he value of 'g' at a certain height h above the free surface of Earth is `x/(16)` where x is the value 16 of 'g' at the surface of Earth. The height h is

Options

A. R
B. 2R
C. 3R
D. 4R

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We know that the value of 'g' at a height 'h' above the Earth's surface is given as \( g' = \frac{x}{16} \), where \( x \) is the value of 'g' at the surface of the Earth.

### Step 2: Write the formula for gravitational acceleration at height 'h'
The formula for gravitational acceleration at a height 'h' above the Earth's surface is given by:
\[ g' = \frac{g}{(1 + \frac{h}{r})^2} \]
where:
- \( g' \) = acceleration due to gravity at height \( h \)
- \( g \) = acceleration due to gravity at the surface of the Earth
- \( r \) = radius of the Earth

### Step 3: Substitute the known values
From the problem, we substitute \( g' = \frac{x}{16} \) and \( g = x \):
\[ \frac{x}{16} = \frac{x}{(1 + \frac{h}{r})^2} \]

### Step 4: Cancel \( x \) from both sides
Since \( x \) is common on both sides, we can cancel it out (assuming \( x \neq 0 \)):
\[ \frac{1}{16} = \frac{1}{(1 + \frac{h}{r})^2} \]

### Step 5: Cross-multiply to simplify
Cross-multiplying gives us:
\[ (1 + \frac{h}{r})^2 = 16 \]

### Step 6: Take the square root of both sides
Taking the square root of both sides results in:
\[ 1 + \frac{h}{r} = 4 \]

### Step 7: Solve for \( \frac{h}{r} \)
Rearranging gives:
\[ \frac{h}{r} = 4 - 1 = 3 \]

### Step 8: Solve for \( h \)
Multiplying both sides by \( r \):
\[ h = 3r \]

### Conclusion
The height \( h \) above the Earth's surface where the value of 'g' is \( \frac{x}{16} \) is:
\[ h = 3r \]