The Shell Theorem and A Problem Related to it

Question

The shell theorem states that:

A uniform spherical shell of matter attracts a particle that is outside the shell as if all the shells mass were concentrated at its center.

Now, I have to solve a problem related to this theorem. The statement of the problem is:

A particle is to be placed, in turn, outside four objects, each of mass $m$:

(1) a large uniform solid sphere,

(2) a large uniform spherical shell,

(3) a small uniform solid sphere, and

(4) a small uniform shell.

In each situation, the distance between the particle and the center of the object is $d$. Rank the objects according to the magnitude of the gravitational force they exert on the particle, greatest first.

As far as I understand this problem (or maybe I've not understood it yet) I think all of the objects should be ranked first. This is because, for the large uniform solid sphere if we imagine that its entire mass $m$ is concentrated at its center which is at a distance $d$ from the particle, then it will attract the particle with the magnitude $\frac{Gmm_0}{d^2}$ where $m_0$ is the mass of the particle. And, each of the other three objects' centers distance is $d$ from the particle, and each of their masses is $m$, so it seems to be the case that all of them gravitate the particle with the same magnitude.

Do the objects gravitate the particle with different magnitudes, and hence should be ranked differently? What is the reason behind it if so is the case?

Answer 1

You are correct - the force is constant in all four cases. Since each of the situations describes a "uniform spherical shell of matter," you can assume that the mass is concentrated at the center of that shell, as per the shell theorem cited.

If you've learned Gauss's Law for electric fields, it can be applied to this problem. Gravitational force, following the same inverse square relationship as the Coulomb force, also obeys Gauss's Law. Set up a spherical Gaussian surface concentric with the spherical shells and passing through the particle. The total gravitational flux through this surface is constant in all four cases, since the total mass enclosed is constant. Moreover, since each sphere is uniform, the gravitational force is evenly distributed across the surface. Therefore, the gravitational force on the particle is the same in all four cases.