A3 - Applied lVIath 250 Assignment #3 Spring 2009 due June...

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Unformatted text preview: Applied lVIath 250 Assignment #3 Spring 2009 due June 2nd A/ Problem Set 2 #1, #2, #3, #4, Problem Set 1 #19 Note: For #4 you may begin with the WP mfi—‘t’ = mg — ow, 12(0) = 0, as derived in the course notes. Where part (b) asks you to “write the DE for the velocity as a function of time”, it is asking you to nondimensionalize that equation. 8/ I/ Coulomb’s Law states that the force F between two point charges Q1 and Q2 422621;” where R is the 0 distance between them and 60 is the “permittivity of free space” (whose numerical value is 8.854 * 10'12). Determine the dimensions of £0. acts along the line joining them, with magnitude F = II/ Torricelli’s Law states that the water flowing from a hole in a container exits at the speed that it would have achieved had it fallen from the level of the surface of the water in the container. This can be reformulated in several ways. For example, one rule derived from this is that the water level in a container being drained from the bottom decreases at a rate proportional to the square root of the height, divided by the area of the db 4% dh surface of the water. Et— — AUI) , or A(h)E — —lc\/h (the value of It depends on factors such as the size and shape of the hole). 3 a) Suppose we have water in a cylindrical container, standing upright and being All“): “Y drained from the bottom. 1 Define a characteristic time and length, and nondimensionalize the DE Lo (use 7' and h for your dimensionless variables). Comment: this is a very unusual problem; you’ll find that we have frac- (hlr tional dimensions! This is odd, but it doesn’t cause any problems for us; you can still proceed as usual. This peculiarity is merely a result of the way the constant k: is arrived at in the reformulation of Torricelli’s Law. ii) Solve the nondimensionalized DE, subject to the condition that [1(0) = ho (which will need to be rewritten in terms of h), and then express your solution in terms of the original variables. How long does it take the container to drain? How long does it take if the barrel is full to begin with (call this time tum)? AM 250 — Assignment #3 Page 2 b) Now suppose that the container is placed on its side. In this orientation, the area of the surface depends on h, r, and l: A(h) = 2M2”; —- (L2 (Why? Consider the equation of a circle of radius 7‘, situated with its lowest point at the origin, and you should be able to find the length of the horizontal secant lines as a function of y). When we have many parameters, nondimensionalization is less helpful. It may be impossible to simplify our equations significantly, or, if it is possible, it may be difficult to identify the appropriate scale factors without solving the equation first (would you use 7" or l as your length scale here?) i) ii) iii) Solve the DB, in dimensional form, with h(0) = ho. Note: it will be more convenient to use the diameter instead of the radius here, so solve 2lx/hd — h2 % = —k\/h, h(0) = no. How long does it take the container to drain this way? Verify that your result is dimensionally consistent. Find them, the time required for a full barrel to drain while lying on its side. t Define 7' = and show that if we could have identified this as our thoriz characteristic time to begin with, and identified the characteristic height h as hC = d as well (so h = E), we could indeed have simplified the DE sig— nificantly (you don’t need to solve the DE again; just nondimensionalize it). c) From comparison of tum and thmz, you should be able to see that if l >> d then a full barrel will drain fastest if stood upright, while if d >> l it will drain faster on its side. Find the ratio l/d which will allow a full barrel to drain equally quickly in either orientation. ...
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This note was uploaded on 03/24/2010 for the course STAT 231 taught by Professor Cantremember during the Spring '08 term at Waterloo.

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A3 - Applied lVIath 250 Assignment #3 Spring 2009 due June...

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