If g is acceleration due to gravity at the surface of the earth, then its value at a depth of 1/4 of the radius of the earth is

Options

A. 0
B. g/4
C. g/2
D. 3g/4

Explanation

To find the value of acceleration due to gravity (g) at a depth of \( \frac{1}{4} \) of the radius of the Earth, we can follow these steps:

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

### Step 2: Substitute the given depth
We are given that the depth \( d \) is \( \frac{1}{4} R \). Therefore, we can substitute \( d \) in the formula:
\[ g_d = g \left(1 - \frac{\frac{1}{4} R}{R}\right) \]

### Step 3: Simplify the expression
Now, simplify the expression:
\[ g_d = g \left(1 - \frac{1}{4}\right) \]
\[ g_d = g \left(\frac{3}{4}\right) \]

### Step 4: Final result
Thus, the acceleration due to gravity at a depth of \( \frac{1}{4} \) of the radius of the Earth is:
\[ g_d = \frac{3}{4} g \]

### Conclusion
The value of acceleration due to gravity at a depth of \( \frac{1}{4} \) of the radius of the Earth is \( \frac{3}{4} g \).

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