- #1

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Given a square with a linear mass density of:

λ(x) = a * x (see image below where black is high density and white is low density)

how would you deduce what the surface mass density is?

I get confused for the following reason:

To me it seems that the surface mass density should depend on x, because the further left or right you go the linear density changes, so why wouldn't the surface density change as well. A surface mass density must have units of $$\frac{m}{L^2}$$ so since λ(x) has units of $$\frac{m}{L}$$ I would think I need to divide λ(x) by some length. I don't think you would divide by y (the height coordinate) because the density is translationally invariant in y. But that means you have to divide by x to get the right units, and that would mean that the surface mass density is a constant, which doesn't make physical sense to me.

λ(x) = a * x (see image below where black is high density and white is low density)

how would you deduce what the surface mass density is?

I get confused for the following reason:

To me it seems that the surface mass density should depend on x, because the further left or right you go the linear density changes, so why wouldn't the surface density change as well. A surface mass density must have units of $$\frac{m}{L^2}$$ so since λ(x) has units of $$\frac{m}{L}$$ I would think I need to divide λ(x) by some length. I don't think you would divide by y (the height coordinate) because the density is translationally invariant in y. But that means you have to divide by x to get the right units, and that would mean that the surface mass density is a constant, which doesn't make physical sense to me.