What is the gravitational potential energy of an object located 32 000m above the Earth’s Surface?

Question

BlackTom AI · Accepted Answer

Gravitational potential at distance r from Earth's center is U(r) = −GMm / r, so the object’s U uses r = R_E + h.

1) Relevant concepts
- Gravitational potential energy (with zero at infinity) for a point mass m in Earth’s field: U(r) = −GMm / r, where M is Earth’s mass and r is distance from Earth’s center.
- For an object a height h above the surface, r = R_E + h. Units must be consistent.

2) Step-by-step reasoning
- Start from the standard result obtained by integrating the gravitational force: U(r) = −∫(GMm/r^2) dr = −GMm / r + constant. Choosing U(∞)=0 gives U(r)=−GMm/r.
- Here h = 32 000 m. So r = R_E + h.

3) Intermediate calculations and unit consistency
- Convert height to the same units as the radius you use. If R_E is expressed in meters: h = 32 000 m. If you use kilometers for R_E, convert h to km: 32 000 m = 32 km.
- Therefore r = R_E + 32 km (if R_E is in km) or r = R_E + 32 000 m (if R_E is in m).
- Substitute into the formula: U = −GMm / (R_E + 32 km).

4) Why this is correct
- The potential depends on the full distance from the centre; adding the altitude to R_E is necessary. The negative sign indicates the bound nature of the mass in Earth’s gravitational field.

5) Why the other options are incorrect
- (a) U = −GMm/(R_E − 32 km) is wrong because it subtracts height (would place the object inside Earth if R_E − h were used).
- (c) U = −GMm/(32000 km + R_E) is wrong because it treats 32 000 m as 32 000 km (incorrect unit magnitude).
- (d) U = −GM / R_E omits the mass m and so is not an energy expression.
- (e) U = −GMm / R_E gives the potential energy at the Earth’s surface (h = 0), not at h = 32 km.

6) Conclusion
Thus the correct expression is U = −GMm / (R_E + 32 km), which matches option b.

What is the gravitational potential energy of an object located 32 000m above the Earth’s Surface?单项选择题

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