A3 - Applied Math 250 Assignment #3 Winter 2009 due...

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Unformatted text preview: Applied Math 250 Assignment #3 Winter 2009 due February 3rd A/ Problem Set 2 #1, #2, #3, #4, Problem Set 1 #19 Note: For #4 you may begin with the IVP mZ—‘t’ = mg —av, 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. B/ 1/ 11/ Coulomb’s Law states that the force F between two point charges Q1 and Q2 Q1Q2 471'60 R2 ’ distance between them and so is the “permittivity of free space” (whose numerical value is 8.854 a: 10-12). Determine the dimensions of co. acts along the line joining them, with magnitude F = where R is the 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 —~ h Q = hf, or [400% = —k\/h (the value of It depends dt A(h) dt on factors such as the size and shape of the hole). surface of the water: a) Suppose we have water in a cylindrical container, standing upright and being drained from the bottom. i) Define a characteristic time and length, and nondimensionalize the DE (use 7' and h for your dimensionless variables). Comment: this is a very unusual problem; you’ll find that we have frac- 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 h(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) = 2N2”; — h? (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, nondirnensionalization 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 1 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 zled — h? ‘15:: = —k\/E, h(0) = ho. How long does it take the container to drain this way? Verify that your result is dimensionally consistent. Find than-z, the time required for a full barrel to drain while lying on its side. t Define T = and show that if we could have identified this as our fIl‘ior‘z characteristic tiine to begin with, and identified the characteristic height as hC = d as well (so h = g), we could indeed have simplified the DE sig- nificantly (you don’t need to solve the DE again; just nondimensionalize it). Fiom comparison of tum and thmz, you should be able to see that if I >> d then a full barrel will drain fastest if stood upright, while if (1 >> I 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 03/20/2011 for the course AMATH 250 taught by Professor Ducharme during the Summer '09 term at Waterloo.

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A3 - Applied Math 250 Assignment #3 Winter 2009 due...

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