Levers are an easy way to gain a mechanical advantage. Basically, you can lift an object by using less force, but at the cost of applying the force for a greater distance.
The distances the forces are applied are proportional to the distances to the pivot point. For example, if you want to lift an object weighing 100 Newtons (a mass of approximately 10kg) and you want to use a force of only 50N, the length of lever from your side must be twice the length on the object's side. Mathematically, 100N × d1 = 50N × d2, which gives that 2 × d1 = d2, or d1 = d2 / 2.
Somehow, Archimedes needs an immovable pivot point, and a body with some gravity for him to apply a force to his lever. Even then, he wouldn't be moving it against the normal gravity we all love, but against the gravity of the sun, which would be approximately
If we assume that the planet he is standing on has the same gravity as Earth (9.81m/s^2) and that it is so massive that the distance it's going to move back will be derisive (the two assumptions are, of course, in conflict), he would need to apply a force of 50N for a meter in order to move the Earth 1.41E-27 meters up, which is smaller than the size of an electron (in fact, it is even closer to the Plank length).
To have a better grasp of it, lets put Archimedes on an infinite plane with a gravity of 10m/s^2 (almost that of Earth), and ignore the fact that the gravity will change depending on how high he is (ie, he is somewhere with a constant gravity). He wants to lift an object weighing 6x10^25N (or 6x10^24kg, almost the mass of Earth, under a 10m/s^2 gravity) for a meter. Let's assume that Archimedes is using his whole body weight (let's say a mass of 100kg, so a weight of 1000N). How far down will he have to push his side of the lever?
That distance is more than 60 times bigger than the milky way, and all that in order to lift the Earth for one meter. I'm not even talking about how long this would take him! If he were to freefall, bringing the lever with him (let's assume a terminal velocity of 100m/s, or 360km/h), it would take him 6x10^20 seconds. That's about a thousand times the age of the universe.
This scenario brings a somewhat unrelated, but interesting concept - propagation of matter. When Archimedes starts to bring his lever down, the effect is not felt immediately at the other end. Indeed, the effect is propagated at the speed of light. In our present case, assuming that the pivot is one meter away from the Earth, it would take more than 6 million years for the Earth to start moving up. Exactly the same thing when he stops applying the force, so it would take him another 6 million years for the Earth to lift one meter.
The last image is exaggerated, but gives an idea of how it would look. An interesting thing to understand is that the lever is straight. Should it be made from unbendable metal, it would look the same.
Conclusion: *technically*, Archimedes could lift the Earth. The point here is that, given a long enough lever, you can lift a very heavy load by using a smaller force over a greater distance.
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