Gravitational Force in terms of Density?

[EthanVandals]

Homework Statement

Express the gravitational force equation in terms of density.

Homework Equations

F(Gravity) = ((GravitationalConstant)(Mass1)(Mass2))/radius^2
Density = mass/volume

The Attempt at a Solution

Based on the original equation for gravitational force, there are two masses involved in the calculation of the final force dependent on your radius, etc. I'm confused as how to convert it into terms of density however, because there's no volume in the Force(gravity) equation.

 

[zwierz]

It seems something like that
$$\boldsymbol f(\boldsymbol r)=\gamma\int\frac{\rho(\boldsymbol r')}{|\boldsymbol r'-\boldsymbol r|^3}(\boldsymbol r'-\boldsymbol r) dV(\boldsymbol r')$$
here $\boldsymbol f$ is the mass density of gravity: the mass $\rho(\boldsymbol r)dV(\boldsymbol r)$ is exerted by the force $\boldsymbol f(\boldsymbol r) \rho(\boldsymbol r)dV(\boldsymbol r)$; here $\rho$ is density; $dV$ is the infinitesimal volume element; $\boldsymbol r$ is a radius vector

 

[EthanVandals]

It seems something like that
$$\boldsymbol f(\boldsymbol r)=\gamma\int\frac{\rho(\boldsymbol r')}{|\boldsymbol r'-\boldsymbol r|^3}(\boldsymbol r'-\boldsymbol r) dV(\boldsymbol r')$$
here $\boldsymbol f$ is the mass density of gravity: the mass $\rho(\boldsymbol r)dV(\boldsymbol r)$ is exerted by the force $\boldsymbol f(\boldsymbol r) \rho(\boldsymbol r)dV(\boldsymbol r)$; here $\rho$ is density; $dV$ is the infinitesimal volume element; $\boldsymbol r$ is a radius vector

Thank you for the rapid response! Our class is algebra based physics, and our professor does not want us using things like integrals and other pieces of calculus. Is there a way to determine it algebraically? Thanks!

 

[TomHart]

Density = mass/volume

Can you rearrange that equation, solving for mass, then plug that back into the equation for gravity? That's the only thing I can figure. It seems like an unusual problem.

 

[haruspex]

Express the gravitational force equation in terms of density.

I suspect the question is worded poorly. My guess is it should be "express the equation for gravitational acceleration at the surface of a uniform sphere in terms of the radius and density of the sphere".
In the form given, it can be answered, but you would need to involve two densities and three "radii": the radius of each of two spheres and the distance between the centres.

 

[EthanVandals]

Can you rearrange that equation, solving for mass, then plug that back into the equation for gravity? That's the only thing I can figure. It seems like an unusual problem.

That's most likely what I will do. Thank you for the help! :) yeah, it is somewhat unusual..I'm not sure in what scenario I'd use it, but maybe it'll show up sometime in the fluids section.

 

[PeroK]

Homework Statement

Express the gravitational force equation in terms of density.

What about?

$F = \frac{G\rho_1V_1\rho_2V_2}{r^2}$

 

[haruspex]

What about?

$F = \frac{G\rho_1V_1\rho_2V_2}{r^2}$

No, I think it much more likely what I suggested in post #5.