# Shell theorem: Gravity of a hollow planet

• Carbon123
In summary, the shell theorem states that for gravity outside of a spherical body, the body can be treated as if all its mass is concentrated at the center. This means that the gravity at a distance x from the center is equal to the gravity of a solid planet with radius R minus the gravity of a solid planet with radius r. However, this only applies if the densities of both planets are the same. The issue of a hollow interior only arises when trying to calculate the total mass of the 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 ?

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

Can you state the Shell Theorem?

gneill said:
[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.

WinstonC said:
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.

So both methods are valid ?

WinstonC said:
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.

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

gneill said:
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

## 1. How does the Shell theorem explain the gravity of a hollow planet?

The Shell theorem states that the gravitational pull of a hollow planet is the same as that of a solid planet of the same mass, as long as the observer is outside of the hollow portion. This is because the mass inside the hollow portion has no net gravitational effect on the observer.

## 2. What factors affect the gravity of a hollow planet?

The gravity of a hollow planet is primarily determined by its mass and the distance from the center of mass. The shape of the hollow portion and the distribution of mass within it can also have an impact on the gravitational pull.

## 3. Can a hollow planet have a stronger gravitational pull than a solid planet?

No, according to the Shell theorem, the gravitational pull of a hollow planet will always be equal to or less than that of a solid planet of the same mass. This is because the mass inside the hollow portion does not contribute to the overall gravitational pull on an external object.

## 4. Does the Shell theorem apply to all types of hollow planets?

The Shell theorem applies to any spherical or nearly-spherical hollow planet. For irregularly-shaped hollow planets, the gravitational pull may vary depending on the distribution of mass within the hollow portion.

## 5. How does the Shell theorem affect the gravity experienced by objects on the surface of a hollow planet?

The Shell theorem only applies to objects outside of the hollow portion of the planet. On the surface of a hollow planet, the gravity experienced will depend on the mass and distribution of mass within the hollow portion, as well as the mass of the planet itself.

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