Final answer:
The gravitational potential at a point half the radius of Earth away from the center, considering Earth's surface potential as V₀, is not V₀/4 but would be doubled when using the standard -GM/r formula for gravitational potential. However, due to the inverse square law of gravity, the gravitational force and hence weight would reduce by a factor of 1/4 if Earth's radius were doubled, not the potential.
Step-by-step explanation:
A student inquired about the gravitational potential at a point half the radius of Earth away from the center, given that the gravitational potential on Earth's surface is V₀. The general principle to answer this question involves the understanding that the gravitational potential V at a distance r from the center of an object, like the Earth, is inversely proportional to r. Therefore, if one were at a point half the radius (r/2) away from the center of the Earth, the potential would not be half of V₀, but instead inversely scaled.
Calculating the new potential at this point involves the relation V = -GM/r, where G is the gravitational constant, M is the mass of the Earth, and r is the distance from the Earth's center. When the radius (distance from the center) is halved, the potential's magnitude increases by a factor of 2 (V becomes -2GM/(r/2)). If we compare this to the potential at the Earth's surface, V₀, which is -GM/r, the new potential is double. It's important to realize, however, that gravitational potential is physically meaningful only in terms of differences. Thus, directly comparing potentials as fractions or multiples can lead to confusion if the reference points and sign conventions are not clear.
Based on the inverse square law for gravity, it's easier to understand changes in force rather than potential. If Earth's radius were to double, the gravitational force at the new surface would be (1/2)² = 1/4 of the original force at Earth's surface due to the inverse square property of gravitational force. Therefore, a person would weigh one-fourth as much at the doubled radius.