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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: 1012). 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 thanz, 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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 Summer '09
 Ducharme

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