The gravitational force between two objects, [math: M1] and [math: M2] , a distance of [math: R] apart is found to be [math: F]. Both masses were multiplied by a factor of 10, and the distance between them by a factor of 7. One of the masses was then doubled. What would be the ratio of the new force compared to the initial force [math: FnewFinitial]?

Question

BlackTom AI · Accepted Answer

Newton's law of universal gravitation states F = G M1 M2 / R^2, so the force is proportional to the product of the two masses and inversely proportional to the square of the separation.

1. Relevant knowledge points:
- F ∝ M1 · M2
- F ∝ 1/R^2
- Multiplicative changes to masses and distance scale F by the product of those factors (masses multiply, distance factor squared in denominator).

2. Step-by-step reasoning:
- Start with initial force F = G M1 M2 / R^2.
- Both masses multiplied by 10: M1' = 10 M1, M2' = 10 M2. After this step the product M1'·M2' = (10 M1)(10 M2) = 100 M1 M2.
- One of the masses is then doubled. Doubling one of the already-scaled masses multiplies the product by 2, so final product = 2 × 100 M1 M2 = 200 M1 M2.
- Distance scaled by factor 7: R' = 7 R, so R'^2 = (7R)^2 = 49 R^2.

3. Calculations with intermediate steps:
- F_new = G (200 M1 M2) / (49 R^2).
- Ratio F_new / F = [G (200 M1 M2) / (49 R^2)] ÷ [G M1 M2 / R^2] = (200/49).
- Compute decimal: 200 ÷ 49 = 4 remainder 4 → 4 + 4/49 ≈ 4.08163265 ≈ 4.08 (rounded to three significant figures).

4. Why this answer is correct:
- The mass changes produced an overall ×200 increase in the numerator, while the distance change produced a ×49 increase in the denominator. The net multiplicative factor is 200/49 ≈ 4.08, so the new force is about 4.08 times the original.

5. Why the other options are incorrect:
- 2.04 would correspond to only the ×100 mass change without the additional doubling (100/49 ≈ 2.04).
- 2.45 and 0.41 do not match the correct combined factor (200/49 ≈ 4.08) and therefore are incorrect.

6. Conclusion:
The correct ratio F_new / F_initial = 200/49 ≈ 4.08, matching option a.

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