At what distance from the center of earth, the value of acceleration due to gravity g will be half of the surface (R = Radius of earth)

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We need to find the distance from Earth's center where gravitational acceleration becomes half of its surface value. On Earth's surface, gravity equals GM over R squared. We want to find where this becomes half. Let's establish our formulas. Surface gravity equals GM over R squared, where G is the gravitational constant, M is Earth's mass, and R is Earth's radius. At any distance r from the center, gravity equals GM over r squared. We need to find r where this equals half the surface gravity. Now let's set up our equation. We start with the condition that gravity at distance r equals half the surface gravity. Substituting our formulas gives us GM over r squared equals one half times GM over R squared. We can cancel GM from both sides, leaving us with one over r squared equals one over two R squared. Now we solve for r. From our equation one over r squared equals one over two R squared, we cross multiply to get two R squared equals r squared. Taking the square root of both sides gives us r equals square root of two R squared, which simplifies to R times square root of two. Therefore, the distance from Earth's center where gravity is half the surface value is R square root of two. To summarize: we found that gravity becomes half its surface value at a distance R square root of two from Earth's center. This is approximately 1.414 times Earth's radius, or about 41.4 percent farther than the surface. This demonstrates the inverse square law of gravitation - to halve the gravitational force, we need to increase the distance by a factor of square root of two.