What event will produce the greatest increase in the gravitational force between the two masses?

Question

BlackTom AI · Accepted Answer

Newton's law of universal gravitation: F = G m1 m2 / r^2. Force is directly proportional to each mass and inversely proportional to the square of the separation.

1) Relevant concepts
- If a mass is multiplied by a factor k, the force changes by the same factor k (because F ∝ m1 m2).
- If the distance r is multiplied by a factor k, the force changes by 1/k^2 (because F ∝ 1/r^2).

2) Step-by-step reasoning with intermediate steps
- Convert given masses to kilograms (optional for ratio work): m1 = 250 g = 0.25 kg, m2 = 150 g = 0.15 kg. Initial separation r = 0.50 m.
- Initial force (symbolic): F0 = G (0.25)(0.15) / (0.50)^2 = G(0.0375) / 0.25 = G(0.15).

Evaluate each option relative to F0:

a) Doubling the large mass: m1 → 0.50 kg. New force Fa = G(0.50)(0.15)/(0.50)^2 = 2·F0. Factor change = 2 (force doubles).

b) Doubling the distance: r → 1.00 m. Fb = G(0.25)(0.15)/(1.00)^2 = F0 · (0.25/1) = 0.25·F0. Factor change = 1/4 (force decreases).

c) Reducing the small mass by half: m2 → 0.075 kg. Fc = G(0.25)(0.075)/(0.50)^2 = 0.5·F0. Factor change = 1/2 (force decreases).

d) Reducing the distance by half: r → 0.25 m. Fd = G(0.25)(0.15)/(0.25)^2 = G(0.0375)/0.0625 = G(0.6) = 4·F0. Factor change = 4 (force quadruples).

3) Why the correct answer is correct
Reducing the separation by one-half increases the gravitational force by 1/(1/2)^2 = 4, a larger multiplicative increase than doubling one mass (factor 2). Thus option d gives the greatest increase.

4) Why other options are incorrect
- a (double large mass): only doubles the force (factor 2), smaller than factor 4.
- b (double distance): decreases force to one quarter.
- c (halve small mass): decreases force to one half.

Conclusion: The greatest increase occurs when the distance is reduced by half. Final answer: d.

What event will produce the greatest increase in the gravitational force between the two masses?单项选择题

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