Tunnels through the center of planet, oscillations

In summary, the conversation discusses the potential for a particle to undergo simple harmonic motion if dropped in a tunnel that connects two points on the Earth's surface but does not go through the center. It is concluded that, assuming ideal conditions, the particle would indeed oscillate back and forth forever. This is a common exercise for freshmen studying physics.
  • #1
Albertgauss
Gold Member
292
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Hi all,

I know that if you drill a hole from one side of the planet to the other, through the planet's center, that a particle dropped in this tunnel will oscillate back and forth forever, like a mass on a spring, with the restoring force given by gravity.

What if the tunnel forms a chord that goes from point A, (say New York City) to point B (say London) but does NOT go through the Earth's center. If I drop a particle in this new tunnel, will the particle also undergo simple harmonic motion back and forth forever?
 
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  • #2
Albertgauss said:
What if the tunnel forms a chord that goes from point A, (say New York City) to point B (say London) but does NOT go through the Earth's center. If I drop a particle in this new tunnel, will the particle also undergo simple harmonic motion back and forth forever?
Sure. (Assuming the usual idealizations of uniform density and no friction.) Set up the math to find the restoring force, and you'll see things work out exactly the same in both cases.
 
  • #3
It would fall into the lower wall of the pit and sooner or later stop because of friction, probably at the lowest point of the tunnel. If you could somehow remove all friction though, and consider only the component of the gravitational force that is parallel to the tunnel, then yes, it would oscillate in harmonic motion. Provided your planet's density is the same everywhere.
 
  • #4
I think we did this one recently.
 
  • #5
epenguin said:
I think we did this one recently.
It's a standard freshman exercise.
 
  • #6
Hi all,

I got it. Thanks! That's what I needed to know. Sorry if this was done recently, I couldn't find it when I searched.
 

1. How would building tunnels through the center of a planet affect its oscillations?

The construction of tunnels through the center of a planet would have a significant impact on its oscillations. Since the tunnels would be altering the internal structure and distribution of mass within the planet, it would cause changes in the overall distribution of energy and the way it oscillates.

2. What factors influence the frequency and amplitude of planetary oscillations?

The frequency and amplitude of planetary oscillations are influenced by various factors, such as the planet's size, density, internal structure, and rotation. The type of material present in the planet's core and the presence of a magnetic field also play a role in determining its oscillations.

3. Can building tunnels through a planet's center affect its rotational motion?

Yes, building tunnels through a planet's center can affect its rotational motion. The construction of tunnels would change the distribution of mass within the planet, which in turn would affect its moment of inertia. This change in moment of inertia would alter the planet's rotational speed and direction.

4. What are some potential benefits of building tunnels through a planet's center?

Building tunnels through a planet's center could have several benefits. It could provide a faster and more efficient means of transportation between two points on the planet's surface. It could also allow for easier access to resources deep within the planet, such as minerals and energy sources. Additionally, studying the oscillations within the tunnels could provide valuable insights into the planet's internal structure and composition.

5. Are there any potential risks or drawbacks of building tunnels through a planet's center?

Yes, there are potential risks and drawbacks associated with building tunnels through a planet's center. The construction process itself could cause significant disturbances to the planet's surface and internal structure. There may also be challenges in maintaining the stability and safety of the tunnels, especially if the planet experiences seismic activity. Additionally, the cost and technological challenges of building such tunnels could be significant obstacles.

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