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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 6.055J / 2.038J The Art of Approximation in Science and Engineering Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 6.055 / Art of approximation 28 Next estimate the force. It is proportional to the mass: F ∝ m . In other words, F / m is independent of mass, and that independence is why the proportion- ality F ∝ m is useful. The mass is proportional to l 3 : m ∝ volume ∼ l 3 . In other words, m / l 3 is independent of l ; this independence is why the proportionality m ∝ l 3 is useful. Therefore F ∝ l 3 . The force and area results show that the pressure is proportional to l : F l 3 p ∼ A ∝ l 2 = l . With a large-enough mountain, the pressure is larger than the maximum pressure pressure ∝ l force ∝ l 3 mass ∝ l 3 volume ∝ l 3 area ∝ l 2 that the rock can withstand. Then the rock ﬂows like a liquid, and the mountain cannot grow taller. This estimate shows only that there is a maximum height but it does not compute the maximum height. To do that next step requires estimating the strength of rock. Later in this book when we estimate the strength of materials, I revisit this example. This estimate might look dubious also because of the assumption that mountains are cu- bical. Who has seen a cubical mountain? Try a reasonable alternative, that mountains are pyramidal with a square base of side l and a height l , having a 45 ◦ slope. Then the volume is l 3 / 3 instead of l 3 but the factor of one-third does not affect the proportionality between force and length . Because of the factor of one-third, the maximum height will be higher for a pyrami-....
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feb25b - MIT OpenCourseWare http/ocw.mit.edu 6.055J 2.038J...

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