The water clock < Diskrete Mathematik < Hochschule < Mathe < Vorhilfe
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(Frage) beantwortet  | | Datum: | 22:06 Mi 14.03.2007 | | Autor: | sara_20 |
A 12 hour water clock is to be designed with the dimensions: height 4 ft, and radius 1 ft,shaped like the surface obtained by revolving the curve
Y=f(x) around the y-axis.What should be this curve, and what should be the radius of the circular bottom hole, in order that the water level will fall at a constant rate of 4 inches per hour (in./h)?
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Hmmm, that's a nice question.
I think you should start with Bernoulli's equation which gives you the speed of the water as a function of pressure (-> height) Finally it says that the speed of the flowing water behaves like [mm] $v^2\sim [/mm] p$ and so [mm] $v^2\sim [/mm] y$ or [mm] $v\sim \wurzel{y}$.
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If you multiply the speed with the surface of the opening, you get the volume of water flowing through the opening:
[mm] $\dot [/mm] V=v*A$
and with my 3rd equation:
[mm] $\dot V\sim \wurzel{y}*A$
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Now it is said that the water level decreases in a constant manner:
[mm] $\dot [/mm] y=const=4inch/h$
If you multiply this with the horizontal cross section C(y) of your cavity, you once again get the volume per time!
[mm] $\dot y*C(y)=\dot [/mm] V$
Both volumes have to be the same:
[mm] $\dot y*C(y)=\wurzel{y}*A$
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Now, [mm] \dot{y} [/mm] is given and it is your job to evaluate C(y), the surface of the water at a given level y, which finally has to be transformed into the radius of a circle with Surface C(y).
I think this is the way you have to go. Finally, it is not an algebraic equation but a differential equation you have to solve. So my formulas are not really correct and also do not regard e.g. to the constants in Bernoullis equation.
Maybe I'll write a bit more tomorrow, but I also have to think about it.
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(Frage) überfällig  | | Datum: | 07:09 Do 15.03.2007 | | Autor: | sara_20 |
I have problems in solving the differential equation Y*C(y)=Sqrt(y)*A
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(Mitteilung) Reaktion unnötig  | | Datum: | 15:20 Do 15.03.2007 | | Autor: | matux |
$MATUXTEXT(ueberfaellige_frage)
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