The acceleration due to gravity decreases by `Deltag_(1)` when a body is taken to a small height `h lt lt R.` The acceleration due to gravity decreases by `Deltag_(2)` when the body is taken to a depth h from the surface off the earth, then (R= Radius of the earth)

Options

A. `Deltag_(1)=Deltag_(2)`
B. `Deltag_(1)=2Deltag_(2)`
C. `Deltag_(2)=2Deltag_(1)`
D. `Deltag_(1)=4Deltag_(2)`

Explanation

To solve the problem, we need to find the relationship between the changes in acceleration due to gravity when a body is taken to a height \( h \) above the Earth's surface and when it is taken to a depth \( h \) below the Earth's surface.

### Step-by-Step Solution

1. **Understanding the change in gravity at height \( h \)**:
When a body is taken to a height \( h \) above the Earth's surface, the acceleration due to gravity \( g' \) at that height can be expressed as:
\[ g' = g \left(1 - \frac{2h}{R}\right) \]
where \( g \) is the acceleration due to gravity at the surface and \( R \) is the radius of the Earth.

2. **Calculating the change in gravity \( \Delta g_1 \)**:
The change in acceleration due to gravity when moving to height \( h \) is given by:
\[ \Delta g_1 = g - g' = g - g \left(1 - \frac{2h}{R}\right) = g \left(1 - \left(1 - \frac{2h}{R}\right)\right) = g \frac{2h}{R} \]
Thus, we have:
\[ \Delta g_1 = \frac{2gh}{R} \]

3. **Understanding the change in gravity at depth \( h \)**:
When a body is taken to a depth \( h \) below the Earth's surface, the acceleration due to gravity \( g' \) at that depth can be expressed as:
\[ g' = g \left(1 - \frac{h}{R}\right) \]

4. **Calculating the change in gravity \( \Delta g_2 \)**:
The change in acceleration due to gravity when moving to depth \( h \) is given by:
\[ \Delta g_2 = g - g' = g - g \left(1 - \frac{h}{R}\right) = g \frac{h}{R} \]
Thus, we have:
\[ \Delta g_2 = \frac{gh}{R} \]

5. **Finding the relationship between \( \Delta g_1 \) and \( \Delta g_2 \)**:
Now, we can set up a ratio between \( \Delta g_1 \) and \( \Delta g_2 \):
\[ \frac{\Delta g_1}{\Delta g_2} = \frac{\frac{2gh}{R}}{\frac{gh}{R}} = \frac{2gh}{R} \cdot \frac{R}{gh} = 2 \]
Therefore, we find that:
\[ \Delta g_1 = 2 \Delta g_2 \]

### Final Result
The relationship between the changes in acceleration due to gravity is:
\[ \Delta g_1 = 2 \Delta g_2 \]