If `g` on the surface of the earth is `9.8 m//s^2`, its value at a depth of `3200km` is (Radius of the earth `=6400km`) is

Options

A. `9.8 ms^(-2)`
B. zero
C. `4.9 ms^(-2)`
D. `2.45 ms^(-2)`

Explanation

To find the value of gravitational acceleration \( g \) at a depth of \( 3200 \, \text{km} \) beneath the Earth's surface, we can use the formula for gravitational acceleration at a depth \( d \):

\[ g_d = \frac{GM}{(R - d)^2} \]

Where:
- \( g_d \) is the gravitational acceleration at depth \( d \),
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the Earth,
- \( R \) is the radius of the Earth,
- \( d \) is the depth.

### Step 1: Identify the known values
- Gravitational acceleration at the surface \( g = 9.8 \, \text{m/s}^2 \)
- Radius of the Earth \( R = 6400 \, \text{km} \)
- Depth \( d = 3200 \, \text{km} \)

### Step 2: Convert the depth into the same units as the radius
Since the radius is given in kilometers, we can keep the depth in kilometers for simplicity:
- \( d = 3200 \, \text{km} \)

### Step 3: Calculate \( R - d \)
\[ R - d = 6400 \, \text{km} - 3200 \, \text{km} = 3200 \, \text{km} \]

### Step 4: Use the formula for gravitational acceleration at depth
We know that the gravitational acceleration at the surface is given by:
\[ g = \frac{GM}{R^2} \]
At depth \( d \), we can express \( g_d \) as:
\[ g_d = \frac{GM}{(R - d)^2} \]

### Step 5: Substitute \( R - d \) into the equation
Since \( R - d = 3200 \, \text{km} \), we can substitute this into the equation:
\[ g_d = \frac{GM}{(3200 \, \text{km})^2} \]

### Step 6: Relate \( g_d \) to \( g \)
We can relate \( g_d \) to \( g \) using the ratio of the squares:
\[ g_d = g \cdot \left( \frac{R}{R - d} \right)^2 \]
Substituting the known values:
\[ g_d = 9.8 \cdot \left( \frac{6400}{3200} \right)^2 \]

### Step 7: Calculate the ratio
\[ \frac{6400}{3200} = 2 \]
Thus,
\[ g_d = 9.8 \cdot (2)^2 = 9.8 \cdot 4 \]

### Step 8: Final calculation
\[ g_d = 39.2 \, \text{m/s}^2 \]

### Step 9: Correct the calculation
However, we need to consider that the gravitational acceleration decreases with depth. The correct formula for gravitational acceleration at depth is:
\[ g_d = g \cdot \left(1 - \frac{d}{R}\right) \]
Substituting the values:
\[ g_d = 9.8 \cdot \left(1 - \frac{3200}{6400}\right) = 9.8 \cdot \left(1 - 0.5\right) = 9.8 \cdot 0.5 = 4.9 \, \text{m/s}^2 \]

### Conclusion
Thus, the value of \( g \) at a depth of \( 3200 \, \text{km} \) is:
\[ \boxed{4.9 \, \text{m/s}^2} \]