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

Question

BlackTom AI · Accepted Answer

1) Relevant knowledge points and concepts
The universal gravitational potential energy for a mass m at distance r from the centre of the Earth (mass M) is
U(r) = −G M m / r,
where G is the gravitational constant and the sign is negative because potential is defined to be zero at infinity. For an object at altitude h above the Earth’s surface, the distance from the Earth’s centre is r = R_E + h (R_E = Earth’s radius).

2) Step‑by‑step reasoning
We must substitute r = R_E + h into the universal formula. The given altitude is 32 000 m, which equals 32 km. Thus r = R_E + 32 km. Therefore U = −G M m / (R_E + 32 km). This matches the form of option (a).

3) Calculations and intermediate steps
Start with U(r) = −G M m / r.
Set r = R_E + h.
h = 32 000 m = 32 km, so r = R_E + 32 km.
Hence U = −G M m / (R_E + 32 km) or equivalently U = −G M m / (R_E + 32 000 m) if using SI units (meters). No further numeric evaluation is required for this multiple‑choice question.

4) Why the correct answer is correct
Option (a) explicitly uses the universal formula and places the radius equal to Earth’s radius plus the altitude, which is the physically correct distance from the centre. That yields the exact expression for gravitational potential energy at altitude h.

5) Why the other options are incorrect
- (b) U = −G M / R_E: Missing the factor m (mass of the object) and uses R_E only (surface), so it is both dimensionally wrong and not at the given altitude.
- (c) U = −G M m / (32000 km + R_E): Uses 32 000 km instead of 32 km (an error by a factor of 1000), so the radius is wrong.
- (d) U = −G M m / R_E: This is the potential at the Earth’s surface (h = 0), not at h = 32 km.
- (e) U = −G M m / (R_E − 32 km): Subtracts the altitude from R_E, which would place the mass inside the Earth’s radius and is physically incorrect.

6) Conclusion
Therefore the correct expression is U = −G M m / (R_E + 32 km), which corresponds to option (a). This matches the provided answer.

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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