Tunnel drilled through the Earth and mechanical work

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Homework Statement


Suppose that a straight tunnel is drilled through the Earth as indicated in figure. Assume that the floor of the tunnel is frictionless and that air resistance can be neglected. If the mid-point of the tunnel is 0.8[tex]R_{E}[/tex] from the centre of the Earth, where the [tex]R_{E}[/tex] is the radius of the Earth, determine the mechanical work done by the field force in moving a 1.0 kg mass from the entrance of the tunnel to the mid-point.



Homework Equations


[tex]F=-GmM_{E}d/R_{E}^3[/tex], where the d is distance from the centre of the Earth


The Attempt at a Solution


http://img35.imageshack.us/img35/323/soltry.jpg [Broken]

[tex]G\ =\ 6.673(10)\ \times\ 10^{-11}\ m^{3} kg^{-1} s^{-2}[/tex]
[tex]M_{E}\ =\ 5.98\ \times\ 10^{24}\ kg[/tex]
[tex]R_{E}\ =\ 6.38\ \times\ 10^{6}\ m[/tex]
[tex]m\ =\ 1.0 kg[/tex]

I get [tex]W\ =\ 1.1258\ \times\ 10^{7}\ J[/tex]

According to book [tex]W[/tex] should be [tex]5.1\ \times\ 10^{7}\ J[/tex]

What did I do wrong? Or is the answer in the book wrong?

Thank you for in advance!
 
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Answers and Replies

  • #2
berkeman
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Homework Statement


Suppose that a straight tunnel is drilled through the Earth as indicated in figure. Assume that the floor of the tunnel is frictionless and that air resistance can be neglected. If the mid-point of the tunnel is 0.8[tex]R_{E}[/tex] from the centre of the Earth, where the [tex]R_{E}[/tex] is the radius of the Earth, determine the mechanical work done by the field force in moving a 1.0 kg mass from the entrance of the tunnel to the mid-point.



Homework Equations


[tex]F=-GmM_{E}d/R_{E}^3[/tex], where the d is distance from the centre of the Earth


The Attempt at a Solution


http://img35.imageshack.us/img35/323/soltry.jpg [Broken]

[tex]G\ =\ 6.673(10)\ \times\ 10^{-11}\ m^{3} kg^{-1} s^{-2}[/tex]
[tex]M_{E}\ =\ 5.98\ \times\ 10^{24}\ kg[/tex]
[tex]R_{E}\ =\ 6.38\ \times\ 10^{6}\ m[/tex]
[tex]m\ =\ 1.0 kg[/tex]

I get [tex]W\ =\ 1.1258\ \times\ 10^{7}\ J[/tex]

According to book [tex]W[/tex] should be [tex]5.1\ \times\ 10^{7}\ J[/tex]

What did I do wrong? Or is the answer in the book wrong?

Thank you for in advance!

Not sure your first equation for F is correct. You are applying a linear ratio to the radius distance, but the mass of the part of the Earth inside your radius varies with the cube of the radius, doesn't it? Maybe revisit how you set up that first equation and see if that fixes the solution.

Welcome to the PF, BTW.
 
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