In the interior of a sphere of uniform density, the gravitational strength is proportional to the radius, exactly like Hooke's law for an ideal spring. That's why the object in the bore hole undergoes harmonic motion.
Why is the graviational strength proportional to the radius?
Firstly, you have to know that the field strength is zero inside a hollow sphere. This is part of that shell theorem.
So for a point at a given depth inside the sphere, we can divide the sphere into a hollow sphere consisting of everything less deep, and a solid sphere consisting of everything deeper. Only the deeper sphere matters; we can ignore the hollow sphere.
So as we progress toward the centre, the attraction is due to a smaller and smaller sphere, whose mass is proportional to r^3. The radius is shrinking though, which has the effect of increasing gravity: the gravitational field strength is proportional to 1/r^2. Wen we combine these factors, we get r.
Note that this article is by the same Greg Egan who wrote Permutation City, a (in my opinion) really good, deeply technical, hard science fiction novel exploring consciousness, computation, and the infinite nature of the universe.
If that sounds interesting, I recommend not reading too much about the book before starting it; there are spoilers in most synopses.
You don't necessarily need a background in programming and theoretical computer science to enjoy it. But you'll probably like it better if you already have some familiarity with computational thinking.
Because of the shell theorem mentioned in the article, any straight tunnel between two points on the surface of a sphere would take the exact same amount of time to traverse under gravitational acceleration (assuming no air resistance and uniform density of the sphere). In the case of the Earth, this time would be approximately 42 minutes.
FYI Greg Egan is practically his own genre of ultra hard math heavy sci-fi that I highly recommend to anyone who knows what partial differential equations are.
In the interior of a sphere of uniform density, the gravitational strength is proportional to the radius, exactly like Hooke's law for an ideal spring. That's why the object in the bore hole undergoes harmonic motion.
Why is the graviational strength proportional to the radius?
Firstly, you have to know that the field strength is zero inside a hollow sphere. This is part of that shell theorem.
So for a point at a given depth inside the sphere, we can divide the sphere into a hollow sphere consisting of everything less deep, and a solid sphere consisting of everything deeper. Only the deeper sphere matters; we can ignore the hollow sphere.
So as we progress toward the centre, the attraction is due to a smaller and smaller sphere, whose mass is proportional to r^3. The radius is shrinking though, which has the effect of increasing gravity: the gravitational field strength is proportional to 1/r^2. Wen we combine these factors, we get r.
It’s really odd how often and where damped spring motion comes up
Note that this article is by the same Greg Egan who wrote Permutation City, a (in my opinion) really good, deeply technical, hard science fiction novel exploring consciousness, computation, and the infinite nature of the universe.
If that sounds interesting, I recommend not reading too much about the book before starting it; there are spoilers in most synopses.
https://en.wikipedia.org/wiki/Permutation_City
You don't necessarily need a background in programming and theoretical computer science to enjoy it. But you'll probably like it better if you already have some familiarity with computational thinking.
Because of the shell theorem mentioned in the article, any straight tunnel between two points on the surface of a sphere would take the exact same amount of time to traverse under gravitational acceleration (assuming no air resistance and uniform density of the sphere). In the case of the Earth, this time would be approximately 42 minutes.
FYI Greg Egan is practically his own genre of ultra hard math heavy sci-fi that I highly recommend to anyone who knows what partial differential equations are.
It's a one-dimensional orbit, right?