Calculating Mass Density & Enclosed Mass in Grav Field $\vec{g}(x,y,z)$

In summary: Gauss's theorem for the gravitostatic field and that has a minus in the RHS so it turned out that way on my calculator but its okay cause i got the answer right in the end
  • #1
Phymath
184
0
[tex] \vec{g}(x,y,z) = -kG((x^3 y^2 z^2)\hat{e_x} + (x^2 y^3 z^2)\hat{e_y} + (x^2 y^2 z^3) \hat{e_z})[/tex] given this grav field (k is constant)

find the mass density of the source of this field, and what is the total mass in a cube of side 2a centered about the origin?

hmmm well we all know...[tex]\int \int \vec{g} \bullet d\vec{a} = 4 \pi G m_{enclosed}[/tex]

and density [tex] p = \frac{m}{V} [/tex] at least the overall density of it is (non-differential)

sooooo...[tex]\frac{1}{4 \pi G}\int\int\int (\nabla \bullet \vec{g}) dV = m_{source/enclosed}[/tex]

now the limits i made a cube of side 2a, because the flux through a box is easier when g is given in cart coords...any way i get...

[tex]m_{source} = \frac{-2}{3 \pi}G k a^9[/tex] how do I get a neg mass (unless this is dark matter which it very well could be) and I'm thinking i missed something about density cause why would it ask that first and then the mass enclosed second?...a lil help?
 
Last edited:
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  • #2
Are u sure u set the integratiing limits from -a to +a for all three integrals?

Daniel.
 
  • #3
yes...i am sure
 
  • #4
You forgot about the minus in the RHS of Gauss's theorem for the gravitostatic field...

[tex] \oint\oint_{\Sigma} \vec{g}\cdot d\vec{S}=\mbox{-}4\pi G m_{\mbox{enclosed by}\ \Sigma} [/tex]

Daniel.
 
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  • #5
i see so that just answers the negative mass problem, now how is it that this grav field increases as you move farther away from the "source" this tells me that the "mass" enclosed is ever increasing i suppose so i guess having the mass enclosed as a function of the flux box side is alright what do you think?
 
  • #6
anyone? anyone? buler buler?
 
  • #8
ok if [tex]\frac{1}{4 \pi G}\int\int\int (\nabla \bullet \vec{g}) dV = m_{source/enclosed}[/tex] then...ill just do it and show you

[tex]\nabla \bullet \vec{g} = -9kGx^2y^2z^2[/tex]
[tex]\frac{9k}{4 \pi }\int^a_{-a}\int^a_{-a}\int^a_{-a} (x^2y^2z^2) dxdydz = \frac{2}{3}k \pi a^9 = m_{source/enclosed}[/tex] so as my "flux cube" getts larger obviously it isn't limiting at some value as the function is ever increasing so, the mass should always be increasing yes?
 
  • #9
Of course the mass increases,but you said about the field.It's not the same thing...:wink:


Daniel.
 
  • #10
alright cool thanks man all that for yes u did it right except a negative sign lol
 

Related to Calculating Mass Density & Enclosed Mass in Grav Field $\vec{g}(x,y,z)$

What is mass density?

Mass density is the measure of how much mass is contained within a given volume. It is typically denoted by the symbol ρ (rho) and is expressed in units of mass per unit volume, such as kilograms per cubic meter (kg/m³).

How do you calculate mass density?

Mass density can be calculated by dividing the mass of an object by its volume. The formula is ρ = m / V, where ρ is mass density, m is mass, and V is volume. For irregularly shaped objects, mass density can also be determined by measuring the displacement of water when the object is submerged in it.

What is enclosed mass in a gravitational field?

Enclosed mass in a gravitational field refers to the total mass contained within a given radius or distance from a central point, such as a planet or star. It is an important concept in understanding the strength and effects of gravitational fields.

How do you calculate enclosed mass in a gravitational field?

Enclosed mass in a gravitational field can be calculated using the formula M = g * r² / G, where M is the enclosed mass, g is the gravitational field strength, r is the distance from the central point, and G is the gravitational constant.

What is the relationship between mass density and enclosed mass in a gravitational field?

Mass density and enclosed mass in a gravitational field are directly related. The higher the mass density of an object, the more mass it contains within a given volume and the stronger its gravitational field will be. This means that objects with higher mass density also have higher enclosed mass in a gravitational field.

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