
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------------------------------ 4piR^3 = 4piR^3
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- He realized that he could perform this operation for each and every slice and the sphere and cone would counterbalance the cylinder. Since this was true for each and every slice it is true for the entire sphere and cone. Therefore, you can visualize the sphere and cone hanging from the 2*R point on one side of the lever and the solid cylinder on the other side exactly in balance. Their volumes on both sides times the distances to the fulcrum had to be equal.
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------------------------- ( sphere volume + cone volume)* 2R = ( cylinder volume) * R
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- The cone volume is 1/3 the base times the height. This can be shown by dividing up the base of the cone like a pizza pie and forming tetrahedrons, pyramids with triangular bases. If you convert the cone into a cylinder these tetrahedrons can then form prisms. Careful investigation will show that there are 3 tetrahedrons in each prism therefore the volume of the cone is exactly 1/3 the volume of the cylinder whose volume is base times the height. Putting this all together:
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--------------- 2*R * (sphere volume) + 2*R * (cone volume) = R * (cylinder volume)
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--------------- 2*R * (sphere volume) + 2*R * (1/3*pi*(2R)^2) = R * (pi*(2R)^2)*(2R)
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--------------- 2*R * (sphere volume) + 2*R * (8R^3*pi/3) = 8*pi*R^4
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--------------- 2*R * (sphere volume) + 16R^4*pi/3 = 24*pi*R^4/3
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--------------- 2*R * (sphere volume) = (8*pi*R^4)/3
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--------------- sphere volume = 4/3*pi*R^3
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- If we knew the radius of the Earth we could now calculate the volume, no problem. Here is how the ancients calculated the radius of the Earth:
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- Eratosthenes, a friend of Archimedes, made this calculation in 240 B.C. He was on a camel train traveling between the Egyptian town of Syene ( now called Aswan ) and Alexandria, which is almost directly north of Syene.
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- On June 21 every year he noticed that the Sun shone directly down the well casting no shadow on the sides. Then, when he traveled to Alexandria at the same time of year there was a shadow on the walls of Alexandria’s wells.
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- He used a 10 foot stick and could see that when perpendicular to the ground it cast a 16 inch shadow, which meant that the Sun’s rays are coming down at a 7.5 degree angle.
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---------------------------- Tangent of the angle = opposite side / adjacent side
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--------------------------- Tan = 16 inches / 10 feet = 16 / 120 = .1333
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---------------------------Angle = 7.59 degrees
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- The camel trains traveled 10.4 miles per day on average and it took 50 days to reach Alexandria from Syene. Eratosthenes reasoned that the Sun’s rays were exactly over the well in Syene and that the rays would go to the center of the Earth if the well was deep enough.
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- Yet, 520 miles away, the sun’s rays were at an angle of 7.5 degrees. Since the sun’s rays are parallel, the angle that the 520 segment makes at the center of the Earth is also 7.5 degrees.
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- Since the entire circle around the Earth is 360 degrees and 7.5 goes into 360 48 times, the circumference of the Earth must be 48 times the 520 miles. 48 * 520 = 24,960 miles. This calculation was accurate to within 100 miles of today’s calculations, and it was done with a stick, a camel drive, and a brilliant mind.
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-------------------------- The circumference of the Earth is 2*pi*R = 24,960 miles
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-------------------------- The radius = 3,972.5 miles
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-------------------------- The volume of the Earth = 4/3 * pi * R^2
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-------------------------- Volume = 4/3*3.14*(3,972.5 miles)^2
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-------------------------- Volume = 2.626*10^11 miles^3
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-------------------------- Kilometer = .621371 miles
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-------------------------- Kilometer^3 = .24 miles^3
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-------------------------- Volume = 10.94 * 10^24 kilograms
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-------------------------- Density = mass / Volume
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- Mass of Earth is 6.0 * 10^24 kilograms. (Ask if you want another book review on how the mass of the Earth was calculated.)
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-------------------------- Density = 6.0*10^24 / 10.94*10^20 = 5,483.6 kilograms/m^3
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- This is pretty close to 5,520 kilograms/m^3 using today’s technologies in measurement. Those old guys were pretty smart.
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- Archimedes was killed by a Roman soldier during the Punic Wars. Eratosthenes, blind and weary at age 80 died of voluntary starvation in 196 B.C
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- March 3, 2019. 61
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--------------------- Monday, March 4, 2019 -------------------------
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