Four times
Explanation
According to Newton's Law of Universal Gravitation, every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Mathematical Analysis
The relationship between gravitational force (F) and the distance (r) between two objects is given by the inverse square law:
F ∝ 1 / r²
- Let the initial distance be r. The initial force is proportional to 1/r².
- The new distance is reduced to half, so the new distance r' = r/2.
- Substituting the new distance into the formula:
New Force (F') ∝ 1 / (r/2)²
F' ∝ 1 / (r²/4)
F' ∝ 4 × (1/r²)
Since the original force F is proportional to 1/r², the new force F' is equal to 4F. Therefore, the gravitational force becomes four times the original value.
Key Takeaway
Inverse Square Law: Gravitational force is inversely proportional to the square of the distance. Halving the distance results in a fourfold increase in the force, while doubling the distance reduces the force to one-fourth.