Difficult Gravitational Force problem

1. Jun 28, 2013

PsychonautQQ

I posted this in the intro section and realized the answer might be more difficult than anticipated, sorry I won't double post anymore X_x

1. The problem statement, all variables and given/known data
The effect of gravity on a point mass is given as F(r), where r is the radius between the point mass and the earths center. If you could place the point mass at R/2, where R is the radius of the earth, what would the relationship be between F(R)/F(R/2)

2. Relevant equations
Fg = Gm1m2 / r^2

3. The attempt at a solution
I tried just plugging 1/2R and R and getting of course 4 since the radius is squared, but the correct answer is suppose to be 2. I realized the flaw in my logic was that if the point mass was half way to the center of the earth, then some of the earth's mass would be pulling in the opposite direction as the stuff still "under" it. Help please :D

Last edited: Jun 28, 2013
2. Jun 28, 2013

omiros

Without being too sure, as long as you are talking about a point mass, being 'into' a massive object, things change. The equation changes inside earth, because you have 'layers' of mass that do not contribute to the 'gravity' of the center, but they also pull that object out. Think of an object, laying in the center of the earth. Its acceleration is zero. Cause everything pulls that object in every direction at the same rate. This should help you.

However I am still not very sure what do you mean with the 1/2R. Is that the depth? Is that the Altitude? Is that the distance from the center?

3. Jun 28, 2013

PsychonautQQ

Yeah, the object is placed as I mean't R/2 not 1/2R, oops. Yes, R is the radius of the earth and the object is halfway to the center.

4. Jun 28, 2013

Infrared

There is an eighth of much mass attracting the point and the distance is halved, so the change in the gravitational force is a decrease by a factor of 2. It is not obvious, but the gravitational force from everything in the Earth a distance greater than R/2 from the center cancels.

Edit: Your original method would have given you an answer of 1/4, not 4.

5. Jun 28, 2013

PsychonautQQ

The answer is supposed to be 2 according to the answer key, but I feel like that answer and the logic you used doesn't account for the mass pulling in the opposite direction that the point mass is beneath. Edit: I just read that the mass above the object cancels itself out, can somebody provide me a proof of this?
Edit2: Nevermind i understand, THANKS FOR THE HELP ALL <3!!

6. Jun 28, 2013

Infrared

Again, it isn't obvious, but the force from that mass will be zero. See http://en.wikipedia.org/wiki/Shell_theorem

7. Jun 28, 2013

omiros

Mate, calm down. Just plug these things in the equation. It give 2. Your logic, is not the logic of solving a problem in physics. It is an equation that just varies. Nothing more.

8. Jun 28, 2013

PsychonautQQ

yeah but i also want to make sure the equation makes sense to me, especially these classical mechanics stuff that's so tangible