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We have an object of mass $m$ which can go through matter like a ghost. It's only affected by gravity, not by any other force, so it's in free fall.

At the instant $t=0$ it is at the surface of the Earth which is assumed to be a perfect sphere of radius $R=6.371 \times 10^6$ m, of mass $M=5.972 \times 10^{24}$ kg and of uniform density $\rho=5.515 \times 10^3$ kg/m$^3$. $G=6.674 \times 10^{-11}$ m$^3$ kg$^{-1}$ s$^{-2}$.

The object has an initial velocity $v_0$ directed precisely at the center of the Earth. Unlike previous questions the velocity need not be zero.

My question: depending on the variables $m$ and $v_0$, how much time will it take to reach the opposite side of the Earth?

David Z
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Quantum Force
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  • As soon as it goes past the centre, the gravitational force would pull it back towards the centre, so you would get some sort of simple harmonic motion occurring. DOnt think it would reach the other side. – lagrange103 Feb 17 '15 at 12:26
  • I don't think it will be in SHM, since in SHM the restoring force increases with the distance from the mean position, whereas the gravitational force decreases? – Hritik Narayan Feb 17 '15 at 12:33
  • @HritikNarayan Gravitational force increases linearly with distance from the center so long as you remain inside the gravitating body (assuming constant density). It only starts decreasing with the inverse square law outside the body. So inside Earth, this absolutely would be SHM – Jim Feb 17 '15 at 12:35
  • Yeah, thanks! I forgot considering the "being inside the Earth" part. Sigh. – Hritik Narayan Feb 17 '15 at 12:37
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    But even if it had a zero initial velocity, its velocity would increase during the first half of the route. Then, once it gets pass the center of the Earth, gravity would slow it down, but since its velocity would have been previously increased it should be able to reach the opposite side of the Earth. – Quantum Force Feb 17 '15 at 13:01
  • @Rob Jeffries: This is not the same question. In my question I have an initial velocity which is not necessarily zero. In the question you've linked the initial velocity is zero. So it's a different question. – Quantum Force Feb 17 '15 at 13:05
  • So it's the same SHM problem with a different initial condition. – ProfRob Feb 17 '15 at 14:07
  • But the Earth isn't of uniform density, nor I imagine is any planet (i.e. any body in hydrostatic equilibrium). So this isn't a question about Earth or other real or hypothetical planets? – RedGrittyBrick Feb 17 '15 at 14:14
  • @Rob Jeffries: So it's a different question. (But actually, it's not necessarily a simple harmonic motion, it depends on the initial velocity.) – Quantum Force Feb 17 '15 at 14:17
  • @RedGrittyBrick: This is obviously to simplify the problem. It's already hard enough considering uniform density, let's not make it even harder... It will be a good enough approximation. – Quantum Force Feb 17 '15 at 14:18
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    No you are wrong. SHM is defined by the fact that the restoring force is proportional to the displacement. The initial conditions are irrelevant. What is relevant is that you probably should not assume that the Earth has a constant density (depending how accurate you want the answer). It is not such a terribly wrong assumption if you are interested in getting it to say the nearest 5 minutes. – ProfRob Feb 17 '15 at 14:20
  • Possibly duplicate of http://physics.stackexchange.com/q/99636/26076; this latter itself a duplicate. – Selene Routley Feb 17 '15 at 14:30
  • If I'm not mistaken, I think you can actually look this up in "Alice in Wonderland" as Alice falls down the rabbit hole she wonders how long she will take to pass through the Earth and I seem to recall that the thought of three quarters of an hour came to her. – Selene Routley Feb 17 '15 at 14:31
  • @Rob Jeffries: But it's not necessarily periodic,is what I meant, if the speed of the object is greater than the escape velocity than it will not be periodic since the object will leave indefinitely. But anyway, like I stated above uniform density will be a good enough approximation, but if you can find the average density of the Earth at every different depth levels and can use this to find the answer than you're welcome to use non-uniform density. – Quantum Force Feb 17 '15 at 14:33
  • @WetSavannaAnimal aka Rod Vance: Thanks. But surely Alice had an initial velocity of zero. I would like the answer depending on the initial velocity. And so no, the "possible duplicate" you gave is not the same question as mine. – Quantum Force Feb 17 '15 at 14:34
  • If it gets re-opened I may expand, but it is SHM (for uniform density). You need to insert the boundary conditions appropriately and you arrive at a remarkably simple answer. – ProfRob Feb 17 '15 at 17:29
  • I still don't understand why it was closed even though it was clearly a different question than the other two questions... – Quantum Force Feb 17 '15 at 20:28
  • In its original version, it wasn't clearly a different question. It became clearly different when you edited it to mention previous questions and say how this one was different. (Though perhaps not clear enough for some voters? Anyway, I edited it to make that even more explicit.) Now, the other issue is that this runs afoul of our policy on homework-like questions, so if I were to unmark it as a duplicate, I would then immediately close it as homework-like. So I'd like to work out that issue before I do reopen it. I've posted a link in [chat]. – David Z Feb 17 '15 at 20:55

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