Shell theorem: Gravity of a hollow planet

[Carbon123]

Homework Statement

There is a planet (spherical) with a hollow that is concentric with the planet.if the inner radius is r and outer radius is R and mass of the planet is M what would the gravity be outside of the planet at distance x from the center ?

Homework Equations

Shell theorem
Universal law of gravity

The Attempt at a Solution

I am confused about whether you can treat hollow sphere as if the mass is concentrated in the middle or if you have to subtract the gravity of the large sphere with gravity of the cavity. So , is it correct to say that the gravity at distance x is actually gravity of a planet with radius R minus gravity of planet with radius r ?

 

[gneill]

[Note: I've changed your thread title to be a bit more descriptive of the problem.]

Can you state the Shell Theorem?

 

[Carbon123]

[Note: I've changed your thread title to be a bit more descriptive of the problem.]

Can you state the Shell Theorem?

The shell theorem states that for gravity outside of a spherical body it can be treated as if the mass was concentrated on the center.

 

[gneill]

The shell theorem states that for gravity outside of a spherical body it can be treated as if the mass was concentrated on the center.

Okay, that's the relevant part of the shell theorem, if a bit loosely stated. Can you see how this directly answers your question?

A bit more precisely (quote from the Wikipedia article on the Shell Theorem):

A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point at its centre.

The "spherically symmetric" stipulation is important: A uniform spherical shell is spherically symmetric in terms of mass distribution, despite the "missing" center.

 

[Carbon123]

So both methods are valid ?

 

[gneill]

So both methods are valid ?

Outside the body the gravity behaves as though all the mass were concentrated in a point at the center; That's it. So think in terms of the total mass.

The issue of the hollow interior only comes up if you don't actually know the total mass and you need to calculate it. For example, suppose you are given the density of the material and the two radii. In order to find the mass of the shell you might consider subtracting the volume of the interior cavity from the volume of the surrounding sphere. Multiply the residual volume by the density and you have the total mass of the shell.

Your statement, "the gravity at distance x is actually gravity of a planet with radius R minus gravity of planet with radius r", is true so long as the densities of both are the same.

 

[gneill]

I think I might be overcomplicating things for you. Let me answer more succinctly:

I am confused about whether you can treat hollow sphere as if the mass is concentrated in the middle or if you have to subtract the gravity of the large sphere with gravity of the cavity. So , is it correct to say that the gravity at distance x is actually gravity of a planet with radius R minus gravity of planet with radius r ?

Yes you can treat the hollow sphere as if all the mass is concentrated at a central point.

Yes, you can find the gravity of a hollow planet by subtracting the effect of the smaller planet from the larger solid planet.

Both are true.

 

[Carbon123]

Outside the body the gravity behaves as though all the mass were concentrated in a point at the center; That's it. So think in terms of the total mass.

The issue of the hollow interior only comes up if you don't actually know the total mass and you need to calculate it. For example, suppose you are given the density of the material and the two radii. In order to find the mass of the shell you might consider subtracting the volume of the interior cavity from the volume of the surrounding sphere. Multiply the residual volume by the density and you have the total mass of the shell.

Your statement, "the gravity at distance x is actually gravity of a planet with radius R minus gravity of planet with radius r", is true so long as the densities of both are the same.

Thanks for clearing my confusion