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Homework Help: Rotating cone filled with water

  1. Dec 6, 2008 #1
    I have a cone filled with liqid with radius R and height H rotating with \omega. Where do we have to drill a hole that the water would spray to the maximum distance from the cone?

    I used the Bernoulli equation obtainig
    p_0+0.5 \rho {v_1}^2=p_0+0.5 \rho v^2
    v is the speed at the hole, getting

    where tg{\alpha}=R/H.
    I taught using Lagrange multiplicator, where the constraint is the water falling on the floor prom the upward cone:

    \psi=v sin{\alpha} t+gt^2/2-h=0.

    Further more:
    F=v_x t+\lambda(v sin{\alpha} t+gt^2/2-h)
    =v cos{\alpha}+\lambda(v sin{\alpha} t+gt^2/2-h)
    Solution should be obtained by
    \frac{\partial F}{partial t} and

    \frac{\partial F}{partial v}, v=v(h),
    but i can't solve it.
    Did I make the concept wrong? Any ideas would be helpfull.

    PS: The cone is standing on its tip and it is opened at the top.
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Dec 8, 2008 #2
    Two things will help to spray that water; the pressure which increases as we go down the cone, and the angular momentum which increases as we go up the cone. With a very flat cone I should imagine the best place is near the top; With a very sharp cone I guess the best place is near the bottom. There should be an angle at which it doesn't matter.
  4. Dec 8, 2008 #3
    Thank you. I agree. But the problem is solving the equation. Is the use of Bernoulli eq. even correct. Do I incorporate the rotation velocity in Bernoulli eq. or as a separate contribution?
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