I was having an argument with someone concerning this problem:
Initial conditions:
1. You have two spheres of identical mass.
2. The spheres are made of material with the same density (say, steel).
3. One mass is a solid sphere inside the other sphere which is hollow and there is a void space between the two spheres.
4. The void space and the space outside the larger sphere is in vacuum.
5. There are no external sources of gravity or other forces at work in this system other than what each sphere provides.
Question:
Supposing the two spheres are at rest and share the same center of gravity, what would happen to the outer sphere should a rotational force be imparted to the inner sphere?
The argument:
He says that because both masses are identical and since the initial system was static that the net gravitational force is zero that the outer sphere would match rotation with the inner sphere.
I believe that he is ignoring Newtonian mechanics. To get to what I believe to be the answer requires a small amount of setup: (Since I can't actually show a picture, I will have to infer one through writing.)
1. Start by drawing a sphere.
2. Draw another larger sphere centered on the first sphere (since both spheres have equal mass make sure to draw a slightly larger or smaller sphere than the second sphere to represent the overall thickness of the second sphere).
3. From the center of the first sphere, draw a vector through the outer shell of the second sphere.
4. (This may or may not be relevant, but it helps explain my view.) Momentarily ignoring the rest of both spheres and concentrating on only the parts of the spheres affected by the vector drawn, The mass column of the first sphere should be larger than the mass column of the second sphere, giving the first sphere higher gravitational force than the second. (I realize that gravity acts more like a well and that the point source outlined above would actually not be linear but more like a cone of force.)
Using the above points what I say would happen is this: since the net gravitational effect is stronger in sphere 1 (from the perspective of the point sources only) if you expand the model back to the entirety of both spheres and have a force instantaneously impart a rotation to sphere 1, sphere 1 will slowly accelerate sphere 2 into a spin matching the direction of rotation of sphere 1 while sphere 2 provides a negative acceleration to sphere 1 gradually slowing it down until after a period of time, the rates of rotation match up and they both move in unison.
To date, I've only covered Newtonian mechanics equivalent to College Physics for Engineers & Scientists I. While I'm familiar with the concepts, the mathematics behind this is presently out of the range of my capabilities (without doing extensive research into the subject).
What I would like to know is who is right (if either of us are), and how to show the correct solution to this argument (Mathematical formulas with solutions would be considered a great bonus.) Thanks!
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