If the mass of Earth increased, with no change in radius, your weight would _____.

Question

BlackTom AI · Accepted Answer

Newton’s law of universal gravitation: F = G (M_earth m)/R^2, so your weight (the gravitational force on you) depends directly on Earth’s mass M_earth and inversely on the square of the Earth’s radius R.

1) Relevant concepts
- Gravitational force between two masses: F = G M m / R^2.
- Your weight near Earth’s surface is this gravitational force (often written as W = mg, where g = GM/R^2).
- Proportionality: for fixed R and m, F ∝ M.

2) Step-by-step reasoning
- Start with F = G (M_earth m)/R^2.
- Treat m (your mass), G (gravitational constant), and R (Earth’s radius) as constants in this scenario.
- If Earth’s mass increases to some new value M' = k·M_earth (where k > 1), then the new force F' = G (M' m)/R^2 = G (k·M_earth·m)/R^2 = k·[G (M_earth m)/R^2] = k·F.
- Thus F' is k times the original weight F; weight increases in direct proportion to the increase in Earth’s mass.

3) Example calculation (intermediate steps)
- Suppose Earth’s mass doubled (k = 2). Original weight F = G (M m)/R^2. New weight F' = G (2M m)/R^2 = 2·G (M m)/R^2 = 2F. So your weight would double.

4) Why the correct answer is correct
- Because weight is proportional to Earth’s mass when radius is fixed, any increase in Earth’s mass produces a proportional increase in the gravitational force on you — i.e., your weight increases.

5) Why the other options are incorrect
- a. “stay the same” is wrong because changing M in F = G M m / R^2 changes F when R is constant.
- b. “decrease” is wrong because increasing the source mass cannot reduce the attractive gravitational force at the same radius.

Conclusion: Your weight would increase, so the correct choice is c. increase also.

If the mass of Earth increased, with no change in radius, your weight would _____.单项选择题

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