Why is water at rest in a rotating cylinder?

In summary, the question was asking how to find the equation of the surface of water rotating around a point at an angle to the vertical. The answer is to use the equation of equilibrium, which is dh/dr = Centrifugal Force/Gravitational Force.
  • #1
Pi-Bond
302
0
A question I was doing asked to find the equation describing the surface of water placed in a cylinder rotating about its central axis. The question asserts that in the rotating frame, the water is at rest, and centrifugal force and the gravitational force of a volume element are perpendicular to each other. I don't see why this is the case - can anyone explain?
 
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  • #2
Hi Pi-Bond! :smile:
Pi-Bond said:
The question asserts that in the rotating frame, the water is at rest, and centrifugal force and the gravitational force of a volume element are perpendicular to each other. I don't see why this is the case - can anyone explain?

Which part don't you get? :confused:

In the rotating frame, obviously the water is at rest.

And obviously the gravity is vertical, and the centrifugal force is horizontal.
 
  • #3
The geometrical axis of the cylinder, which is also the axis of rotation, must be vertical so the centrifugal force must be horizontal: this is clearly at right angles to gravity.

The water must have reached equilibrium so it is rotating with the cylinder (friction forces and viscosity will eventually achieve this). "Rotating with" means that the water is at rest wrt a rotating frame of reference about the axis of rotation of the cylinder.
 
  • #4
Tiny-tim, what I was confused about is the thing MrAnchovy explained. I think I was getting confused by "at rest in the rotating frame" somehow. But now since both of you have written it out, it makes more sense. Thanks!

As a follow on, I was wondering if something similar to this method can be used to find the equation of the surface if the cylinder is rotated around a point at some angle to the vertical. For example, rotating a bucket of water by using a rope. The question previously mentioned used

dh/dr = Centrifugal Force/Gravitational Force
h: height of water w.r.t centre
r: distance from centre
 
  • #5
You should work out the bucket/rope problem. It might be helpful for your understanding of this type of problem. But first ask yourself what the angle to the vertical will be for the rope.
 
  • #6
Hi Pi-Bond! :smile:

(just got up :zzz:)
Pi-Bond said:
… I was wondering if something similar to this method can be used to find the equation of the surface if the cylinder is rotated around a point at some angle to the vertical. For example, rotating a bucket of water by using a rope. The question previously mentioned used

dh/dr = Centrifugal Force/Gravitational Force
h: height of water w.r.t centre
r: distance from centre

Yes, so long as the water keeps the same shape inside the bucket, you can use a rotating frame whose axis is the vertical line through the top of the rope.

Since the surface has a shape such that an object placed on top of the surface will not move relative to the water, it will be in equilibrium in the rotating frame.

Since the only three forces on it are gravity centrifugal and normal, you can take tangential components and get dh/dr = Centrifugal Force/Gravitational Force :wink:
 
  • #7
After thinking it through, I think I understand it now. Thanks for the help!
 

What is the purpose of studying water in a rotating cylinder?

The purpose of studying water in a rotating cylinder is to understand the behavior of fluids in a rotating environment, which has many real-world applications such as in the study of ocean currents and atmospheric systems.

How does the rotation of the cylinder affect the behavior of water?

The rotation of the cylinder creates a centrifugal force, causing the water to move towards the outer wall of the cylinder. This creates a curved surface in the water, known as the free surface, which has a parabolic shape due to the balance between gravity and centrifugal force.

What is the Coriolis effect in a rotating cylinder?

The Coriolis effect is the apparent deflection of the path of a moving object due to the rotation of the Earth. In a rotating cylinder, it causes the water to move in a circular motion, known as Taylor circulation, due to the Coriolis force acting on the water.

What is the Rossby number and how does it relate to water in a rotating cylinder?

The Rossby number is a dimensionless quantity that represents the ratio of inertial forces to Coriolis forces. In the case of water in a rotating cylinder, a low Rossby number indicates that the rotation of the cylinder dominates the behavior of the water, while a high Rossby number indicates that the inertial forces are more significant.

What are some practical applications of studying water in a rotating cylinder?

Studying water in a rotating cylinder has many practical applications, such as in the design of rotating machinery, understanding weather patterns, and predicting ocean currents. It also has applications in the study of fluid mechanics and thermodynamics.

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