Could Archimedes have lifted the Earth?

Archimedes was a native of Syracuse, Sicily. It is reported by some that he visited Egypt and there invented a device now known as Archimedes screw. This is a pump, still used in many parts of the world.

It is presumed that, when Archimedes was a young man, he studied with the successors of Euclid in Alexandria. Certainly he was completely familiar with the mathematics developed there, but what makes this conjecture much more certain, is that he knew personally the mathematicians working there.

"Give me a place to stand and I will move the earth!"

"Archimedes," Plutarch says, "Once wrote to King Hiero of Syracuse, whose kinsman and friend he was, that this force could be used to move any weight. Carried away by the power of argument, he added that, were there another earth, he would go there and lift our own planet from it."

King Hiero, who was absolutely astonished by the statement, asked him to prove it. In the harbor was a ship that had proved impossible to launch even by the combined efforts of all the men of Syracuse. Archimedes, who had been examining the properties of levers and pulleys, built a machine allowing him to single-handedly move the ship from a distance away.

Archimedes knew that by applying a lever, one could lift the heaviest of weights by applying even the weakest of forces. One had only to apply this force to the levers longer arm and cause the shorter one to act on the load.

He therefore thought that by pressing with his hand on the extremely long arm of a lever he would be able to lift a weight, the mass of which would be equivalent to that of the earth. For conceptual clarity, we shall take the "moving" of the earth to mean the lifting on the earth's surface of a weight whose mass would be equivalent to that of the earth.

But, if this great scholar of antiquity would have known what an enormous mass the earth possesses, he would have most likely "eaten his words".

Let us imagine that he had at his disposal "another earth" and also the point of support he sought. Imagine that he was even able to manufacture a lever of the required length. if he needed to lift a load equivalent in mass to that of earth by at least a centimeter, he would've needed no less than thirty million million years.

Astronomers know the earth's mass. On earth a body possessing such a mass would weigh in round numbers, around 6*10²¹ tons.

Supposing a man could lift only 60 kg directly, to "lift the earth" he would need a lever with a long arm that would be longer than the shorter arm by
10²¹ times!!

You can easily figure it out that to have the end of the short arm rise by one centimeter; the other end must describe through space the huge arc of 10¹⁸ km. That is the colossal distance Archimedes would have had to push the lever to lift the earth by just one centimeter.

Presuming Archimedes could have lifted 60 Kg one meter in one second- the work of almost one horsepower! - to lift the earth by just one centimeter, even then he would need 10²¹ seconds or 30 million million years.

Though he lived to a ripe old age, Archimedes and his lever wouldn't have lifted the earth by so much as even the thinnest of hair.

According to the "golden rule" of mechanics, the mechanical advantage derived will always be accompanied by a loss in displacement, or, in other words, in time.

Even if Archimedes had been able to push the lever with a speed of 0.34 km/sec the speed of sound, he would have lifted the earth by one centimeter only after 93,264,094,069,895.84265 years.

If he had pushed the lever with the speed of light, 300,000 km/sec, nature's fastest possible - he would have lifted the earth by one centimeter only after ten million years of pushing.

References:-
- Lewis, Albert C. "Archimedes." Encyclopedia of World Biography. New York: McGraw- Hill, Inc., 1973. vol. 1, pp.219-223.
- Various history books