At a distance 320 km above the surface of the earth , the value of acceleration due to gravity will be lower than its value on the surface of the earth by nearly ( radius of earth = 6400 km )

Options

A. `2%`
B. `6%`
C. `10%`
D. `14%`

Answer

To solve the problem of how much lower the acceleration due to gravity is at a height of 320 km above the Earth's surface compared to its value on the surface, we can follow these steps:

### Step 1: Understand the formula for acceleration due to gravity
The acceleration due to gravity \( g \) at a distance \( r \) from the center of the Earth is given by the formula:
\[ g = \frac{GM}{r^2} \]
where \( G \) is the universal gravitational constant and \( M \) is the mass of the Earth.

### Step 2: Determine the radius at the height of 320 km
The radius of the Earth is given as \( R = 6400 \) km. When we are at a height of 320 km above the surface, the distance from the center of the Earth becomes:
\[ r = R + h = 6400 \text{ km} + 320 \text{ km} = 6720 \text{ km} \]

### Step 3: Calculate the acceleration due to gravity on the surface of the Earth
The acceleration due to gravity at the surface of the Earth \( g_s \) is:
\[ g_s = \frac{GM}{R^2} = \frac{GM}{(6400 \text{ km})^2} \]

### Step 4: Calculate the acceleration due to gravity at the height of 320 km
The acceleration due to gravity at the height of 320 km \( g_a \) is:
\[ g_a = \frac{GM}{(6720 \text{ km})^2} \]

### Step 5: Find the ratio of \( g_a \) to \( g_s \)
To find how much lower \( g_a \) is compared to \( g_s \), we can take the ratio:
\[ \frac{g_a}{g_s} = \frac{GM / (6720 \text{ km})^2}{GM / (6400 \text{ km})^2} = \frac{(6400 \text{ km})^2}{(6720 \text{ km})^2} \]

### Step 6: Simplify the ratio
This simplifies to:
\[ \frac{g_a}{g_s} = \left(\frac{6400}{6720}\right)^2 \]
Calculating this gives:
\[ \frac{g_a}{g_s} = \left(\frac{64}{67.2}\right)^2 \approx \left(0.9524\right)^2 \approx 0.907 \]

### Step 7: Calculate the percentage decrease
To find the percentage decrease in \( g \):
\[ \text{Percentage decrease} = \left(1 - \frac{g_a}{g_s}\right) \times 100\% \approx (1 - 0.907) \times 100\% \approx 9.3\% \]

### Conclusion
Thus, the value of acceleration due to gravity at a height of 320 km above the Earth’s surface is approximately 9.3% lower than its value at the surface.