Two small objects of masses m1 and m2 are originally separated by a fixed distance r, resulting in a mutual attractive force of magnitude F0. Four separate hypothetical alterations (I–IV) are proposed, each applied independently to the original setup:
I. m1 is doubled, m2 is unchanged, and r is halved.
II. Both masses are tripled, and r is tripled.
III. m1 is halved, m2 is doubled, and r is doubled.
IV. Both masses are reduced to one-quarter of their original values, and r is reduced to one-half.
Assume no other forces act on the objects.
(a) For each alteration (I–IV), express the new attractive force magnitude F as a multiple of the original force F0. Show the proportional reasoning that leads to each multiple.
(b) Rank the four alterations from greatest to least resulting force magnitude. If any alterations produce equal forces, indicate this clearly.
(c) For which alteration does the gravitational potential energy of the two-object system experience the greatest magnitude change compared with the original configuration? Justify your choice qualitatively by referring to the combined effect of mass and separation changes.
(d) A student argues that alteration II leaves the force unchanged because both numerator and denominator in the underlying relationship are tripled. Identify and correct the flaw in this reasoning.