Gravity Decrease in Discs: Rate Explained

[gonegahgah]

Just wondering what the rate of gravitational decrease is in a disc - as opposed to a sphere.

I know the decrease inside a sphere is at a constant rate g₁/r but that once you leave the sphere it then decreases at a squared rate g₂/r².

So, while inside a disc at what rate does the gravity decrease if you are moving out from the centre of the disc towards the rim and then when you continue to move away out from the rim?

Also, a what rate does the gravity decrease if you move from the centre away from the disc in a direction perpendicular to the disc?

Can anyone help me with the answer? Thanks if you can.

 

[gonegahgah]

Wouldn't the rate of gravitational decrease within a disc from centre to edge be at a constant rate also; just like for a sphere?

A sphere would create a greater dimple in space-time than a disc but essentially within their confines the change in gravity would be by a linear amount wouldn't it?

If that is so, wouldn't this effect essentially be why galaxies seem to have a constant rate of decrease in gravity outwards.

It changes from a bulge at the centre to a flatter spiral disc shape outwards. So the centre might act more sphere like graduating out to acting more disc like but essentially this would still be a gradual process so that the rate of gravitational decrease would still be fairly linear wouldn't it?

Or are there problems with with I am saying - such as varying density? Just like there is for a planet scenario where the planet is most dense at it's centre. (Are planets most dense at their centre?) Either way planets vary in density in different places so I guess a galaxy would be similar.

Does the above make sense or am I completely off track?

 

[Nik_2213]

http://en.wikipedia.org/wiki/Alderson_disk

This may help...

 

[gonegahgah]

Hi Nik. Thanks, but doesn't really tell me my answer.

 

[IsometricPion]

Gravity works according to an inverse square law: $$\vec{F}=\frac{GMm\vec{r}}{|\vec{r}|^3}\Rightarrow{}|\vec{F}|=\frac{GMm}{|\vec{r}|^2}$$where G is the universal gravitational constant, M is the mass of one (spherically symmetric or very small) object, m is the mass of the another (similar) object, and r is the vector from one to the other. There is a mathematical property of inverse square laws called http://en.wikipedia.org/wiki/Shell_theorem" of the disk. Note that the acceleration is a constant independent of one's distance from the center of the disk (this property depends on the density being a constant in terms of r and assumes the disk to be of infinitesimal thickness).