What will be the acceleration due it gravity at a depth in Earth. where g is acceleration due to gravity on the earth ?

Options

A. `(g)/([1 + (d)/(R)]^(2))`
B. `g [1 - (2h)/(R)]`
C. `(g)/([1- (d)/(R)]^(3)]`
D. `g [1-(d)/(R)]`

Explanation

To find the acceleration due to gravity at a depth \( d \) inside the Earth, we can derive the formula step by step.

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find the acceleration due to gravity \( g' \) at a depth \( d \) from the surface of the Earth, where \( g \) is the acceleration due to gravity at the surface.

2. **Define Variables**:
- Let \( R \) be the radius of the Earth.
- Let \( g \) be the acceleration due to gravity at the surface.
- Let \( d \) be the depth from the surface.

3. **Distance from the Center**:
- At depth \( d \), the distance from the center of the Earth is \( r - d \), where \( r \) is the radius of the Earth.

4. **Acceleration due to Gravity at the Surface**:
- The acceleration due to gravity at the surface is given by:
\[ g = \frac{GM}{R^2} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

5. **Acceleration due to Gravity at Depth \( d \)**:
- The acceleration due to gravity at depth \( d \) can be expressed as:
\[ g' = \frac{GM'}{(R - d)^2} \]
where \( M' \) is the mass of the Earth enclosed within the radius \( R - d \).

6. **Mass Enclosed**:
- The mass \( M' \) can be expressed in terms of the density \( \rho \) of the Earth:
\[ M' = \rho \cdot \frac{4}{3} \pi (R - d)^3 \]

7. **Substituting Mass into the Equation**:
- Now substitute \( M' \) into the equation for \( g' \):
\[ g' = \frac{G \cdot \rho \cdot \frac{4}{3} \pi (R - d)^3}{(R - d)^2} \]

8. **Simplifying the Equation**:
- This simplifies to:
\[ g' = \frac{4}{3} \pi G \rho (R - d) \]

9. **Relating \( g' \) to \( g \)**:
- We know that at the surface:
\[ g = \frac{4}{3} \pi G \rho R \]
- Therefore, we can express \( g' \) in terms of \( g \):
\[ \frac{g'}{g} = \frac{R - d}{R} \]

10. **Final Formula**:
- Rearranging gives us:
\[ g' = g \left(1 - \frac{d}{R}\right) \]

### Conclusion:
Thus, the acceleration due to gravity at a depth \( d \) inside the Earth is given by:
\[ g' = g \left(1 - \frac{d}{R}\right) \]