physics question

Question

consider the following pairs of objects with varying masses and separation distances. which of these pairs has the smallest gravitational potential energy? (each "-" stands for 1 unit between r and m) 1. {m-r-m} 2. {4m----4r-----m} 3. {m--2r--m} 4. {2m----4r----2m} a) 1. b) 2. c) 3. d) 4. e) 3 and 4 are equally small

Answer 1

The potential gravitational energy between two masses A and B separated by distance R is

U = -GAB/R

1. U = -Gm²/r

2. U = -4Gm²/4r = -Gm²/r

3. U = -Gm²/2r

4. U = -4Gm²/4r = -Gm²/r

So 1., 2., 4. have the same potential gravitational energy -Gm²/r.

That is smaller than the potential gravitational energy, -Gm²/2r, of 3.

On the other hand, if the question is about the absolute value of potential gravitational energy,

then 3. is the smallest.