Small mass element for laminar (moment of inertia)

AI Thread Summary
The discussion focuses on the moment of inertia for a 2D lamina, specifically the formulation of the small mass element dM. The user initially uses dM = ρdxdy, where ρ is the area density, but later proposes M(x,y) = ρxy, leading to a different expression for dM. This new formulation suggests that mass increases with x and y, which is not typically desired in modeling. The key point raised is that if mass density varies with position, then ρ should be a function of x and y rather than a constant. Understanding these differences is crucial for accurate calculations in the moment of inertia tensor.
jamie.j1989
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Hi, I was just going over the moment of inertia for a 2D lamina, I've been happy with writing the small mass element dM as dM = ρdxdy where ρ is the area density, but for some reason decided on doing it like this,

M(x,y) = ρxy

so

dM = \frac{∂M}{∂x}dx + \frac{∂M}{∂y}dy
= ρ(ydx + xdy)

This is obviously not the same, and does not give the same answer when substituting in for dM in the moment of inertia tensor. I would like to know if there's a difference between the dM's, and if so why? Thanks.
 
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M(x,y) = ρxy

This means the mass increases with increasing x and y which I don't think is what you want to model. But if it was then ρ is not constant and should be a function of x and y instead.
 

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