Some more thoughts here.
(1) If the density of the earth is dependent on radius, becoming larger as radius becomes small, how would it affect the round-trip time? Ans: It will shorten the time.
(2) In this case, will the shell theorem still applicable? Ans: Yes. See my lecture note about the exact statement of the shell theorem. Newton's shell theorem is applicable when the density is the function of r only. The density does not need to be uniform. As long as the density is not orientation dependent (again an assumption here), the shell theorem is valid.
(3) In all these "calculations," simple and "stupid" assumptions like uniform density is made. Why should I trust the calculation? Ans: In engineering, the inaccuracy of these results may be viewed as great deficiency. In physics, the inaccuracy of these results may be viewed as problematic in some cases (if you are trying to build a high precision experimental equipment based on these estimates) but it would not prevent the results from being interpreted as insightful in other cases (if you are after only an order of magnitude estimate, to be compared with other numbers based on other views). Generally, these order of magnitude estimates are very valuable, and so "stupid" does not really mean that. Even if you are into doing fancy computerized calculations, these simple calculations give you an idea what the answer should be approximately, and play a great role of preventing mistakes in super duper computer calculations. Physicists like to simplify things and draw a rough sketch first. More often than not the sketch already contains all essential physics!
When I say "stupid" or "crazy" in class, I actually likely to mean "smart" or "cool"!
(4) Just to make sure people are on board on this. To avoid Coriolis force to mess up with our "travel through the center of mass" (since the earth is spinning), we need to do it from the north pole to the south pole, or the other way around.
Those were great questions!
4 comments:
Imagine there were a bunge cord, one end attached to the foot of the person and the other end attached to the center of the earth. If the person were at rest at the center of the earth, where radius = 0, would they experience 0 gravity and just float there?
Problem #4 in the practice exam is very similar to this problem. I am a bit confused with the formula given. What is little r?
It is actually the same problem! r is the position of the falling object and the center of the object for that problem #4. in my lecture, it was slightly different, r was the distance. they are in essence the same thing, though. in practice exam #4, r can be negative, but in my lecture r can be only positive. if this subtlety bothers you, just change r to y or something, and then consider it as positive or negative, depending on where the falling mass is. it could be simpler this way.
Yes, Alex, at the center of the earth, there would not be any gravity! That is due to the shell theorem of Newton!
* Not only that, if the object is at rest at the center, there would not be any other force (centrifugal or Coriolis).
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