Ratio of Distances using Acceleration of Gravity

[Kabal]

Homework Statement

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

Homework Equations

a_g= GM/R²

ρ=M/V

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.

 

[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.

 

[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.

 

[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.

 

[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.

 

[haruspex]

GM/R² = GρV/2r²

No, this is backwards. a_s = GM/R², and you want an r₁ for which a_r1 = a_s/2.
All that is for r₁ < R. Please post your working for the r₂ > R case.