Re: [math-fun] The highest mountain in the solar system.
Another mystery spam filter misfile. --Rich ---------- Date: Mon, 20 Jul 2015 15:28:26 -0400 Subject: Re: [math-fun] The highest mountain in the solar system. From: wba <wbackerman@gmail.com> A long time ago (a long, long, long time ago) I attended a physics lecture/colloquium at MIT titled "How High is a Mountain?" The lecture was a virtuoso performance of the kind of approximation and hand-waving that some professors are spectacularly good at. He used fundamental constants, like Planck's constant, the gravitational constant, particle masses, strength of the electromagnetic force, etc., to work out that, for "things" made of "matter" (that is, the kind of stuff that familiar matter seems to be made of--you know, atoms and molecules and all that) the product of the highest feature and the object's diameter would be 40,000 miles. (Please excuse the non-metric units.) For Earth, it's 8000 miles times 5 miles, which is the height of Mount Everest. The square root is 200 miles, meaning that the largest arbitrarily-shaped "thing" (grand piano, person, carrot, desk, etc.) that you can have is about 200 miles across. Of course exotic things like carbon fiber can stretch that, but only so much. On 07/20/2015 01:35 PM, Henry Baker wrote: [Hide Quoted Text] Similar question, inspired by ancient Egyptians & Romans: The definition of a planet(oid) seems to be a body large enough to force itself into a nearly spherical shape. Suppose you wanted to hack this defn. What weird shape could you build out of standard materials (bricks, cement, steel, carbon fiber?) that would have the same mass as a real planet but retain its weird shape. Alternatively, assuming the exact same recipe of elements as found in the entire Earth, how big of an object could be built that retained its shape under a) self gravity; and b) rotational stresses; and b) the various tidal forces due to the Moon, planets, Sun. Fractal structures aren't so farr-fetched: http://www.london-institute.org/people/farr/fractals.shtml At 10:23 AM 7/20/2015, Bill Gosper wrote: Maybe one of the deviations from sphericity of https://en.wikipedia.org/wiki/21_Lutetia ? How would you rule it out? Related: An earth-size cube of "solid" rock would immediately collapse to a sphere with eight mountains. About how tall? Maybe mountains should be ranked by gravitational potential energy. --rwg
There seems to be a problem of units here. Distance times distance gives area, not distance. In this example, a simple unit change will allow you to produce any result you wish, since "units" and "units^2" don't scale the same. There would have to be a divide by another distance, with units, for this to work. Tom rcs@xmission.com writes:
Another mystery spam filter misfile. --Rich
---------- Date: Mon, 20 Jul 2015 15:28:26 -0400 Subject: Re: [math-fun] The highest mountain in the solar system. From: wba <wbackerman@gmail.com>
A long time ago (a long, long, long time ago) I attended a physics lecture/colloquium at MIT titled "How High is a Mountain?" The lecture was a virtuoso performance of the kind of approximation and hand-waving that some professors are spectacularly good at. He used fundamental constants, like Planck's constant, the gravitational constant, particle masses, strength of the electromagnetic force, etc., to work out that, for "things" made of "matter" (that is, the kind of stuff that familiar matter seems to be made of--you know, atoms and molecules and all that) the product of the highest feature and the object's diameter would be 40,000 miles. (Please excuse the non-metric units.) For Earth, it's 8000 miles times 5 miles, which is the height of Mount Everest. The square root is 200 miles, meaning that the largest arbitrarily-shaped "thing" (grand piano, person, carrot, desk, etc.) that you can have is about 200 miles across.
Of course exotic things like carbon fiber can stretch that, but only so much.
On 07/20/2015 01:35 PM, Henry Baker wrote:
[Hide Quoted Text] Similar question, inspired by ancient Egyptians & Romans:
The definition of a planet(oid) seems to be a body large enough to force itself into a nearly spherical shape.
Suppose you wanted to hack this defn. What weird shape could you build out of standard materials (bricks, cement, steel, carbon fiber?) that would have the same mass as a real planet but retain its weird shape.
Alternatively, assuming the exact same recipe of elements as found in the entire Earth, how big of an object could be built that retained its shape under
a) self gravity; and b) rotational stresses; and b) the various tidal forces due to the Moon, planets, Sun.
Fractal structures aren't so farr-fetched:
http://www.london-institute.org/people/farr/fractals.shtml
At 10:23 AM 7/20/2015, Bill Gosper wrote: Maybe one of the deviations from sphericity of https://en.wikipedia.org/wiki/21_Lutetia ? How would you rule it out? Related: An earth-size cube of "solid" rock would immediately collapse to a sphere with eight mountains. About how tall? Maybe mountains should be ranked by gravitational potential energy. --rwg
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