Finding Density of a Point in Sphere Using Mass & Radius

[Yanah]

Could someone please tell me how to find the density of a point in a sphere using the overall mass and radius of the sphere, due only to gravitational forces? I can't for the life of me find any actual numbers relating density and pressure.

*EDIT* Oh yeah, it's all the same material, so don't worry about that.

 

[berkeman]

Is the material compressible or incompressible? Are the gravitational forces due just to the material in the sphere (like for a sun or planet), or are you talking about a small sphere of material sitting on a table in front of you here on the surface of the earth? Is this a homework problem?

 

[Yanah]

I just wanted to know because it's something I don't know.

I mean that how would you find the density of a point in a sphere with no other influence, using only the mass and radius of the object and depth of the point. Just pretend that it can be compressed infinitely.

 

[Integral]

$$D = \frac m V$$

D = Density
m = mass
V = Volumn

If you want something different from that then you must explain yourself a bit more.

 

[berkeman]

I mean that how would you find the density of a point in a sphere with no other influence, using only the mass and radius of the object and depth of the point. Just pretend that it can be compressed infinitely.

Well, assuming the sphere is all by itself with no other gravitational infulences, then you have a gaseous sphere whose density increases as you get closer to the center. There are two things to use in calculating the density as a function of radius r. First, for any radius r, the pressure at that radius comes about from the weight of the gas above it. The mass of the gas above a point will depend on the density function that you are trying to derive.

The second thing to use is the ideal gas law: PV = nRT. You can google ideal gas law for more info on what the variables stand for and how to use it. The P in the equation is the pressure of the gas, and V is the volume.

You'll end up with a couple simultaneous equations, including an integration of the pressure function over the radius. In the end, you should be able to find the density as a function of radius, based on the total mass of the gas sphere.

 

[rbj]

Well, assuming the sphere is all by itself with no other gravitational infulences, then you have a gaseous sphere whose density increases as you get closer to the center. There are two things to use in calculating the density as a function of radius r. First, for any radius r, the pressure at that radius comes about from the weight of the gas above it. The mass of the gas above a point will depend on the density function that you are trying to derive.

The second thing to use is the ideal gas law: $PV = nRT$. You can google ideal gas law for more info on what the variables stand for and how to use it. The P in the equation is the pressure of the gas, and V is the volume.

You'll end up with a couple simultaneous equations, including an integration of the pressure function over the radius. In the end, you should be able to find the density as a function of radius, based on the total mass of the gas sphere.

the OP will have to set up a differential equation to solve for P(r), and will have to make an assumption regarding temperature as a function of r

the gas law $PV = nRT$ can be rewritten to be

$$PV = m \frac{n}{m}RT = m \hat{R}T$$

where $$\hat{R} = \frac{n}{m}R$$

is another form of the gas constant that is a function of the molecular weight of the gas (mass per mole or m/n). that gets turned around to represent density or specific volume

$$P = \hat{R} \frac{m}{V} T = \hat{R} \rho T$$

after a little more scribbling on a piece of paper, i'll set up the diff eq.

if this is not what the OP wants, i'd appreciate knowing about it, because this is where the work begins.

 

[Yanah]

Thank you very much, that's exactly what I needed.

I really should have Googled it, but I didn't want to spend that much time searching through endless search results with no relevancy to my query.

 

[silver-rose]

lol...any A level/ IB/ AP textbook would've done the job =)

 

[berkeman]

lol...any A level/ IB/ AP textbook would've done the job =)

I disagree. This is a non-trivial problem, IMO.