[Math Lair] Archimedes and His Lever

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The following was printed in 1915 in Scientific American magazine:

Archimedes and his Lever

He Might Move the World, but He Would Certainly Be a Very Long Time Doing It.

"Give me a fulcrum and a place on which to stand and I will raise the earth from its place!" That is a saying popularly attributed to Archimedes.

If the required conditions were possible the feat might be performed, but in addition to providing a fulcrum and a place on which to stand Archimedes would also have to be furnished an indefinite lease on life.

To raise the earth a height of one inch by the force which Archimedes would have been capable of exerting would take not only an extremely long lever but an extremely long time, as can be readily shown.

We shall assume the following data in our calculations: That the earth is a sphere 7,926 miles in diameter and that 5.5 is its mean density; also that the lever has no weight. Should we design a real lever it would be of such enormous size and weight that Archimedes' weight would be a negligible.

If the earth is 7,926 miles in diameter, the volume is about 261,000,000,000 cubic miles, or 38,400,000,000,000,000,000,000 cubic feet, and if the density is 5.5 the weight per cubic foot would be about 344 pounds, which, multiplied by the volume, would give us the weight of the earth 13,209,600,000,000,000,000,000,000 pounds.

We shall assume, further, that Archimedes weighed 150 pounds, and that "the place on which to stand" was some distant star; then, if the fulcrum is one mile from the point of application of the lever to the earth, the length of the power arm of the lever, or the distance of Archimedes' "standing place," would have to be 88,064,000,000,000,000,000,000 miles.

To move the earth a distance of one inch Archimedes, end of the lever would have to move through a distance of 1,388,000,000,000,000,000 miles.

Now, if Archimedes should take hold of the end of the lever and apply his weight of 150 pounds to it, and should move off into space with the velocity of light, of 186,000 miles a second, it would take him 237,000 years to finish the job he proposed, so that now, nearly 2,200 years after making the famous dictum, he would barely have started the undertaking.

Should Archimedes, while flying through space at the above rate of speed, encounter the atmosphere of some planet, the effect would be that of striking a solid, and he, for an instant would perform the function of a swiftly moving hammer with a very long handle.—Scientific American.