Understanding Gravity and Depth

* Newton's Law of Universal Gravitation: The force of gravity between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

* Acceleration due to Gravity (g): This is the acceleration experienced by an object due to the Earth's gravitational pull. At the Earth's surface, it's approximately 9.8 m/s².

Setting up the Problem

1. Let 'R' be the radius of the Earth.

2. Let 'g' be the acceleration due to gravity at the surface.

3. We want to find the depth 'd' where the acceleration due to gravity is 'g/4'.

Calculations

* At the surface: The acceleration due to gravity is g = GM/R² (where G is the gravitational constant and M is the Earth's mass).

* At depth 'd': The distance from the center of the Earth is (R - d). The acceleration due to gravity at depth 'd' is g' = GM/(R - d)².

Solving for Depth (d)

We are given that g' = g/4. So:

g/4 = GM/(R - d)²

Substitute the expression for g:

(GM/R²)/4 = GM/(R - d)²

Simplify and solve for d:

1/4 = (R - d)²/R²

√(1/4) = (R - d)/R

1/2 = (R - d)/R

R/2 = R - d

d = R - R/2

d = R/2

Answer:

The depth from the surface of the Earth where the acceleration due to gravity is one-fourth of that at the surface is half the radius of the Earth.