A what height above the earth's surface the value of g becomes 25% of its value on the earth if radius of the earth is 6400km.

Answer

To solve the problem of finding the height above the Earth's surface where the value of \( g \) becomes 25% of its value on the Earth's surface, we can follow these steps:

### Step 1: Understand the relationship of \( g \) with height
The acceleration due to gravity at a height \( h \) above the Earth's surface is given by the formula:
\[ g' = g \cdot \frac{R^2}{(R + h)^2} \]
where:
- \( g' \) is the acceleration due to gravity at height \( h \),
- \( g \) is the acceleration due to gravity at the Earth's surface,
- \( R \) is the radius of the Earth,
- \( h \) is the height above the Earth's surface.

### Step 2: Set up the equation for \( g' \)
We know that \( g' \) is 25% of \( g \):
\[ g' = \frac{g}{4} \]
Substituting this into the equation gives:
\[ \frac{g}{4} = g \cdot \frac{R^2}{(R + h)^2} \]

### Step 3: Simplify the equation
We can cancel \( g \) from both sides (assuming \( g \neq 0 \)):
\[ \frac{1}{4} = \frac{R^2}{(R + h)^2} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ (R + h)^2 = 4R^2 \]

### Step 5: Take the square root of both sides
Taking the square root results in:
\[ R + h = 2R \]

### Step 6: Solve for \( h \)
Rearranging the equation gives:
\[ h = 2R - R = R \]
Thus, the height \( h \) is equal to the radius of the Earth.

### Step 7: Substitute the value of \( R \)
Given that the radius of the Earth \( R = 6400 \) km, we find:
\[ h = 6400 \text{ km} \]

### Final Answer
The height above the Earth's surface where the value of \( g \) becomes 25% of its value on the Earth is **6400 km**.

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