Calculating Moment of Inertia for Hollow Cylinder w/ Water

[haruspex]

The surface of that liquid should adopt certain angle (tan a/g) respect to the horizon.

That would be a steady state condition, but if we are taking the water as inviscid it will never be achieved.
As a simplification, I would treat the water as a frictionless solid, as though it were ice (but not stuck to the cylinder, as it was in the video you posted). It might be possible to solve this model, but still tricky enough. As @jbriggs wrote, it will behave like a pendulum, but with the complication of the interaction with the cylinder, which will roll in a jerky fashion.

It is not clear to me whether that is what @mostafaelsan2005 means by "sloshing dynamics", or if he is thinking of more complex motion.
Certainly the ice model overlooks that the surface of the water would not remain flat; a lower centripetal force would be required along the centre line of the water surface than at the edges. But intuitively I feel that is quite a minor effect. It could be studied in isolation merely as water slopping back and forth in a bowl.

 

[erobz]

That would be a steady state condition, but if we are taking the water as inviscid it will never be achieved.
As a simplification, I would treat the water as a frictionless solid, as though it were ice (but not stuck to the cylinder, as it was in the video you posted). It might be possible to solve this model, but still tricky enough. As @jbriggs wrote, it will behave like a pendulum, but with the complication of the interaction with the cylinder, which will roll in a jerky fashion.

It is not clear to me whether that is what @mostafaelsan2005 means by "sloshing dynamics", or if he is thinking of more complex motion.
Certainly the ice model overlooks that the surface of the water would not remain flat; a lower centripetal force would be required along the centre line of the water surface than at the edges. But intuitively I feel that is quite a minor effect. It could be studied in isolation merely as water slopping back and forth in a bowl.

The water hardly rocks at all in the test I conducted.

 

[haruspex]

The water hardly rocks at all in the test I conducted.

It will depend greatly on the angle of the slope. Tests on steep slopes of sufficient length would be nontrivial to conduct.

 

[Lnewqban]

That would be a steady state condition, but if we are taking the water as inviscid it will never be achieved.

Sorry, could you explain that reasoning of @jbriggs444 and yourself a little further?
To me, the only difference the zero-viscosity condition makes is the lack of internal friction within the fluid; which means no friction with the cylinder either, if the boundary layer is not considered.

 

[haruspex]

Sorry, could you explain that reasoning of @jbriggs444 and yourself a little further?
To me, the only difference the zero-viscosity condition makes is the lack of internal friction within the fluid; which means no friction with the cylinder either, if the boundary layer is not considered.

When released from rest, the water surface is horizontal. In steady state, it would be at a constant angle to the horizontal. In transitioning from the former orientation to the latter, it would overshoot, then reverse its path relative to the cylinder. (This would be observed externally as a temporary increase in the acceleration of the cylinder.) If there are no frictional losses, this (I feel sure) would lead to a persistent and non diminishing oscillation.

 

[Lnewqban]

Thank you, @haruspex

 

[mostafaelsan2005]

That would be a steady state condition, but if we are taking the water as inviscid it will never be achieved.
As a simplification, I would treat the water as a frictionless solid, as though it were ice (but not stuck to the cylinder, as it was in the video you posted). It might be possible to solve this model, but still tricky enough. As @jbriggs wrote, it will behave like a pendulum, but with the complication of the interaction with the cylinder, which will roll in a jerky fashion.

It is not clear to me whether that is what @mostafaelsan2005 means by "sloshing dynamics", or if he is thinking of more complex motion.
Certainly the ice model overlooks that the surface of the water would not remain flat; a lower centripetal force would be required along the centre line of the water surface than at the edges. But intuitively I feel that is quite a minor effect. It could be studied in isolation merely as water slopping back and forth in a bowl.

The jerking motion is precisely the sloshing dynamics that I plan to discuss in the research paper, although it's out of my scope mathematically speaking. I plan to write about it in terms of my visual findings upon conducting the experiment.

The way I see it (perhaps erroneously, as many other times):
The liquid inside should be only affected by the linear acceleration of both centers of mass, cylinder and liquid.
The surface of that liquid should adopt certain angle (tan a/g) respect to the horizon.

The value of that linear acceleration (a) should be lower than of an equivalent mass freely sliding down the slope, because the rotational acceleration of the cylinder alone should be slowing that rate of linear acceleration down.

As the level of liquid changes, the location of its CM respect to the central axis of the cylinder and point of contact and moment should change as well.

Yes, I will also discuss it in terms of the process that the CM changes depending on the amount of water within the cylindrical shell and at what point it is in (in terms of the rotation on the ramp--which is the reason that the moment of inertia cannot be easily found as a particular value as it changes as it goes down the ramp based on how much water is in the shell).

It will depend greatly on the angle of the slope. Tests on steep slopes of sufficient length would be nontrivial to conduct.

That's what I believe as well, though I plan to conduct a preliminary experiment in order to make sure of it. Conceptually I see it as if it is a less steep slope it won't behave in terms of sloshing dynamics due to the initial angle it is displaced at.

 

[erobz]

which is the reason that the moment of inertia cannot be easily found as a particular value as it changes as it goes down the ramp based on how much water is in the shell).

I think you are still confused. The moment of inertia does not change as the cylinder rolls down the ramp based on how much water is in the cylinder (under the inviscid assumption).

Conceptually I see it as if it is a less steep slope it won't behave in terms of sloshing dynamics due to the initial angle it is displaced at.

It’s the large “step” change in acceleration that starts the water rocking. It’s sitting there at rest, then it almost instantly goes to some value $a$. The steeper angle scales the magnitude of that step change, which likely scales the effect of “sloshing”.

 

[jbriggs444]

the moment of inertia cannot be easily found as a particular value as it changes as it goes down the ramp based on how much water is in the shell

The point of a having a "moment of inertia" is that you can then multiply it by a rotation rate to get a number for angular momentum. For a rigid object, this works out well. A rigid object has a rotation rate. Its moment of inertia can be computed with an integral or found in a table on Wikipedia.

For non-rigid objects the notion is problematic. If you have, for instance, a sloshing mass of water, what is the rotation rate of that water? Worse, if you have a moving mass element in that water, its contribution to the total angular momentum of the water will not, in general be equal to $m \omega r^2$. Nor in general, would any "moment of inertia" remain constant from one instant to the next.

You could calculate a "moment of inertia" by taking the angular momentum of the non-rigid object and dividing by the imputed rotation rate. But what is the point? Angular momentum is the figure you are after. The moment of inertia is only a calculational aid.

Why do you want to find the moment of inertia of the container plus contents?

 

[mostafaelsan2005]

Hello all, thanks for all your previous help; it is much appreciated. I have come to the conclusion that visually modeling the initial angle that the water makes with respect to the can based on how much water is in it will be the best way to conduct the theoretical basis of the experiment to then simply treat it as a translational model with an inviscid fluid. I was wondering if the forces acting would be quite identical to the full cylinder model but just with the added variable of there being an angle of the water and how I could represent that mathematically.

 

[haruspex]

modeling the initial angle that the water makes with respect to the can based on how much water is in it

I don’t see how the angle the eater surface makes to the wall of the can is interesting.

identical to the full cylinder model but just with the added variable of there being an angle of the water

If it is full, what angle is there?

I can only repeat my earlier advice: assume, at least as a start, that the water maintains its shape, i.e. behaves as a frictionless solid.
Draw a FBD with the surface of the water at some angle to the horizontal. Analyse the forces and obtain equations for the angular acceleration of the water and the linear acceleration of the can.