The depth at which the effective value of acceleration ot gravity is `(g)/(4)` us

Options

A. R
B. `(3 R)/(4)`
C. `(R)/(2)`
D. `(R)/(4)`

Explanation

To find the depth at which the effective value of acceleration due to gravity is \( \frac{g}{4} \), we can use the formula for the acceleration due to gravity at a depth \( d \) below the surface of the Earth. The formula is given by:

\[ g' = g \left(1 - \frac{d}{R}\right) \]

where:
- \( g' \) is the acceleration due to gravity at depth \( d \),
- \( g \) is the acceleration due to gravity at the surface,
- \( R \) is the radius of the Earth.

### Step-by-Step Solution:

1. **Set up the equation**: We want to find the depth \( d \) such that \( g' = \frac{g}{4} \). Therefore, we can substitute \( g' \) in the formula:
\[ \frac{g}{4} = g \left(1 - \frac{d}{R}\right) \]

2. **Cancel \( g \)**: Since \( g \) is not zero, we can divide both sides of the equation by \( g \):
\[ \frac{1}{4} = 1 - \frac{d}{R} \]

3. **Rearrange the equation**: We can rearrange the equation to isolate \( \frac{d}{R} \):
\[ \frac{d}{R} = 1 - \frac{1}{4} \]
\[ \frac{d}{R} = \frac{3}{4} \]

4. **Solve for \( d \)**: Now, we can solve for \( d \) by multiplying both sides by \( R \):
\[ d = \frac{3R}{4} \]

5. **Conclusion**: The depth at which the effective value of acceleration due to gravity is \( \frac{g}{4} \) is:
\[ d = \frac{3R}{4} \]

### Final Answer:
The depth \( d \) is \( \frac{3R}{4} \).

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