Unusual gravitational observation....answers?

[Celes123]

Hello,
I noticed something odd today while removing two identical tubs of cottage cheese from the fridge. They were stacked one on top of the other with no visual way to tell if a container was half full or completely full. When picking them up at the same time by only grasping "the lid of the bottom container," I felt that the bottom tub was more full than the tub on top. When I checked I found that I was right.
I decided to put the half full tub on the bottom and tried picking them up together again... only by grasping the lid of the bottom container. I could tell immediately that the tub on the bottom was lighter.
What was confusing wasn't how I was able to tell which tub was more full than the other with the two experiments, but why the total weight of the two tubs held together seemed lighter when the half full tub was on the bottom. I could have held it indefinitely without support. While when the heavy tub was on the bottom, it honestly felt as if I needed to eventually support my wrist.
Disregarding that it was merely psychological ...what would explain the total weight of both tubs seeming lighter only when holding the half full tub on the bottom?

 

[A.T.]

I decided to put the half full tub on the bottom and tried picking them up together again...

You should have some else prepare them, in a way that you have way to tell other than the supposed effect.

Disregarding that it was merely psychological ...what would explain the total weight of both tubs seeming lighter only when holding the half full tub on the bottom?

In general, the mass distribution can matter for how it feels to hold something in a stable manner. To prevent swinging or toppling you sometimes have to apply torques through relatively short lever arms.

 

[Celes123]

Thanks AT,
I failed to include that I had someone else hand me the two tubs already stacked on top of each other. I probably should have randomized which way they were handed to me in order for the testing to be more thorough. The reason why I didn't was because the tubs felt so much lighter with the one that was half full on the bottom. It just was so obvious.
I agree that the mass distribution can matter for how it feels to hold something in a stable manner. I think that it was part of the reason why I could tell whether the half full tub was on the bottom or the top when handed to me. I'm not sure about the toppling over adjustments that might have had an effect, because the tubs sit very firmly together on top of each other without any risk of falling off. I still can't see why with the half full tub on the bottom, the two tubs together always feel significantly lighter. Thank you for your answers.

 

[sophiecentaur]

I think there are two possibilities. 1. Dynamics: Moment of Inertia. Turning with the axis nearer the CM requires less torque than when the CM is further from the axis.
2. Statics: The system is not in equilibrium because you are holding it on the side. Once it tilts by a finite amount, the moment about your hand is greater when the perpendicular distance between weight force and hand is greater. (I.e. More torque for higher CM when tilted)
Your brain is very good at this sort of thing.

 

[A.T.]

I think there are two possibilities. 1. Dynamics: Moment of Inertia. Turning with the axis nearer the CM requires less torque than when the CM is further from the axis.

Moment of inertia depends on the overall mass distribution, not on whether the CM is closer or further from the axis.

2. Statics: The system is not in equilibrium because you are holding it on the side. Once it tilts by a finite amount, the moment about your hand is greater when the perpendicular distance between weight force and hand is greater. (I.e. More torque for higher CM when tilted)
Your brain is very good at this sort of thing.

There are also potential aspects of biomechanics and neuromuscular control, that make some mass distributions subjectively easier to stabilize. This could even vary between people, depending on their previous training.

 

[sophiecentaur]

Moment of inertia depends on the overall mass distribution, not on whether the CM is closer or further from the axis.

Do you want to reconsider that?
The minimum value of MOI is about an axis through the CM.

 

[sophiecentaur]

There also potential aspects of biomechanics and neuromuscular control, that make some mass distributions subjectively easier to stabilize. This could even vary between people, depending on their previous training.

Yes. It's very complex. In this case, though. We would have to assume that the experimenter would be holding the two identical tubs in exactly the same way - else the answer could be anyone's guess.

 

[A.T.]

The minimum value of MOI is about an axis through the CM.

Only if the mass distribution stays the same, which is not the case here.

 

[sophiecentaur]

Only if the mass distribution stays the same, which is not the case here.

Are you saying that there is a significant difference between mass distribution with the full one on top and the full one below? Would it really make all that much difference if you put both tubs upside down - keeping the mass distribution identical in both cases. The fact is that MOI is not the same through all axes and that's what you were implying:

not on whether the CM is closer or further from the axis.

 

[A.T.]

... a significant difference ... all that much difference...

I wasn't making any quantitative statements about this scenario, for which we don't have any numbers. I was pointing out the fallacy of assuming that the MOI must decrease, if the COM moves closer to the axis due to mass redistribution.

 

[sophiecentaur]

of assuming that the MOI must decrease, if the COM moves closer to the axis due to mass redistribution.

Perhaps you would quote a good example that counters that. It would not involve two tubs in a fridge, I think. My point has been that the minimum MOI is about an axis through the CM. I hope you wouldn't disagree with that but what you wrote had a strong implication that the position of the CM is not the main factor. For similar objects (tubs in any order), the main factor has to be the CM position. At the very least, you could acknowledge that your statement could have been worded better to make it clear that it was a general comment and explained in the context of the actual situation. People don't always get the message when comments are made with minimal wording.
We all know a lot about the actual geometry of this particular situation and we can safely assume that there is no whacky redistribution of masses during the experiment.

 

[A.T.]

It would not involve two tubs in a fridge, I think.

Why not?

 

[sophiecentaur]

Why not?

