A point mass inside a spherical shell

[amiras]

I am having some difficulties understanding something here, it seems to me that the book at some point deny itself or I clearly do not get it.

So it firstly states that:
Inside the spherical shell the potential energy does not depend on radius from the center of the shell to the point of mass m and has the same value everywhere inside the shell.

So the potential energy between the spherical shell and the point mass m is:
U = -GMm/R, where M is the mass of the shell, and m - mass of a point mass inside the shell, R is the radius of the shell. So clearly this potential energy does not depend on where it is inside the shell and is always constant.

Since the force is the negative gradient of the potential energy function, and U=const. the Force must be zero (everywhere inside the shell?)

But now the book says:
More generally, at any point in the interior of any spherical symmetric(radius R) mass distribution, at a distance r from its center, the gravitational force on a point mass m is the same as though we removed all the mass at points farther than r from the center and concentrated all the remaining mass at the center.

So now it states that the force is not zero and can be calculated. First it said that the force on m at any point inside the shell must be zero. What am I missing here?

 

[Fightfish]

Since the force is the negative gradient of the potential energy function, and U=const. the Force must be zero (everywhere inside the shell?)

That is correct. This was proven by Newton in his Shell Theorem.

But now the book says:
More generally, at any point in the interior of any spherical symmetric(radius R) mass distribution, at a distance r from its center, the gravitational force on a point mass m is the same as though we removed all the mass at points farther than r from the center and concentrated all the remaining mass at the center.

If you would read this part carefully, it is not referring to just spherical shells but any spherical mass distribution, thus there is no contradiction. In fact, this statement is a rewording of the Shell Theorem, because we can consider any spherical mass distribution to be composed out of many spherical shells. That we can "remove all the mass at points farther than r from the center" is a consequence of the earlier result that the force on the point mass due to these outer spherical shells is zero because the point mass lies within these shells!

 

[Michael C]

I am having some difficulties understanding something here, it seems to me that the book at some point deny itself or I clearly do not get it.

So it firstly states that:
Inside the spherical shell the potential energy does not depend on radius from the center of the shell to the point of mass m and has the same value everywhere inside the shell.

So the potential energy between the spherical shell and the point mass m is:
U = -GMm/R, where M is the mass of the shell, and m - mass of a point mass inside the shell, R is the radius of the shell. So clearly this potential energy does not depend on where it is inside the shell and is always constant.

Since the force is the negative gradient of the potential energy function, and U=const. the Force must be zero (everywhere inside the shell?)

Yes

But now the book says:
More generally, at any point in the interior of any spherical symmetric(radius R) mass distribution, at a distance r from its center, the gravitational force on a point mass m is the same as though we removed all the mass at points farther than r from the center and concentrated all the remaining mass at the center.

So now it states that the force is not zero and can be calculated. First it said that the force on m at any point inside the shell must be zero. What am I missing here?

There's no contradiction. The second case is not just a shell: there can also be mass within the radius r. Consider a solid sphere of radius R. If we have a point mass at radius r from the centre, we can imagine the sphere divided into two parts:
1. A sphere of radius r
2. A spherical shell of inner radius r and outer radius R
The gravitational force of the sphere of radius r on the point mass is the same as if all its mass were concentrated at its centre. The gravitational force of the remaining shell on the point mass is zero.

 

[torquil]

But now the book says:
More generally, at any point in the interior of any spherical symmetric(radius R) mass distribution, at a distance r from its center, the gravitational force on a point mass m is the same as though we removed all the mass at points farther than r from the center and concentrated all the remaining mass at the center.

So now it states that the force is not zero and can be calculated.

No, this also gives zero in the case of your example with the massive shell. :-)

 

[amiras]

Thank you for your explanation, all this starts to make sense to me now! Thanks :)