Calculating Moment of Inertia for Hollow Cylinder w/ Water

[mostafaelsan2005]

Homework Statement

I am doing an experiment with the overall research question of: To what extent does the amount of fluid within a hollow cylindrical can affect its dynamics while rolling down an inclined plane

w I have been treating it as an inviscid fluid for the purpose of the derivations that I have made.

Relevant Equations

I have found the equations for how to represent acceleration for a solid and hollow cylinder along an inclined ramp assuming "pure roll" for now but I want to find an equation to represent different MOI's and perhaps hydrodynamic pressures?

I was able to derive an equation for acceleration for the case of a fully solid cylindrical shell and then used law of conservation of energy to determine equations for the velocity of a solid and hollow cylinder and I understand that the moment of inertia's of the aforementioned cans can also be found easily using the radius. However, I am finding difficulty in finding a method to determine the moment of inertia for cans that are partially filled other than the parallel axis theorem which I am not sure can be applied to this scenario. Furthermore, I am also unsure about the actual experiment that I am doing to answer the initial research question (I have access to a motion sensor and photogate sensor) and was wondering if I should simply measure the final velocities and relate it to varying moment of inertias if it is possible to calculate but would appreciate any alternatives to this experiment. Finally, I am not sure if I should treat the water as an inviscid fluid as that would mean there would be no effects on the moment of inertias but the mass would still change so that would affect it in that regard, right? For now I have been treating it as an inviscid fluid for the purpose of the derivations that I have made.

 

[Chestermiller]

You are talking about a very complicated fluid mechanics problem if you want to consider the fluid viscous. Just imagine the simpler case of only rotating a horizontal cylinder filled with liquid at a constant angular velocity without allowing the cylinder to translate. You have a free surface and, if the rate of rotation is not high enough. some of liquid dragged up to the top will release and fall to the pool before it can even complete the rotation. Even if the viscosity and rotation rate are high enough, the thickness of fluid coating the wall will not be constant as a function of angular position. And this is with constant rotation rate and steady state operation. With the accelerating angular velocity that would be present in rolling down an incline, this unsteady state situation would be an order of magnitude more complicated.

 

[mostafaelsan2005]

Thanks for the reply. I am writing a research paper on this so in terms of the experiment that I will conduct for it what can I do if I have a photogate sensor and motion sensor to measure? I will now be assuming water as an inviscid fluid so there will be no friction between the water and the walls of the can. However, will that change the moment of inertia of the cylindrical shell based on how much liquid is in the can? Furthermore, for the theoretical background of the research paper I have a section on derivations of formulas for acceleration for completely solid and hollow cylinders as well as derivations for velocity... what else can I add to the theoretical background that will relate to what I will do in the experiment preferably?

 

[erobz]

I think if the fluid was truly inviscid ( save you the trouble, its not...) it won't factor into the moment of inertia. But I believe you are going to have to account for the forces (maybe the pressure distribution - which is a function of the acceleration?) acting on the cylinder (and fluid) via Newtons 3
^rd Law which impart the required translational kinetic energy to the fluid as the system accelerates down the ramp.

I doubt your experimentation is going to agree with the model well (even if the "simplified" version is solvable). Viscosity is going to be an issue.

 

[mostafaelsan2005]

Yes, I planned on researching the differences in hydrodynamic forces that would vary based on the amount of water in the can but the research papers I went through had very complicated derivations and I couldn't quite understand it. Since viscosity will be an issue, I believe that it will be best to treat it as inviscid and find a way to express the differences in hydrodynamic pressures although it will be harder. Is there any way I can start with figuring it out because I figured out the acceleration formulas for a completely solid and hollow cylinders but have not been able to find expressions for them for being partially filled as it depends on the moment of inertia to find the translational acceleration at that point which I found would be extremely hard to find directly. Any thoughts of what I can do in terms of pressure and how I could reflect that in my experiment?

 

[erobz]

Yes, I planned on researching the differences in hydrodynamic forces that would vary based on the amount of water in the can but the research papers I went through had very complicated derivations and I couldn't quite understand it. Since viscosity will be an issue, I believe that it will be best to treat it as inviscid and find a way to express the differences in hydrodynamic pressures although it will be harder. Is there any way I can start with figuring it out because I figured out the acceleration formulas for a completely solid and hollow cylinders but have not been able to find expressions for them for being partially filled as it depends on the moment of inertia to find the translational acceleration at that point which I found would be extremely hard to find directly. Any thoughts of what I can do in terms of pressure and how I could reflect that in my experiment?

I’d neglect forces from hydrostatic pressure for the time. If the fluid is to be modeled as inviscid it won’t rotate, but it will translate. There must be a force acting on the fluids center of mass in the direction of motion if it’s accelerating down the ramp with the cylinder. This equates to a retarding force on the cylinder by Newtons a third law. Solve that system separating the bodies.

