# Mass, Volume, MOI question

• I
• Randy Cotteleer

#### Randy Cotteleer

I am looking to prove or disprove the following statement:

Two objects, of the same homogeneous material, the same mass, the same volume, the same center of mass and the same moment of inertia will be dimensionally the same.

If there is a way to generate a mathematical proof, that would be extremely helpful.

Regards,

Randy

I am looking to prove or disprove the following statement:

Two objects, of the same homogeneous material, the same mass, the same volume, the same center of mass and the same moment of inertia will be dimensionally the same.

If there is a way to generate a mathematical proof, that would be extremely helpful.

Regards,

Randy
That doesn't seem correct to me. You haven't been able to find any counterexamples?

Randy Cotteleer
berkeman, I haven't found any counterexamples, but I am skeptical as well. The statement seems plausible, I'd like to see if there is a definitive answer.

You haven't been able to find any counterexamples?
For example, take 2 dumbbells and tie them rigidly at the centerpoint where they cross at a right angle. Then instead of joining them at a 90 degree angle, join them at a 45 degree angle. Will the two configurations not have the same mass, volume and MOI? But they are very different dimensionally, no?

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• gym-crossed-dumbbell-barbell-weight-athletics-1c.png
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Okay, but imagine that it is not 2 dumbbells tied together, but rather a single item essentially "cast" in those 2 shapes (one @90 and one @45) because of the change in angles at the center, the volume would be different. Am I thinking about it correctly?

Am I thinking about it correctly?
I don't think so. Use a bolt through a hole in the center of both bars to tie them together.

berkeman, that would still be 2 pieces joined together. I am considering monolithic items. I did some math, if the shapes were cast monolithically, one at 90 degrees and one at 45 degrees, and assuming that the crossection of the arms was 1" x 1" the volume of the interface for the 90 degree example would be 1 in^3 the volume of the interface for the 45 degree example would be 1.414 in^3.

I appreciate you helping me to noodle this out...

Okay, but imagine that it is not 2 dumbbells tied together, but rather a single item essentially "cast" in those 2 shapes (one @90 and one @45) because of the change in angles at the center, the volume would be different. Am I thinking about it correctly?
If the rods have a flat face where they touch you can essentially rotate them to any angle then heat them slightly so that they fuse. I was also going to suggest a bolt, but @berkeman beat me to it.

You only have a handful of constraints and otherwise complete freedom to specify the surface of your object. I find it very difficult to imagine that this problem is anything other than enormously under-determined.

Randy Cotteleer and berkeman
I am considering monolithic items.
Just weld the joint after it's bolted.

Plus, this counterexample was pretty easy to think of. Can you extend it to a more general counterexample?

EDIT -- @Ibix has suggested a good alternative.

Randy Cotteleer
Actually, if you consider the MOI in all 3 axes, our counterexamples so far don't work. Need to think more about counterexamples to all 3-axis MOIs...

Randy Cotteleer and Ibix
Pondering...

you are right, in 3 axes, MOI does change.

Consider a cylinder with its axis aligned with the z axis. It has four identical grooves in it, circling it at four different constant z values.

Can you lay the grooves out in two distinct ways to satisfy your constraints? You've got four variables (the positions of the grooves) and three constraints (center of mass, moment of inertia around z, moment of inertia around x or y - they're degenerate by symmetry) so it should be possible. You shouldn't need more than the parallel and perpendicular axis theorems and standard results for moment of inertia of cylinders and rings.

berkeman
Ibix are you effectively saying that if you move the, for lack of a better term, outer grooves out and the inner grooves in the appropriate amount to offset the outer groove displacement. that the MOI's stay the same? I think I understand.

Ibix
Okay, I agree and that is valuable to know. question: without a physical measurement or inspection is there a way to tell the 2 objects in post #14 apart? Is there a constraint that could be added to identify the difference?

In principle you can keep adding new things to measure. For example you may find that the impedance of the two is different, since you'll get a bit of reflection from the grooves and interference between them will give you different resonances. Or you will find that they displace different amounts of water if submerged to a depth that covers different numbers of grooves. But each time you add a constraint I can add another pair of grooves to allow another variable to fox that test too.

Why are you asking? As a theoretical exercise the answer is that it's possible to find a counter example for pretty much any combination of tests, I think. The basic argument is that there are far more variables (crudely, the position of every atom in the object) than you can possibly constrain. But there will be practical and cost limitations to the precision with which we can make and test these objects.

Ibix, Thank you for your help. I understand your assertion. we can always have more variables than constraints, however, the more constraints we have the more likely the objects are dimensionally identical. The question is as much a mental exercise and anything else. Thanks again.