My question is, if we were to dig a hole to the centre and carve out a sphere there, would the very central point have no gravity?
My thinking is that the mass is in all directions so the gravity would cancel out in the middle. You’d then have an inside surface you could run around on which would have gravity.
Yes! In fact, as you go down the borehole, gravity will decrease linearly, until it reaches zero at the center. Strictly speaking, not quite linearly, since the earth’s density is not uniform. Thinking of the earth as made up of concentric shells, it turns out that once you’re inside a spherical shell of uniform density, that shell contributes zero to the gravitational field. Outside of it, OTOH, the shell produces gravity as if all its mass were located at the center.
One of my favorite fun facts is that if you were to dig a frictionless borehole to the other side of the world, you could hope in, be accelerated towards the center of the earth, decelerate on your way up to the other side, and pop out with 0 velocity 42 minutes later.
Now, this is the fun part, the 42 minute time frame is true for using gravity to coast through a straight hole between any two locations on Earth's surface!
(Assuming a perfectly uniform density sphere and no friction on your journey of course)
It’s fun to think about, but frictionless means you’d have to pump out the air. So you’d need to wear a space suit, and heat shielding. It’s kinda hot down there.
"... For here, at the very center of this watery globe, there seemed to be no gravity. There was collossal pressure, certainly, pressing in from every side, but one was in effect weightless ..."
My thinking is that the mass is in all directions so the gravity would cancel out in the middle. You’d then have an inside surface you could run around on which would have gravity.