 

[mostafaelsan2005]

Thank you, I will attempt to do that and report back to the thread. For now though I need to do an experiment ASAP by Friday so should I stick with measuring both the final velocity and change in displacement of the can with varying amounts of water with a photogate sensor and motion detector respectively?

 

[erobz]

Thank you, I will attempt to do that and report back to the thread. For now though I need to do an experiment ASAP by Friday so should I stick with measuring both the final velocity and change in displacement of the can with varying amounts of water with a photogate sensor and motion detector respectively?

Thats what you are hoping to model with this.

 

[jbriggs444]

If the can is released from rest, I would expect the fluid within to "slosh" back and forth. In the inviscid case, I would expect this slosh to persist indefinitely.

 

[mostafaelsan2005]

Yes I have looked into sloshing dynamics as well. Would that be better to look into or hydrostatic pressure? I have a word limit so I guess it depends on what I understand most, right?

 

[jbriggs444]

Yes I have looked into sloshing dynamics as well. Would that be better to look into or hydrostatic pressure? I have a word limit so I guess it depends on what I understand most, right?

One experiment is worth 1000 theories. How about a trip to the grocery store for a jar of baby food?

The model in my mind's eye is the fluid delineated by a chord through the container moving back and forth as if it were a rigid body. Now what you have is almost exactly a pendulum. If this model is apt, fluid dynamics is irrelevant. It is purely a mechanical problem.

 

[mostafaelsan2005]

Unfortunately, I do not think that I can treat the sloshing dynamics like a pendulum system since it acts rather erratically. Is there a method that I could model this being as a pendulum system as I am unsure if it is too much of an assumption though if there is any source that corroborates this model it could be good; ill try to search for one now. I guess the worst case scenario would be that I would generally speak about the moments of inertia without calculating a specific quantity for each value and just calculating (and experimentally finding) the velocities of the completely filled cylinder (regarded as solid) and completely hollow cylinder and then just finding the final velocities of the partially filled ones. Not sure if the displacement-time graph would be of any use though, perhaps if I discuss it in terms of a pendulum system it could be utilized.

 

[erobz]

Unfortunately, I do not think that I can treat the sloshing dynamics like a pendulum system since it acts rather erratically. Is there a method that I could model this being as a pendulum system as I am unsure if it is too much of an assumption though if there is any source that corroborates this model it could be good; ill try to search for one now. I guess the worst case scenario would be that I would generally speak about the moments of inertia without calculating a specific quantity for each value and just calculating (and experimentally finding) the velocities of the completely filled cylinder (regarded as solid) and completely hollow cylinder and then just finding the final velocities of the partially filled ones. Not sure if the displacement-time graph would be of any use though, perhaps if I discuss it in terms of a pendulum system it could be utilized.

It would be good to get yourself a clear cylindrical vessel so you can observe (record) the motion of the fluid on the decent.

 

[mostafaelsan2005]

Alright, I'll record as much data as I can on it. Thanks

 

[mostafaelsan2005]

I am still working on the theoretical model of this scenario and have not been able to accurately determine how to treat it as a pendulum system with the equipment I have available. Any further ideas that I can explore? Here's a link to my research paper thus far to see what I have done so far:
https://docs.google.com/document/d/...SNinjcOlh2Yh_N9Cr0nfHQ/edit#heading=h.1fob9te

 

[Lnewqban]

Why is the moment of inertia of a solid cylinder considered if the ideal liquid is not rotating at all?

 

[haruspex]

Why is the moment of inertia of a solid cylinder considered if the ideal liquid is not rotating at all?

Why do you think its surface will stay level?
.

 

[Lnewqban]

Why do you think its surface will stay level?
.

Why do you think that I think so?

 

[haruspex]

Why do you think that I think so?

Because if it does not stay level then it must rotate within the cylinder. Or did you mean something else by the liquid's not rotating?

 

[erobz]

Because if it does not stay level then it must rotate within the cylinder. Or did you mean something else by the liquid's not rotating?

I think he is asking why the author is considering a solid cylinder for which the entire mass would rotate. If the cylinder was fully filled with truly inviscid fluid, the fluid would acquire no angular kinetic energy, just translational KE. Since they are ignoring the mass of the cylinder (the container), the water should be effectively sliding down a frictionless ramp.

I think the author should instead account for the rotational KE of the container and translational energy of the inviscid fluid inside it for a starting point.

 

[haruspex]

If the cylinder was fully filled

But it isn’t.

 

[erobz]

But it isn’t.

They seem to be considering that in the first part of the paper under "Theoretical Background"

In the context of a hollow cylinder fully–filled with water rolling down an inclined plane...

The primary source of rotational motion is a result of a quantity called the moment of force or torque that is often denoted by which produces the rotational acceleration of a rotating object. Therefore, in regards to a rotational motion to the center of mass of a cylinder...

