Shell Theorem Q: Understand Gravity Inside/Outside Hollow Sphere?

[mgkii]

I've just watched half a dozen or so videos on shell theorem and I just can't get my head around something that none of the videos address directly, but seems so counter-intuitive I am assuming my understanding is incorrect. Can anyone help me out here?

With all the usual simplifying conditions applying such as a "the Earth is uniformly dense, a perfect sphere, etc", I was guided through hairy calculus (hairy to me anyway!) to demonstrate:

1) If I stand on or above the surface of the Earth and you secretly replaced the Earth with a hollow sphere of the same mass... I would not notice (shhh!) - i.e. both the uniformly dense Earth and hollow-sphere Earth act on my body as if gravity emanates from a point in the very centre of the earth.

2) If I were inside the hollow earth, then I would be effectively weightless as gravitational forces balance out at all points inside the sphere. (Shell Theorem)

Extending the conditions so that the thickness of the hollow shell is negligible, then the counter-intuitive conclusion I draw, is that as I approach hollow-earth from a distant point my weight increases to a maximum at the surface of the sphere, and then instantly goes to zero as I cross the threshold. I'm clearly missing something key here!

Thanks for any insight you can give
Matt

 

[Geofleur]

Think of it this way. The moment you cross the boundary, there is mass on both sides of you. There is the relatively small amount of Earth very nearby (that you just passed through to get into the sphere), but its gravity is large because you are near it (think of Newton's law of gravitation with small $m$ but also small $r^2$), and there is the large amount of Earth that is far away, but its gravity is strong because there is more of it (big $r^2$ but also big $m$). The forces pull in opposite directions and, in the end, they cancel each other. Also, when you imagine that the shell can be infinitely thin, realize that you are also imagining it to be infinitely dense, because the mass of the Earth has to be packed in there!

 

[mgkii]

Of course! Like so many things... completely counter-intuitive at first sight (and sixth for me), but makes perfect sense when someone "turns the light on" for you! Many thanks.