Two bodies, A of mass , and B of mass , are separated by a distance . If the distance between the two bodies is reduced to , the gravitational force exerted by A on B

Question

BlackTom AI · Accepted Answer

Gravitational force follows the inverse-square law: F = G m1 m2 / r^2. With A of mass m and B of mass 4m, the force depends on the product m1·m2 and inversely on the square of the separation r.

Step-by-step reasoning and intermediate calculations:
1. Initial separation r1 = d. Initial force:
   F1 = G·(m)·(4m) / d^2 = 4 G m^2 / d^2.
2. New separation r2 = 0.5 d. New force:
   F2 = G·(m)·(4m) / (0.5 d)^2 = 4 G m^2 / (0.25 d^2).
   Evaluate the denominator: (0.5 d)^2 = 0.25 d^2, so
   F2 = 4 G m^2 ÷ (0.25 d^2) = (4 / 0.25)·(G m^2 / d^2) = 16 G m^2 / d^2.
3. Compare F2 to F1:
   F2 / F1 = (16 G m^2 / d^2) / (4 G m^2 / d^2) = 16 / 4 = 4.

Thus the gravitational force becomes four times larger when the separation is halved. The masses remain the same, so the change in force comes solely from the r^2 dependence: halving r multiplies force by (1 / 0.5^2) = 1 / 0.25 = 4.

Why the other choices are incorrect:
- "Will double": doubling would correspond to a factor of 2, but halving distance gives factor 4, not 2.
- "Will remain the same": incorrect because force depends on r and r changed.
- "Will increase eight times": would require r to be reduced by a factor of 1/√8 ≈ 0.3536, not 0.5.

Conclusion: The gravitational force will increase four times, which matches the provided answer: will increase four times.

Two bodies, A of mass , and B of mass , are separated by a distance . If the distance between the two bodies is reduced to , the gravitational force exerted by A on B单项选择题

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