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- Thread starter Levi Tate
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A tunnel is drilled straight through the earth from Detroit to Zxxiuw. At the middle of the tunnel it is at 1/2 earth radius from the center. Assuming that a mass m placed in the tunnel will slide without friction, show that it experiences a force that is directly proportional to its distance from the center of the tunnel. Give an expression for this force in terms of the distance and the weight mg at the surface. Assume the earth is not rotating, a bad approximation, but hey, you're not going to drill this tunnel anyhow.

I am thinking I need to use energy and then take the derivative to get the force, but I am having trouble finding out the right assumptions to make such that I can do that.

That there is the problem, I am on an iPad so in the attachment is a picture of the problem as well as what I've been able to work out.

Thank you in advance

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davenn

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id imagine one would get stuck in the middle

why do you think that ?

Dave

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when you reached the very center of the tunnel the majority of gravitational force applied on you would be equal on all sides (theoretically speaking, if it was perfectly in the center) then as you moved to towards one end the majority of the gravity from the earth applied to you would be coming from the opposite side of the tunnel you were on, so you would begin to fall back the direction you came. (This is why the movie total recall confused me so much.)

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SteamKing

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ehild

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## Homework Statement

A tunnel is drilled straight through the earth from Detroit to Zxxiuw. At the middle of the tunnel it is at 1/2 earth radius from the center. Assuming that a mass m placed in the tunnel will slide without friction,show that it experiences a force that is directly proportional to its distance from the center of the tunnel.Give an expression for this force in terms of the distance and the weight mg at the surface. Assume the earth is not rotating, a bad approximation, but hey, you're not going to drill this tunnel anyhow.

The problem is

ehild

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- #8

Dick

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## Homework Statement

A tunnel is drilled straight through the earth from Detroit to Zxxiuw. At the middle of the tunnel it is at 1/2 earth radius from the center. Assuming that a mass m placed in the tunnel will slide without friction, show that it experiences a force that is directly proportional to its distance from the center of the tunnel. Give an expression for this force in terms of the distance and the weight mg at the surface. Assume the earth is not rotating, a bad approximation, but hey, you're not going to drill this tunnel anyhow.

I am thinking I need to use energy and then take the derivative to get the force, but I am having trouble finding out the right assumptions to make such that I can do that.

That there is the problem, I am on an iPad so in the attachment is a picture of the problem as well as what I've been able to work out.

Thank you in advance

What you really need to know is some facts about Newton's law of Gravitation, and you need to assume the mass density of the earth is constant everywhere.

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Dick

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Nothing you've said proves that, now does it?

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I know that the gravitation stems from the center of the sphere, and I know he initial PE because I know the radius, then I am just visualizing the radius of the sphere constantly decreases continuously, so that to approaches 1/2 radius of sphere.

I played around with the mass density, which I am not sure of, is it just d=m/v? Or the infinitesimal in case I end up integrating something? I'm trying to look at this equality here, because right at the center I reason the mass has zero speed instanteously, so..

MgRe= 1/2mv^2+mgr= 1/2mgRe

And somehow using that double equality, that is as far as I can think, but I don't see how that would fit in with the mass density, that seems to imply using an integral and force methods, but I could very well be wrong.

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sorry should have explained this more, before i was explaining the end result. The reason for the acceleration is because the force of gravity is based on the amount of mass on the other side of the object. From the surface of the earth there is an acceleration of 9.8 m/s/s as it moves closer to the center of the earth it decreases. then becomes zero at the center of the tunnel. then as it is at the other end of the tunnel it has an acceleration of 9.8 m/s/s in the other directionNothing you've said proves that, now does it?

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Right xirow that's the general idea of what's going on but the hard part is solving the problem.

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Dick

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Right xirow that's the general idea of what's going on but the hard part is solving the problem.

Then pay attention to ehild's more detailed clue. The acceleration of a mass at radius r is only coming from the mass inside of r. There is a calcuation to do.

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ehild

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Derive the expression for the force of gravity at distance r from the Earth centre.

ehild

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Cos I have to write v=dv/dt in one dimension don't i?

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ehild

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What expression do you have?

Cos I have to write v=dv/dt in one dimension don't i?

v can not be equal to dv/dt, they are not of the same dimension.

Is "t " time? What do you call "v" at all?

ehild

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V is a velocity in 1d, t is time.

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V=dr/dt I meant there

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