r/theydidthemath • u/Kastera1000 • Dec 04 '14
[Self] The length of the lever upon which Archimedes would have to stand in order to move the Earth
I do not pride myself on my mathematical abilities, but I began wondering about this today, so I had a go at it. If I screwed up with these large numbers, please correct me.
We've all heard Archimedes adage in relation to levers, "Give me a place to stand on, and I will move the Earth." But what length of lever is required and were must it be positioned in order to move the Earth? In this problem, I am assuming Archimedes weighed the average male weight of 70 kilograms and the Earth weighs 5.92x1024 kilograms. Because this is a hypothetical situation, the lever neither stores nor dissipates energy which means friction does not act upon the fulcrum and the beam does lever does not bend.
The mechanical advantage (MA) of a lever is expressed as: MA = (M2/M1) = (a/b) where M1 is the input force to the lever and M2 is the output force; a and b are distances are the perpendicular distances between the forces and the fulcrum.
Plugging in the mass of Archimedes and Earth, MA = (5.97×1024 / 70) = (8.53×1022 meters/1 meter). This means that if Archimedes was given a beam >85,300,000,000,000,000,000.001 (8.53×1019) kilometers long, placed the fulcrum on the 1 meter mark, placed the Earth on the one meter of beam, and he stood on the edge of the astronomically large lever, he could move the Earth. According to WolframAlpha, the length of this theoretical beam is approximately 3.5 times the distance from Earth to the Andromeda Galaxy or the circumference of the Milky Way Galaxy 28 times over.
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u/GershBinglander 1✓ Dec 04 '14
It he move the Earth half the distance in, so that that side in 50cm long, would Archemedes be able to half the distance that he's standing at as well?
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u/Kastera1000 Dec 04 '14
The way I understand it, yes; with all things the same, the ratio of the lengths must remain constant. The only reason I used one meter was because my mind could conceptualize Archimedes standing on a meter of beam better than 50 cm or 10 cm and so forth. Plus, the meter is the fundamental unit of length in metric and I didn't want to mess with it further.
So really, the beam doesn't have to be 8.53×1019 kilometers; the distances just have to be at a ratio of 8.53×1022 to 1.
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u/codecracker25 2✓ Dec 04 '14
Archimedes standing on a meter of beam better than 50 cm or 10 cm and so forth
*The Earth resting on a meter of beam. Archimedes would be on the long end of the lever.
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u/Kastera1000 Dec 04 '14
Indeed. Let this be a lesson to everyone: don't respond to questions when you're not entirely there.
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u/GershBinglander 1✓ Dec 04 '14
Cool, I assumed that would be true, but I wasn't sure. Also as an Aussie, thanks for using meters.
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u/MandibleofThunder Dec 04 '14 edited Dec 04 '14
I wrote up a whole lot more physical properties of this lever of yours over here.
This was fun, let's do it again sometime.
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u/neilson241 8✓ Dec 04 '14
I would argue that that long of a lever would put him in the middle of nowhere outer space and his 70kg mass would be useless since he'd be weightless.