For continuous mass distribution they introduce linear, surface and volume mass density (represented by $\lambda$, $\sigma$ and $\rho$ constants). I just want to know what they mean. Also, in books they say to use linear density for rings, linear bodies; surface density for hollow sphere, plates. So, on what basis do they use these constants on different bodies, like let's say I want to use them on solid sphere so which one should I use?
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3 Answers
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For a 1D object, use linear mass density. For example, along the x-axis, one can specify a mass density $\rho(x)$ for every point on the x-axis.
For a 2D object, use surface mass density. For example, on the 2D surface of a sphere of radius $R$, one can specify a mass density $\rho(\theta,\phi)$ for every point on the sphere, where $\theta $ & $\phi$ are the azimuthal angles.
For a 3D object, use volume mass density. For example, for a 3D solid sphere, one can specify a mass density $\rho(r,\theta,\phi)$ for every point within or on the solid sphere.
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$\begingroup$ Hmm, "on the 2D surface of a sphere of radius R", isn't sphere a 3D shape even if you consider a hollow sphere? $\endgroup$– AnkitCommented Dec 15, 2020 at 11:35
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1$\begingroup$ @Ankit The hollow sphere is a 3D object. But the surface of the hollow sphere is a 2D object. It's similar to saying a piece of paper is a 3D object. But if you draw on one side of the paper, you are drawing on a 2D surface. $\endgroup$– TaeNyFanCommented Dec 15, 2020 at 11:52
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Linear density means the change in mass or charge per unit length (1D)
For surface charge density mass or charge per unit area (2D)
Volume density is charge or mass density per unit volume (3D)
These are the properties of a material
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It depends on what you want. Are you looking for something along 1D, 2D or 3D geometry? Some exapmles:
2D: Do you want to know the amount of paint that you must use to paint the roof? Possibly certain parts of the woodwork suck in more of the tint and require more paint. You could talk about the paint concentration or surface density being uneven.
1D: Are you making candles, dipping the wick in molten stearine wax, then you might want to know if you kept it down in that bath for too long at some point along the wick. That point might have accumulated a higher concentration. The linear density is not even.
3D: Finally, the most typical one is the 3D version called volume density (often simply called density) which is what you are interested in when stuffing your pillow with goose feathers. It should be even with the same concentration, the same volume density, throughout.
