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I came up with a problem, which is to find a way to measure the mass of a 'certain part' of a stick without breaking it.

For instance, given a stick 10cm long, I want to know how to find out the mass of the stick between x=0 and x=5.

The density might not be uniform, so the answer isn't simply half the total mass.

I couldn't find a way, and am guessing that such a method doesn't exist.

If gravity can be completely simplified as a sigle pull on the center of mass, it would mean that two objects with the same mass, shape, and center of mass would be 'physically identical', thus leading to the conclusion that finding a solution to the problem is impossible.

(It is possible for two objects to have a different density distribution and still have the center of mass at the same position, so we won't be able to tell them apart)

However, that such a simple task would be impossible seemed too bizarre, and I suspected that there would be a slight difference between two objects mentioned above. Is there a difference?

Conditions are: The stick is in a uniform gravity field, and density does not affect anything other than the gravital pull.

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You are correct in asserting that in a homogeneous gravitational field, two infinitely stiff objects with the same shape, mass and center of mass will behave identically when accelerated, thrown or dropped.

The behavior of objects with different mass distributions can differ however when they are rotated. The moment of inertia depends strongly on the mass distribution.

There are however still many possible mass distributions that give the same moment of inertia. It is therefore not possible to determine the exact mass distribution from the moment of inertia. So even though the moment of inertia does give some extra information there remains uncertainty.

(side note: In an attempt to get more information on the object one might come up with the idea to measure the moment of inertia around different points on the stick. Unfortunately the parallel axis theorem tells us that all these momenta are related and therefore do not give any extra information)

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  • $\begingroup$ Thanks! Is it possible to mesure the mass of the stick from x-0 to x-5 given one more condition that the stick is infinitely narrow and straight, by using moment of inertia? $\endgroup$ – Gyeonghyo Min Nov 22 '16 at 12:46
  • $\begingroup$ No, with the exception of some special cases. Special cases could be for instance: center of mass at x=5, or moment of inertia so big that all mass has to be at ends of stick, or moment of intertia zero so that all mass has to be at center. The last two options are only theoretically possible. $\endgroup$ – Crimson Nov 22 '16 at 14:06

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