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I'm having trouble solving this situation.

Imagine that a hole is drilled through the center of the Earth to the other side. An object of mass m at a distance r from the center of the Earth is pulled toward the center of the Earth only by the mass within the sphere of radius r. Write Newton's Law of gravitation for an object at the distance r from the center of the Earth, and show that the force on it is of HOoke's law form, F=-kr, where the effective force constant is k= (4/3)pi(density)Gm

And show that a sack of mail dropped into the hole will execute simple harmonic motion if it moves without friction. When will it arrive at the other side of the earth.

Ok, so far, I think I got the first part, where I used density= Mass/Volume

and volume of a sphere is (4/3)pi(r^3), and I isolated the M.

I replaced the M value with (4/3)pi(r^3)(density) in the law of gravitation formula, and then by grouping some terms together I get F= Kr

BUT I HAVE NO CLUE as to where the negative is coming from.

And also, I have no idea where to start proving that dropping a sack of mail into the hole will be SHM, and When will it arrive at the other side of earth.

I tried using x(t)=Acos(wt) and making A=r of earth

w (angular frequency)= [4(pi^2)r]/T BUT now I'm getting myself even more confused. A hint would be really appreciated!!! I really need help here and I need a clue as to where I can start finding the amount of time, etc...

Thank you very much for your time.