Newton's law of gravitation, find the mass and radius of the sphere

[Sunsethorizon]

Homework Statement

Two fully equal sphere's of lead are placed next to each other so that the gravitational force between sums up to 10N. Calculate mass and radius of the two sphere's.

F=10N , ρ_lead=11300kg/m²

Homework Equations

F=gm, F=GMm/r² , V=4πr³/3 , ρ=m/V

The Attempt at a Solution

This has bugged me for quite some time now. With only gravitational force and density known, any attempt i do to calculate either mass or radius end's up with either two unknown variables or the wrong answer.
I am missing something crusial. Please Point me in the right direction :).

 

[Steven Thomas]

Homework Statement

Two fully equal sphere's of lead are placed next to each other so that the gravitational force between sums up to 10N. Calculate mass and radius of the two sphere's.

F=10N , ρ_lead=11300kg/m²

Homework Equations

F=gm, F=GMm/r² , V=4πr³/3 , ρ=m/V

The Attempt at a Solution

This has bugged me for quite some time now. With only gravitational force and density known, any attempt i do to calculate either mass or radius end's up with either two unknown variables or the wrong answer.
I am missing something crusial. Please Point me in the right direction :).

When it says placed next to each other, does that mean the spheres are touching one another?
If so you can rewrite the one radii in terms of the other and you only have one unknown.

 

[Sunsethorizon]

When it says placed next to each other, does that mean the spheres are touching one another?
If so you can rewrite the one radii in terms of the other and you only have one unknown.

It acctually states that the spheres are placed so that the gravitational force sums up to 10N. It does not state that the two spheres are touching.

 

[ehild]

It acctually states that the spheres are placed so that the gravitational force sums up to 10N. It does not state that the two spheres are touching.

The problem states that

Two fully equal sphere's of lead are placed next to each other

What do you think "next to each other " can mean in this context?

 

[Steven Thomas]

@Sunsethorizon If you attempt to solve assuming that they are touching, do you get the correct answer? Otherwise, as you say before, I think you will have two unknowns, both the mass / radius of he lead spheres and their separation.

 

[Sunsethorizon]

@Sunsethorizon If you attempt to solve assuming that they are touching, do you get the correct answer? Otherwise, as you say before, I think you will have two unknowns, both the mass / radius of he lead spheres and their separation.

If i assume that the two sphere's are touching. How can i use that to go forward when the gravitational forces between originate from each sphere's center.
it the two spheres touch, does that eliminate r from F=GMm/r2?

The problem states that
What do you think "next to each other " can mean in this context?

It probably means that the two spheres does touch. But i don't know how to proceed.

The correct answer for the radious is r=4.04m.

 

[ehild]

If i assume that the two sphere's are touching. How can i use that to go forward when the gravitational forces between originate from each sphere's center.
it the two spheres touch, does that eliminate r from F=GMm/r2?

No. The gravitational force between two spheres is as if all mass of each sphere was concentrated in the center.

 

[Steven Thomas]

Let's assume they are touching. We call the radius of the spheres r and the separation between the two spheres (centre to centre) R. When they touch we have R = 2r. You can substitute this into the equation for the force of gravity, and substitute for the mass in terms of density and volume, then sub volume of a sphere. You will now have only one unknown, r.

Whether or not this turns out to be the correct answer I'm not sure, calculate it and let me know. If not, then our assumption that they touch was incorrect.

 

[Sunsethorizon]

Let's assume they are touching. We call the radius of the spheres r and the separation between the two spheres (centre to centre) R. When they touch we have R = 2r. You can substitute this into the equation for the force of gravity, and substitute for the mass in terms of density and volume, then sub volume of a sphere. You will now have only one unknown, r.

Whether or not this turns out to be the correct answer I'm not sure, calculate it and let me know. If not, then our assumption that they touch was incorrect.

The two sphere's did touch, i finally solved it. Thanks for your kind help :)

Since m₁=m₂ , equation for force can be stated F=GM²/r²

with r=2r and substitute for mass the final equation is F=G(ρ4πr³/3)²/(2r)² , solved for r does give the correct anwer for radius.