At what depth from earth's surface does the acceleration due to gravity becomes `1/4` times that of its value at surface ?

Answer

To find the depth from the Earth's surface at which the acceleration due to gravity becomes \( \frac{1}{4} \) times its value at the surface, we can follow these steps:

### Step 1: Understand the formula for gravity at depth
The acceleration due to gravity \( g_d \) at a depth \( d \) below the Earth's surface is given by the formula:
\[ g_d = g \left(1 - \frac{d}{R}\right) \]
where:
- \( g \) is the acceleration due to gravity at the surface,
- \( R \) is the radius of the Earth,
- \( d \) is the depth below the surface.

### Step 2: Set up the equation
According to the problem, we want to find the depth \( d \) where the acceleration due to gravity \( g_d \) is \( \frac{1}{4} g \):
\[ g_d = \frac{1}{4} g \]

### Step 3: Substitute into the formula
Substituting \( g_d \) into the formula gives:
\[ \frac{1}{4} g = g \left(1 - \frac{d}{R}\right) \]

### Step 4: Cancel \( g \) from both sides
Since \( g \) is not zero, we can divide both sides by \( g \):
\[ \frac{1}{4} = 1 - \frac{d}{R} \]

### Step 5: Rearrange the equation
Rearranging the equation to isolate \( \frac{d}{R} \):
\[ \frac{d}{R} = 1 - \frac{1}{4} \]
\[ \frac{d}{R} = \frac{3}{4} \]

### Step 6: Solve for \( d \)
Now, multiply both sides by \( R \) to find \( d \):
\[ d = \frac{3}{4} R \]

### Conclusion
Thus, the depth \( d \) from the Earth's surface at which the acceleration due to gravity becomes \( \frac{1}{4} \) times that at the surface is:
\[ d = \frac{3}{4} R \]