What is the force of gravity acting on an astronaut of mass 74.3 kg on a planet of mass 6.9 × 1024 kg and radius 6000 km? Give your answer to the nearest whole number. G is 6.67 × 10-11 in SI units.

Question

BlackTom AI · Accepted Answer

Newton's law of universal gravitation: F = G M m / r^2, so use G = 6.67×10^-11, M = 6.9×10^24 kg, m = 74.3 kg, r = 6000 km = 6.00×10^6 m.

1) Relevant knowledge points and concepts
- Gravitational force between two masses is given by F = G M m / r^2.
- All quantities must be in SI units (mass in kg, distance in m).
- Convert radius from km to m: 6000 km = 6000×1000 m = 6.00×10^6 m.

2) Step-by-step reasoning and intermediate calculations
- Compute the product M·m:
  M·m = (6.9×10^24 kg) × (74.3 kg) = 6.9 × 74.3 × 10^24 = 512.67 × 10^24 = 5.1267 × 10^26.
- Multiply by G:
  G·M·m = (6.67×10^-11) × (5.1267×10^26) = (6.67 × 5.1267) × 10^( -11 + 26 )
         = 34.194669 × 10^15 = 3.4194669 × 10^16 (in SI units before dividing by r^2).
- Compute r^2:
  r^2 = (6.00×10^6 m)^2 = 36.00 × 10^12 = 3.600 × 10^13 m^2.
- Divide to get force:
  F = (3.4194669×10^16) / (3.600×10^13) = (3.4194669 / 3.6) × 10^3
    ≈ 0.9498525 × 10^3 ≈ 949.8525 N.

3) Rounding and final value
- To the nearest whole number: F ≈ 950 N.

4) Why this answer is correct
- All steps followed the universal law, used correct unit conversions (km → m), and maintained scientific notation to avoid rounding error until the final step. The arithmetic leads directly to ≈ 949.85 N, which rounds to 950 N.

5) Brief note on common incorrect approaches
- Forgetting to convert the radius to meters (using 6000 instead of 6.00×10^6) would give an answer off by a factor of 10^12.
- Using g = 9.8 m/s^2 incorrectly assumes Earth's gravity; you must compute g from G·M/r^2 for this planet.

6) Conclusion
- The calculated gravitational force is 950 N, which matches the provided correct answer.

What is the force of gravity acting on an astronaut of mass 74.3 kg on a planet of mass 6.9 × 1024 kg and radius 6000 km? Give your answer to the nearest whole number. G is 6.67 × 10-11 in SI units.数值题

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