Final answer:
If Earth's radius shrank to 0.25 of its original size, the gravitational force at Earth's surface would increase by a factor of 16, and thus a person would weigh 16 times their current weight, due to the inverse-square law of gravitation.
Step-by-step explanation:
To understand the impact of Earth's radius shrinking to 0.25 of its original size, we need to apply the law of universal gravitation, which states that the force of gravity is directly proportional to the product of the two masses and inversely proportional to the square of the distance between their centers.
If the Earth were to shrink to a quarter of its radius, the distance in the gravitational force equation would decrease, thereby increasing the gravitational force.
Given that the gravitational force varies inversely with the square of the radius, if Earth's radius decreased to 0.25 (or 1/4) of its original size, the gravitational force would increase by a factor of (1/0.25)², which is equivalent to 4² or 16.
Therefore, the relative strength of Earth's gravity would increase by a factor of 16, meaning a person would weigh 16 times as much as they do now, assuming Earth's mass remains constant.