Rotating cone filled with water

In summary, the conversation discusses the location to drill a hole on a rotating cone filled with liquid in order to spray the water to the maximum distance. The Bernoulli equation is used to obtain the solution, where the speed at the hole is calculated using the constraint of the water falling on the floor from the upward cone. The use of Lagrange multiplier is also mentioned, but the speaker is unable to solve the equation. The two factors that affect the spraying of water are pressure and angular momentum, and the best location for the hole may vary depending on the shape of the cone. The speaker also raises a question about incorporating the rotation velocity in the Bernoulli equation.
  • #1
largich
2
0
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.
 
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  • #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.
 
  • #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?
 

1. What is a rotating cone filled with water?

A rotating cone filled with water is a scientific apparatus used to demonstrate centripetal force. It consists of a cone-shaped container filled with water and attached to a rotating platform, allowing the water to spin around the cone in a circular motion.

2. How does a rotating cone filled with water work?

The water inside the cone is held in place by centrifugal force, which is created by the rotation of the cone. This force is directed away from the center of rotation and keeps the water from spilling out. The water also experiences centripetal force, which pulls it towards the center of rotation and causes it to form a concave surface against the walls of the cone.

3. What is centripetal force?

Centripetal force is a force that acts towards the center of rotation of a body in circular motion. It is responsible for keeping the rotating cone filled with water in motion and preventing the water from flying outwards due to its inertia.

4. What are some real-world applications of a rotating cone filled with water?

A rotating cone filled with water can be used to demonstrate and study centripetal force in various fields such as physics, engineering, and meteorology. It can also be used to understand the concept of inertia and its effects on rotating bodies.

5. Can a rotating cone filled with water be used to create artificial gravity?

No, a rotating cone filled with water cannot create artificial gravity. While it may seem like the water is being pulled towards the bottom of the cone, this is actually due to the centripetal force acting on it. Artificial gravity would require a much larger and faster rotating object to simulate the gravitational pull of a planet.

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