It must be noted that these equations only apply to cylindrical shells that are assumed to be either fully solid due to the fact that the shell has been fully-filled with water or fully hollow; therefore, they can be treated as a simple cylinder with a known moment of inertia.

They are considering that the fully filled cylinder (a cylinder which they characterize as having negligible inertial properties earlier on) of inviscid water is rotating as it goes down the slope. They are not getting the basic idea as far as I can see.

 

[haruspex]

They seem to be considering that in the first part of the paper under "Theoretical Background"

They are considering that the fully filled cylinder (a cylinder which they characterize as having negligible inertial properties earlier on) of inviscid water is rotating as it goes down the slope. They are not getting the basic idea as far as I can see.

ok, thanks.
Maybe this is why @mostafaelsan2005 is reluctant to treat the part-filled case as a pendulum, i.e. as a solid mass of water oscillating freely within the cylinder.

 

[erobz]

I don’t believe they are ready for that level of analysis. I think they should get a clear cylindrical water bottle, partially fill it, roll it down an incline and tell us what they observe with their eyes for a low angle ramp, and with video footage for a steeper angle?

 

[mostafaelsan2005]

They seem to be considering that in the first part of the paper under "Theoretical Background"

They are considering that the fully filled cylinder (a cylinder which they characterize as having negligible inertial properties earlier on) of inviscid water is rotating as it goes down the slope. They are not getting the basic idea as far as I can see.

So in the case of the fully hollow and filled cylinders; they will not rotate along the ramp due to the fact that water is being treated as inviscid? That would be simply untrue since, experimentally, as the cylinder isn't just sliding down with only the translational motion to account for.

I don’t believe they are ready for that level of analysis. I think they should get a clear cylindrical water bottle, partially fill it, roll it down an incline and tell us what they observe with their eyes for a low angle ramp, and with video footage for a steeper angle?

Yes, I was planning on doing this already, but then it would be worthy of me to consider the oscillatory motion of the cylinder which again does not assume water as inviscid.

If I were to go down the route of considering the oscillatory motion I would base my paper off of this here as this is where I got the idea to consider the moment of inertia in the first place and it delves into oscillatory motion (visually, not in a formulaic sense, which I believe is better in this case).

 

[haruspex]

So in the case of the fully hollow and filled cylinders; they will not rotate along the ramp due to the fact that water is being treated as inviscid? That would be simply untrue since, experimentally, as the cylinder isn't just sliding down with only the translational motion to account for.

The cylinder would rotate, but not the inviscid fluid. There is no force exerting a torque on the fluid.
Scatter some powder on a glass of water and rotate the glass slowly. The powder won't move much.

 

[erobz]

The powder won't move much.

Just to grind this point in for the OP. If water were truly inviscid it wouldn’t rotate at all.

 

[mostafaelsan2005]

Oh, I understand the point that since water is being treated as inviscid it wouldn't rotate as the cylinder rotations along the inclined plane; however, it is still subject to sloshing dynamics in this scenario and a type of oscillatory motion where the water moves without sliding I believe. Sloshing dynamics though would unfortunately not fit into the scope of the paper and my expertise all the same and thus I may broadly speak in terms of moment of inertia's depending on the increasing mass of water resulting in a higher final velocity and visual data to observe the oscillatory sloshing phenomenon that occurs for different amounts of fluid within the can.

 

[erobz]

Oh, I understand the point that since water is being treated as inviscid it wouldn't rotate as the cylinder rotations along the inclined plane; however, it is still subject to sloshing dynamics in this scenario and a type of oscillatory motion where the water moves without sliding I believe. Sloshing dynamics though would unfortunately not fit into the scope of the paper and my expertise all the same and thus I may broadly speak in terms of moment of inertia's depending on the increasing mass of water resulting in a higher final velocity and visual data to observe the oscillatory sloshing phenomenon that occurs for different amounts of fluid within the can.

I set up a ramp, partially filled up a clear sports drinking bottle with tap water. A string was placed at the point of contact to hold it until the water settled and then I pulled the string while taking a slow-mo video on my phone. I suggest you try this experiment before you try to analyze sloshing, because it might not (hint) be as critical to the motion as you expect. I tested this for a two relatively shallow ramps ( 3 to 5 degrees inclination), you could try steeper. The point I (and @jbriggs444 in post #11 ) am trying to make is to experiment so you can be efficient in the modeling. Don't get me wrong, I enjoy a good thought experiment too, but you could be chasing mice that you believe are fire breathing dragons. Also, it could be that at some point during you probing the mouse will turn into a fire breathing dragon. Thats the fun of it!

 

[Lnewqban]

Because if it does not stay level then it must rotate within the cylinder. Or did you mean something else by the liquid's not rotating?

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.

 

[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.