 I learned yesterday that the falloff of gravity for a spherical mass is exactly the same as for a point mass outside of the mass, and just reduces linearly to zero at the center inside the mass. This was surprising to me; I wouldn't have expected such a non-smooth function. (This assumes a uniform distribution of mass of course, which Earth does not have.) 8 comments
@runevision I don't know for discs or anything like that (university said, that's too complicated), but isn't a sphere like a point in the sphere center but with some of the initial distance being inside the source? I'd say that's approriate. @runevision A consideration here is that when you get far enough away from the source of sound you are probably in space and there sound doesn't propagate at all :) @breakin Indeed! but my concern is with controlling the volume of a sound source without it becoming arbitrarily high as the listener approaches the center. And I'm hoping for a better solution than the typical one of just arbitrarily capping the value to some maximum value. @runevision oh that's interesting indeed, I first thought this would be a fairly straight standard thing but colour me surprised @runevision Another (possibly related) fact is that if you want to remove 25% of the surface area of a sphere you only need to cut through at a plane that's half the radius away from the centre, i.e. one that from a 2D perspective is 25% up the sphere. I reckon this and your gravity one come down to similar maths.  @runevision This is because the gravity field anywhere inside a sphere (due to the sphere) is zero - all the directions cancel each other out. The gravity field of a sphere you're not inside is as if all the mass is at the center. So as you travel out, all you're feeling is the sphere "below" you, which is increasing in mass according to r^2, but also its center is getting further away by r. Divide one by the other = linear. It's neat that two simple-but-surprising results give a third. @TomF Yeah I learned about the Shell Theorem as part of learning this. :) It's neat indeed. |
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Gravity is special in that it's a vector and they begin to cancel out inside the sphere, but this is not the case for all things that follows falloffs based on distance, for example light or sound.
I haven't been able to find out what the falloff function is for sound or light for a non-point source (for example a sphere or disc). Anyone know?