Free fall down (Chord) Tunnel through Earth

In summary, the conversation discusses the problem of showing that any tunnel through the Earth will have the same free-fall time, regardless of its position. The relevant equation is the acceleration of free-fall, which is equal to GM / r^2 where r is changing. The solution involves trigonometry and calculus, and the attempt at solving it involves considering the acceleration at an arbitrary position and angle within the tunnel. However, there is a mistake in the calculation, which is eventually caught and corrected.
  • #1
Agrasin
69
2

Homework Statement



I'm trying to show that any tunnel through the Earth (not necessarily through the center) will have a free-fall time that is the same. I heard this was true somewhere.

Homework Equations



acceleration of free-fall = GM / r^2 where r is changing

I believe this involves trig and calculus.

The Attempt at a Solution



I attempted a solution (attached-- please bear with me. I'm sorry if it's not very clear or if my process is weird).

I considered the acceleration of free-fall for a particle at an arbitrary position in a tunnel x that makes an arbitrary angle theta with the radius of the Earth. I eventually solve for the acceleration as a (complicated) function of x and theta.

Not sure what to do after that... If it involves integration, I feel like it's hopelessly complicated.
 

Attachments

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  • #2
I caught my mistake. It's in the "Solving for r" stage. I foolishly assume d is an arc instead of a flat chord.

Admins, please feel free to delete this thread. Not sure how to.
 

1. What is free fall down chord tunnel through Earth?

Free fall down chord tunnel through Earth is a hypothetical scenario in which an object is dropped from one side of the Earth and falls through a tunnel that goes straight through the center of the Earth, emerging on the other side.

2. How does free fall down chord tunnel through Earth work?

In this scenario, the object experiences free fall due to the gravitational force of the Earth. As it falls through the tunnel, the force of gravity decreases until it reaches the center of the Earth, where it becomes zero. The object then continues to fall towards the other side of the Earth, accelerating due to the increasing gravitational force on that side.

3. Is free fall down chord tunnel through Earth possible?

No, it is not possible for several reasons. The first is that the Earth is not a perfect sphere, so there would not be a straight tunnel through the center. Additionally, the Earth's core is composed of molten iron, making it impossible to create a tunnel that goes through it. Finally, the intense pressure and heat at the Earth's core would make it impossible for any object to survive the journey.

4. How long would it take for an object to free fall down chord tunnel through Earth?

If it were possible, it would take approximately 42 minutes and 12 seconds for an object to fall from one side of the Earth to the other, assuming a straight tunnel through the center and no air resistance.

5. What would happen to an object in free fall down chord tunnel through Earth?

If an object were to somehow survive the intense heat and pressure at the Earth's core, it would eventually reach the other side of the Earth and begin to slow down due to the increasing gravitational force. It would then continue to oscillate back and forth between the two sides, eventually coming to a stop at the center of the Earth.

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