# Homework Help: Rotating cone filled with water

1. Dec 6, 2008

### largich

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

v^2=2g(H-h-h^2-r^2\frac{\omega^2}{2g})=2g(H-h-h^2\frac{(tg{\alpha})^2}\omega^2}{2g}),
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. Dec 8, 2008

### Carid

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.

3. Dec 8, 2008

### largich

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?