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Unformatted text preview: AMath 250 ASSIGNMENT 4 Fall 2008 Submit all problems by Noon on Tuesday, October 14th in the drop boxes across from
M04066.
All solutions must be clearly stated and fully justiﬁed. FROM PROBLEM SET 1 IN THE COURSE NOTES : #14 Hint: There ’3 no need to solve the DE. Don’t pay much attention to the restriction on
h at ﬁrst; it will become clear why it is restricted after you’ve found the equilibrium
solutions. #19 ‘ FROM PROBLEM SET 2 IN THE COURSE NOTES : #1,2,3 Note: force = mass x acceleration, pressure = force/area,, work = force x distance,
gradient means ”rate of change with respect to distance”——e.g. d—T is the temperature d3:
gradient. ALSO, DO THE FOLLOWING PROBLEM I I. 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 surface of the water: 5% = or AM)??? = —k\/h (the value of It depends on factors such as the size and shape of the hole). “r i) Deﬁne a characteristic time and length, and nondimensionalize the DE (use
7' and h for your dimensionless variables). 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)? continued on next page... b) Now suppose that the container is placed on its side. In this orientation1 the area
of the surface depends on it, r, and l: A(h) = 2lv2rli — h? (Why? Consider the equation of a circle of radius r, 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). I 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 r or i as your length scale here?). i) ' ii) iii) Solve the DB, in dimensional form, with 31(0) = 1%. Note: it will be more con
venient to use the diameter instead of the radius here, so solve 2N he! — h? 53—? =
ﬁlm/5‘ 21(0) = ho. How long does it take the container to drain this way? Verify that your result
is dimensionally consistent. Find thmz, 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 charac— hen{z teristic time to begin with, and identiﬁed the characteristic height as he = d E), we could indeed have simplified the DE Signiﬁcantly (you d.
don’t need to solve the DE again; just nondimensionalize it). as well (so it = c) From comparison of ﬁver; and thanz, you should be able to see that. if I >> d then a
full barrel will drain fastest if stood upright, while if d. >> i it will drain faster on
its side. Find the ratio [/d which will allow a full barrel to drain equally quickly
in either orientation. ...
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 Fall '08
 ANDREWCHILDS
 Algebra, container, appropriate scale factors

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