Gravitational Self Potential Energy

[Rmehtany]

Note: I know this question has been asked before, but I wasn't allowed to ask my question on that thread

1. Homework Statement
The gravitational self potential energy of a solid ball of mass density ρ and radius R is E. What is the gravitational self potential energy of a ball of mass density ρ and radius 2R?

Homework Equations

$$PE = -\frac{GMM}{R}$$

The Attempt at a Solution

My attempt was dimensional analysis, because I had no other idea on how to approach this. The energy was somehow going to be related to G, radius, density, and other constants. PE has units $$kg \frac{m^2}{s^2}$$, density $$\frac{kg}{m^3}$$, and G in terms of $$\frac{m^3}{s^2}$$. I tried using E = $$k G^a \rho^b R^c$$, but I couldn't eliminate enough masses from the equation.

 

[TSny]

G in terms of $\frac{m^3}{s^2}$.

These are not the units for G.

Otherwise, your approach should work.

See if it works out once you have the correct units for G. If not, post the details of your attempt.

 

[Rmehtany]

These are not the units for G.

Otherwise, your approach should work.

See if it works out once you have the correct units for G. If not, post the details of your attempt.

I will try that, but are there other ways to solve this question? I kinda think my method is a bit sloppy.

 

[TSny]

I will try that, but are there other ways to solve this question? I kinda think my method is a bit sloppy.

Yes. Look at the equation you wrote for the PE of two point masses. This equation tells you how to dimensionally get PE from G, M and R. Thus, you need to think about relating M to density and R so that you can see how PE is related to G, ρ, and R.

 

[Rmehtany]

Yes. Look at the equation you wrote for the PE of two point masses. This equation tells you how to dimensionally get PE from G, M and R. Thus, you need to think about relating M to density and R so that you can see how PE is related to G, ρ, and R.

Uh huh, so let me see if I understand you:

The mass will be 8 times the original mass of the ball due to unvarying density, and radius is doubled. Since $$PE = \frac{-GMM}{R}$$, this equals $$\frac{8^2}{2}$$ = 32 times? Is that correct?

 

[TSny]

Yes, that's correct. The actual formula for the gravitational PE of a solid sphere will have some numerical factor out front, but it must be proportional to GM²/R.

 

[Rmehtany]

Thank you for helping