What Would Happen If We Ignored the Rotation of Earth When Using an Elevator?

[kkrizka]

Hi all,

Some time ago I watched this http://www.youtube.com/watch?v=FAFUSbIs5KA".

However in it I assumed that the Earth is not rotating and I would like to know what would happen if I did not make that assumption

I do believe that if the Earth is rotating, it will affect the speed of the elevator because as it travels it changes it's radius and so will have to change its tangential speed, since the angular velocity matches that of the Earth (the walls of the shaft will prevent it going faster/slower than the Earth). Now there are two cases:

If the path of the elevator is straight through the core, the changes in the tangential speed are irrelevant, because this speed is always perpendicular to the path of motion. So it will take 42 minutes no matter what.

If the path of the elevator is at some weird angle such that a component of the tangential speed is parallel to it, then I am not sure what would happen. I am thinking that I can split the net force into two components. One would be the centrifugal force, which keeps the elevator going in circle. This force does not change the speed of the elevator, since it's perpendicular to the path of motion. The second force would be the force that does change the speed, which is due to the changes in potential energy as the elevator changes the radius. Since I am only using these changes in potential energy for my solutions, then the equations for the speed should be right and so the equation for the time should be right. However it somehow seems that some of my reasoning applies only to inertial frames, but rotation is non-inertial...

What do you think, would it take 42 minutes to use the elevator even if the Earth was rotating?

 

[pallidin]

It would take more time, because the forces you introduced would resist an "ideal" scenario. Looked at another way, the elevator is not only moving linearly, but tangentally as well, thus covering more distance collectively.

Not an expert, just my thoughts...

 

[kkrizka]

It would take more time, because the forces you introduced would resist an "ideal" scenario.

What do you mean they would "resist" an ideal scenario? I'm assuming that by ideal, you mean the scenario mentioned in my assumption (no rotation, uniform density...), but I'm not sure how a force could "resist" something. So I hope you could clarify that point.

Looked at another way, the elevator is not only moving linearly, but tangentally as well, thus covering more distance collectively.

Well, that's what I though too. But then the tangential movement is always perpendicular to the gravitational force, so there is no work being done there. But then I'm unsure about the normal force, which is doing some work but then I'm not sure if it changes the speed in the direction I'm interested in (note that this force is always perpendicular to the elevator shaft).

 

[pallidin]

Yeah, I see what you mean. You have my statements correct, but I think I presented them improperly. My bad.
When I was referring to rotation, it was not a reference to a "clean" rotation, rather one which has a "wobble", thus cyclically slamming the elevator against the shaft during descent.
Even still, perhaps someone more knowledgeable than I can jump in here and clarify this.

 

[Gear300]

I'm not sure what the effect is but the velocity of rotation does change across radial displacements, so the elevator would apparently feel an acceleration as it goes through different radial positions because the velocity is apparently changing over different radial positions...which is why the elevator may not be in an inertial frame.