Altitude at which acceleration due to gravity decreases by `0.1%` approximately : (Radius of earth = 6400 km)

Options

A. 3 . 2 km
B. 6 . 4 km
C. 2 . 4 km
D. 1 . 6 km

Explanation

To solve the problem of finding the altitude at which the acceleration due to gravity decreases by 0.1%, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find the altitude \( h \) at which the acceleration due to gravity \( g' \) decreases by 0.1% compared to the acceleration due to gravity at the surface of the Earth \( g \).

2. **Using the Formula for Gravitational Acceleration**:
The acceleration due to gravity at a height \( h \) above the Earth's surface is given by:
\[ g' = g \left(1 - \frac{2h}{R}\right) \]
where \( R \) is the radius of the Earth.

3. **Setting Up the Equation**:
We know that the decrease in gravity is 0.1%, which can be expressed as:
\[ \frac{g' - g}{g} \times 100 = -0.1 \]
This can be rearranged to:
\[ g' - g = -0.001g \]
Therefore, we can write:
\[ g' = g - 0.001g = 0.999g \]

4. **Substituting \( g' \) in the Formula**:
Now substituting \( g' \) in the formula:
\[ 0.999g = g \left(1 - \frac{2h}{R}\right) \]

5. **Canceling \( g \)**:
Since \( g \) is not zero, we can divide both sides by \( g \):
\[ 0.999 = 1 - \frac{2h}{R} \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ \frac{2h}{R} = 1 - 0.999 = 0.001 \]

7. **Solving for \( h \)**:
Now, multiplying both sides by \( R \):
\[ 2h = 0.001R \]
Thus,
\[ h = \frac{0.001R}{2} \]

8. **Substituting the Value of \( R \)**:
Given that \( R = 6400 \) km, we substitute:
\[ h = \frac{0.001 \times 6400}{2} \]
\[ h = \frac{6.4}{2} = 3.2 \text{ km} \]

### Final Answer:
The altitude at which the acceleration due to gravity decreases by 0.1% is approximately **3.2 km**.