Tunnels through the center of planet, oscillations

[Albertgauss]

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?

 

[Doc Al]

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.

 

[someGorilla]

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.

 

[epenguin]

I think we did this one recently.

 

[Doc Al]

I think we did this one recently.

It's a standard freshman exercise.

 

[Albertgauss]

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.