B Mass estimate only through mass rate of change

[FranzS]

TL;DR Summary

How to estimate the mass of partially non-homogeneous bodies.

Hello,

do you ever get the feeling that you cannot grasp an apparently easy concept? Like setting up the problem, writing down some equations and not knowing how to go on?
Well, I'm in that situation right now.

So, I would like to estimate the mass of certain objects by knowing only and precisely how the mass changes when the object dimensions change. The problem is the objects do not have a homogeneous density.
This smells like solving an indefinite integral and getting stuck with an unknown $+C$, so to speak.
Therefore, a part of my intuition tells me I have no chance of finding a solution (not even an approximate one).

However, I'll explain what another part of my intuition tells me with the following simplified example.
Let's suppose I have a cylinder (meant as the basic geometrical shape) with base radius $R$ and height $x_0$. The cylinder is made of steel but has some holes ("air bubbles") inside, hence I cannot calculate its exact mass $m_0$ using its outside volume (and the steel density $\rho_s$). I can only give a rough estimate of it, and I'll call it $\tilde{m}_0$.
Then, I can somehow extend this cylinder to a height $x>x_0$, thus making it longer/taller by an amount $\Delta x = x - x_0$, and I'll do that by adding pure steel to the original cylinder.
Therefore, I'm adding an exact mass $\Delta m (x) = \rho_s \cdot \left( \pi R^2 \Delta x \right)$ and my estimate for the overall mass of the extended cylinder will be $\tilde{m} (x) \approx \tilde{m}_0 + \Delta m (x)$, whereas the actual mass is $m(x) = m_0 + \Delta m (x)$.
Now, as I extend the cylinder more and more, always by adding pure steel, the overall mass will be approximately equal to the added mass only. In other words:
$$
\lim_{x\to\infty} \frac{\tilde{m}(x)}{m(x)} = 1
$$
On the $x, y$ plane (with $y$ being the mass), $\tilde{m}(x)$ and $m(x)$ would be two parallel lines that always differ by the quantity $\lvert \tilde{m}_0 - m_0 \rvert$ and it's only the relative/percentage error between $\tilde{m}(x)$ and $m(x)$ that decreases as $x$ increases.

However, as said above, part of my intuition tells me that I should be able to use the more accurate values of $\tilde{m}(x \gg x_0)$, which are acceptable estimates, in order to correct the less accurate values $\tilde{m}(x \sim x_0)$, in a sort of feedback / retrofit style.
Is my intuition wrong?

Thanks to everyone who'll be willing to chime in.

 

[Mark44]

So, I would like to estimate the mass of certain objects by knowing only and precisely how the mass changes when the object dimensions change. The problem is the objects do not have a homogeneous density.
This smells like solving an indefinite integral and getting stuck with an unknown +C, so to speak.
Therefore, a part of my intuition tells me I have no chance of finding a solution (not even an approximate one).

In essence, the problem you're trying to solve is a differential equation of the form $\frac {dm}{dt} = \text{some function of t}$. (I'm assuming that the rate of change of mass is with respect to time t.) The solution will be a family of functions that differ from each other by a constant. To nail down the precise function you need to know the mass at some specific time.

 

[Steve4Physics]

... However, as said above, part of my intuition tells me that I should be able to use the more accurate values of $\tilde{m}(x \gg x_0)$, which are acceptable estimates, in order to correct the less accurate values $\tilde{m}(x \sim x_0)$, in a sort of feedback / retrofit style.
Is my intuition wrong?

I believe it is wrong. Consider two example cases for the initial cylinder of length $x_0$:
1) the cylinder is made pure steel;
2) the cylinder is made of steel with air bubbles.

In both cases, adding known amounts of extra pure steel will increase the total length in exactly the same way.

So, knowing only the added mass of steel and the new length, there is no way to distinguish between 1) and 2). You can never find $m_0$ this way.

 

[FranzS]

In essence, the problem you're trying to solve is a differential equation of the form $\frac {dm}{dt} = \text{some function of t}$. (I'm assuming that the rate of change of mass is with respect to time t.) The solution will be a family of functions that differ from each other by a constant. To nail down the precise function you need to know the mass at some specific time.

Thanks for offering this point of view. In this case, not knowing $m(t_0) = m_0$, it makes no sense to use $m(t \gg t_0) \approx \Delta m(t \gg t_0)$ as an estimate for the value at a certain point, because I would conclude that $m_0 =0$ and that's useless.

 

[FranzS]

I believe it is wrong. Consider two example cases for the initial cylinder of length $x_0$:
1) the cylinder is made pure steel;
2) the cylinder is made of steel with air bubbles.

In both cases, adding known amounts of extra pure steel will increase the total length in exactly the same way.

So, knowing only the added mass of steel and the new length, there is no way to distinguish between 1) and 2). You can never find $m_0$ this way.

Thanks for your reply. The more I think about it, the more I'm convinced that my intuition was completely wrong.