The mass of the Earth is 6 × 1024 kg. Its radius is 6390 km. The gravitational constant G = 6.67 × 10-11 in SI units.What is the force of gravity on a mass of 22.9 kg at the Earth's surface? Give your answer to the nearest whole number.

Question

BlackTom AI · Accepted Answer

F = G M m / r^2.

1) Relevant concepts:
- Newton’s law of universal gravitation: F = G M m / r^2 (force between two point masses or spherically symmetric bodies).
- Equivalent approach: compute gravitational acceleration g = GM/r^2 at Earth's surface, then F = m g.
- Units must be SI: convert radius from km to m.

2) Step-by-step reasoning:
- Given: G = 6.67 × 10^-11 N·m^2/kg^2, M = 6.0 × 10^24 kg, m = 22.9 kg, radius r = 6390 km = 6.39 × 10^6 m.
- Use F = G M m / r^2.

3) Intermediate calculations:
- r^2 = (6.39 × 10^6 m)^2 = 6.39^2 × 10^12 m^2 = 40.8321 × 10^12 m^2 = 4.08321 × 10^13 m^2.
- Compute numerator: G M m = (6.67 × 10^-11)(6.0 × 10^24)(22.9)
  = (6.67 × 6.0 × 22.9) × 10^(-11+24)
  = 91.6458 × 10^13
  = 9.16458 × 10^15 (N·m^2).
- Now F = 9.16458 × 10^15 / 4.08321 × 10^13
  = (9.16458 / 4.08321) × 10^(15-13)
  ≈ 2.245 × 10^2 N
  ≈ 224.5 N.

Alternative (short) route: g = GM/r^2 ≈ 9.801 m/s^2, then F = m g = 22.9 × 9.801 ≈ 224.4 N.

4) Why this is correct:
- All values were converted to SI units and plugged into Newton’s gravitational formula. The numerical result matches the expected surface-weight calculation using g ≈ 9.8 m/s^2, so the computed force is the weight of the 22.9 kg mass at Earth’s surface.

5) Why other answers would be incorrect:
- Common mistakes: forgetting to convert km to m (gives errors by factor 10^3), misplacing exponents when multiplying powers of ten, or using an incorrect value of G. Small rounding differences (e.g., using g = 9.81 instead of 9.801) yield slightly different results but the nearest whole number remains 224.

6) Conclusion:
The force of gravity on a 22.9 kg mass at Earth’s surface is approximately 224 N. This confirms the provided answer: 224.

The mass of the Earth is 6 × 1024 kg. Its radius is 6390 km. The gravitational constant G = 6.67 × 10-11 in SI units.What is the force of gravity on a mass of 22.9 kg at the Earth's surface? Give your answer to the nearest whole number.数值题

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