The Gravitational Force and Periodic Motion: A Tunnel Through Earth

[yasar1967]

Imagine a tunnel that has been drilled through Earth: a
smooth, straight tunnel with a frictionless interior .
The deepest point of the tunnel is at depth d, and the coordinate x
measures the distance along the tunnel from its deepest point to an
arbitrary point P a distance L from Earth's center. The known
parameters are the mass(Me) and radius of Earth(Re), as well as G.
Earth is assumed to have uniform density.
(a) What is the total mass of that portion of Earth that lies within
the distance L in termS of L, Me, Re. and G? (b) What is the grav-
itational force, in direction and magnitude, acting on a ball of
mass at point P? (c) What is the rotal force. in direction and magnitude. acting on the ball as a function of x and of the constants of the problem? Why is there no net force acting perpendicular to the tunnel? (d) What is the period of the motion if the ball
is released at rest at an entrance to the tunnel? Ignore air resistance. (e) What is the period of a satellite in circular orbit around
Earth at a radius equal to Earth's radius?




2. Harmonic, periodic motion and Gravity formulas



3. I'm at total loss

 

[Andre]

Well, aren't there a few Newton laws to jot down?

And what happens to gravity inside a solid sphere?

 

[tiny-tim]

… one step at a time … !

I'm at total loss

Hi yasar!

Well, let's start with (a) - it asks you for the mass of a sphere of radius L.

And that is … ?

 

[yasar1967]

Mass is 4/3 pi R^3 d, d being density but is it asked to "extract" the portion left over the tunnel?? if so how can I do that?

 

[tiny-tim]

Mass is 4/3 pi R^3 d, d being density but is it asked to "extract" the portion left over the tunnel?? if so how can I do that?

No! (a) is the easy part … designed to help you with the other parts!

(a) only asks you for the sphere.

ok, now for (b):

What is the radius at point P?

So what is the gravitational force at P, and what is its angle to the tunnel?

 

[yasar1967]

Am I getting this right Tim? you're saying it's asked for the whole mass of Earth at (a)?? -can't be that easy, I think the portion left is asked here.

raidus is L at point P.

gravitational force is:
F(r)=-GmM(e)/L^2

sine of angle is x/L

 

[tiny-tim]

oh, we're misunderstanding each other.

Yes, they want 4πdL^3/3.

Though you could convert it to (Me)L^3/(Re)^3.

gravitational force is:
F(r)=-GmM(e)/L^2

sine of angle is x/L

No … the gravitational force comes only from the mass inside the sphere of radius L … that's why the question asks you to do part (a) first.

Try again!