At a distance of 2R from the centre of the Earth (where R is the radius of the Earth), a person weight becomes what fraction of the weight at the Earth surface?

Introduction / Context:
Gravitational force and weight depend on the distance from the centre of the Earth. Many competitive exams ask how weight changes with altitude. This question asks what happens to a person weight when the person is located at a distance of 2R from the centre of the Earth, where R is the radius of the Earth, and we compare this with the weight at the Earth surface (distance R from the centre).

Given Data / Assumptions:
- Radius of the Earth = R.- Distance of the person from the centre of the Earth = 2R.- Weight is proportional to gravitational force between the Earth and the person.- We ignore effects of rotation and atmosphere and use the inverse square law for gravity.

Concept / Approach:
According to Newton law of gravitation, the gravitational force F between two masses is given by F = G * M * m / r^2, where r is the distance between their centres. For a body on the Earth surface, r = R. At a distance 2R from the centre, r = 2R. Since weight W is equal to m * g and g is proportional to 1 / r^2, the weight at different distances will scale inversely with the square of the distance from the centre of the Earth.

Step-by-Step Solution:
1. Let the weight at the Earth surface be W1. Then W1 is proportional to 1 / R^2.2. At distance r = 2R from the centre of the Earth, let the weight be W2, so W2 is proportional to 1 / (2R)^2.3. Compute (2R)^2 = 4R^2, so W2 is proportional to 1 / (4R^2).4. Compare W2 and W1: W2 / W1 = (1 / 4R^2) / (1 / R^2) = 1 / 4.5. Therefore, W2 = W1 / 4, meaning the weight becomes one-fourth of the original weight at the Earth surface.

Verification / Alternative check:
A general rule is that if you double the distance from the centre of the Earth, the gravitational force and therefore the weight fall to one fourth. This is a direct consequence of the inverse square law. The same logic applies to satellites and space stations at various altitudes, although they are also in orbital motion. The computed factor of one fourth is consistent with this known rule.

Why Other Options Are Wrong:
- Remains the same: This would imply that gravity does not change with distance, which contradicts the inverse square law.- Becomes half: Half would correspond roughly to a distance of about 1.4R, not 2R, so this is incorrect for 2R.- Becomes twice: Gravity weakens with distance, so weight cannot become twice at a point further from the centre of the Earth.

Common Pitfalls:
Many students confuse distance from the centre with height above the surface. Always read carefully whether the question gives height h or total distance r. In some questions, height equal to the radius of the Earth (h = R) leads to r = 2R, which also yields one-fourth weight. Here the distance is directly given as 2R from the centre, so the same 1 / 4 factor applies.

Final Answer:
At a distance of 2R from the centre of the Earth, the person weight becomes one-fourth of the weight at the Earth surface.