This is a very easy problem to characterise. If you are really suggesting that the MOI would increase with the part-filled tub on top, I would love to know your argument. Put some numbers in and quote an answer to support what you say. Assume that the axis is half way down the lower tub, which is probably how you'd hold it. As far as I can see, you are just arguing for the sake of it but you could prove me wrong.
The model can be 2D, with two rectangles. Equal long dimensions and short dimensions b and c. The normalised MOI of the rectangles is (a²+b²) /12and (a²+c²)/12 and the parallel axis theorem tell us that if the axes of rotation for the two rectangles are d₁ and d₂, from the chosen pivot point, the MOI of each rectangle is increased by d₁² and d₂². I guess we could do the sums . . . . .

 

[A.T.]

...which is probably how you'd hold it.

What if you hold it as the OP actually states?

 

[sophiecentaur]

What if you hold it as the OP actually states?

I have tried. Have you?
It would depend on the shape of the tub, perhaps. I only have rectangular ones, with very little strength to the lip.. They need support or they sag. This is a common problem, where the precise problem is not specified.

 

[sophiecentaur]

. . . . .but at least I have tried a practical approach and the beginnings of a mathematical one.

 

[A.T.]

I have tried.

And what does your model say about MOI vs COM distance from axis, assuming you hold it in the middle as the OP states?

 

[sophiecentaur]

And what does your model say about MOI vs COM distance from axis, assuming you hold it in the middle as the OP states?

I haven't worked it out. Been eating and watching TV. In practical terms I don't seem to be able to hold it as the OP suggests so I see little point in calculating something that may well not apply. Have you any opinion on or even looked into the 'holding it' issue? It seems quite relevant to me. Apart from having stated the rather obvious fact that the arrangement of the masses can affect the MOI and stated that the position of the axis from the CM is not relevant, you are not helping me here. I am looking for an answer and you could possibly contribute to it with some positive input.

 

[A.T.]

I haven't worked it out.

Which of the below configurations has:
- The greater MOI around the axis in the middle (more mass further away from that axis)?
- The greater distance between the axis in the middle and the COM?

 

[sophiecentaur]

Which of the below configurations has:
- The greater MOI around the axis in the middle (more mass further away from that axis)?
- The greater distance between the axis in the middle and the COM?

Yes. I drew that diagram some while ago, along with the expressions for MOI and parallel axis addition factor. I imagine that you have had the same diagram in mind from the very beginning.
Looking at your diagram shows that the COM of the left hand arrangement is further to the mid point and also the MOI is also less because the distance from the mid point axis to the CM of the half tub is also less. The correction factor for MOI, Mr² is less. That was my original point - about a specific arrangement. You seem to have objected to the statement when applied to a general case(?) If, instead of just telling me that I was wrong (in the original scenario), you had given some detail, we could have saved about a dozen posts.
But, as I already commented, I do not believe that the support axis really was at the half way point. Try it and see what I mean. Even holding the rim with pliers will not achieve it as the tub lip sags. To keep it upright, you have to support it some way down the side of the lower tub, making the difference even more and changing the CM position with respect to the pivot in both cases.
The actual pivot point is still an issue which needs to be addressed if we are to answer the OP's question.

 

[sophiecentaur]

@AT. I notice that your diagram seems to assume that the axis is parallel to the x-axis (rectangles on XY plane and axis drawn in 'heavy line'). That doesn't describe the situation adequately. The axis of rotation should be parallel to the z axis, I think. My formula for the MOI (above) is for the zz axis because the aspect ratio is relevant.

 

[A.T.]

The axis of rotation should be parallel to the z axis, I think.

What axis is Z?

 

[sophiecentaur]

What axis is Z?

@AT. I notice that your diagram seems to assume that the axis is parallel to the x-axis (rectangles on XY plane and axis drawn in 'heavy line').

Not x or y. Is there something wrong with my notation?

 

[A.T.]

Is there something wrong with my notation?

The critical axis is the one through the held points. Whether you call it X or Z is irrelevant.

 

[sophiecentaur]

The critical axis is the one through the held points. Whether you call it X or Z is irrelevant.

Yes if course. But your diagram has a thick black line which, in the absence of any other clues about the axis, I took that to be the axis.
I referred to the rectangle in your post as being in the xy plane so it is not to hard to decide which is the z axis. i.e. not in the plane of your diagram.
The formula M(x²+y²)/12 wouldn't apply to your diagram.
By the way, there are no bonus points available for keeping the work count of your posts so low that the meaning can suffer. Feel free to use the occasional extra sentence to make sure that the message is understood from the start.

 

[Celes123]

[QUOTE="sophiecentaur
2. Statics: The system is not in equilibrium because you are holding it on the side. Once it tilts by a finite amount, the moment about your hand is greater when the perpendicular distance between weight force and hand is greater. (I.e. More torque for higher CM when tilted)
Your brain is very good at this sort of thing.[/QUOTE]

I agree that there are imperceptible adjustments that the brain is continuously making of which we are not aware. Thank you for your ideas.

 

[Celes123]

Sophicentaur and AT...
Thank you for your input. I have attached some pics to show the containers. Although seemingly large in the pics, they are only three inches high at best and not hard to hold...no sagging. My hands are small and I still easily can hold them by the lid of the bottom container.
What we tried was to, in a blind test, not only perceive the total weight by holding the tubs by the lid in the middle, but also by placing the two tubs in the palm of our hands. The results were the same that with the half full tub on the bottom, the over all weight was perceived to be less than with the half full tub on the top. I visually noticed the tendons pop up on my daughter's arm when trying told hold the tubs with the full one on the bottom. This did not happen with the reverse of that experiment.
Thank you in advance for any insight you can add.

 

[A.T.]

Thank you in advance for any insight you can add.

Aside from the different balance due to MOI and COM changes, there could be a psychological effect going on like: Content that concentrates at the bottom of a container appears heavier than content distributed more evenly within the volume.