# Ratio of Distances using Acceleration of Gravity

1. May 15, 2014

### Kabal

1. The problem statement, all variables and given/known data

A constant-density planet of radius R has a gravitational acceleration ag =
as at its surface. There are two distances from the center of the planet at
which ag = as/2. Show that the ratio of these distances may be given by
R = 2√2

2. Relevant equations

ag= GM/R2

ρ=M/V

3. The attempt at a solution

This is my attempt at approaching the problem. Given that there are two distances I'm thinking that one of the distances is inside the planet while the other would be outside. I know that this is a constant density planet therefore in some fashion I must use a density equation. The problem I'm receiving though is that when I setup the ratio I simply get 1

GM/R2 = GρV/2r2

GM/R2 = Gρ(4/3)∏r13/2r12

2M/R2 = ρ(4/3)∏r1

3M/2∏R2ρ=r1

At this point I simply do the same thing for r2 then set the ratio.
This definitely doesn't feel right and I think I'm missing a key understanding to be able to completely understand this problem completely. Another idea I had was that the Rs must vary depending on where in relation I'm talking about. Could it be that the radii is R-r1 and R+r2?

Any help is much appreciated.

2. May 15, 2014

### tms

Think about where the two distances at which $a_g = a_s / 2$ must be, in general terms, taking into account the symmetries of the problem. Then write down the expressions for $a_g$ that apply to each distance.

Your last sentence shows that you are on the right track.

3. May 15, 2014

### Kabal

Thanks I'll give that an attempt

Another question. Is it possible that the planet with radius R have different masses?
Ex.
GM/2R2=Gm/R2

I feel that this is wrong because of the fact that the planet is at constant density. Therefore at Radius R it must have mass M.

4. May 15, 2014

### tms

The radius and density distribution uniquely determine the mass. Ignoring such real-world things as infalling mass, of course. In other words, since in the problem the density is constant and the radius is fixed, there is only one possible mass for the planet, or for any planet with those two characteristics. But it isn't the constancy of the density per se that makes that true; any density distribution would also give a single mass, as long as it is the same for the planets under consideration.

5. May 16, 2014

### dauto

What you did for r1 was correct but you can't do the same thing for r2 since r2 is not inside the planet.

6. May 16, 2014

### haruspex

No, this is backwards. as = GM/R2, and you want an r1 for which ar1 = as/2.
All that is for r1 < R. Please post your working for the r2 > R case.