The weight of a body on the surface of the earth is 12.6 N. When it is raised to height half the radius of earth its weight will be

Options

A. 2.8 N
B. 5.6 N
C. 12.5 N
D. 25.2 N

Answer

To solve the problem of finding the weight of a body when it is raised to a height equal to half the radius of the Earth, we can follow these steps:

### Step 1: Understand the given information
The weight of the body on the surface of the Earth is given as \( W = 12.6 \, \text{N} \).

### Step 2: Define the variables
Let:
- \( W \) = weight of the body on the surface of the Earth = \( 12.6 \, \text{N} \)
- \( R \) = radius of the Earth
- \( h \) = height above the Earth's surface = \( \frac{R}{2} \)

### Step 3: Use the formula for weight at height
The weight of the body at a height \( h \) above the surface of the Earth can be calculated using the formula:
\[ W' = W \left( \frac{R}{R + h} \right)^2 \]
where \( W' \) is the weight at height \( h \).

### Step 4: Substitute the values
Substituting \( h = \frac{R}{2} \) into the formula:
\[ W' = W \left( \frac{R}{R + \frac{R}{2}} \right)^2 \]
This simplifies to:
\[ W' = W \left( \frac{R}{\frac{3R}{2}} \right)^2 = W \left( \frac{2}{3} \right)^2 \]

### Step 5: Calculate \( W' \)
Now, substituting \( W = 12.6 \, \text{N} \):
\[ W' = 12.6 \left( \frac{2}{3} \right)^2 = 12.6 \left( \frac{4}{9} \right) \]
Calculating this gives:
\[ W' = \frac{12.6 \times 4}{9} = \frac{50.4}{9} \approx 5.6 \, \text{N} \]

### Final Answer
Thus, the weight of the body when raised to a height equal to half the radius of the Earth is approximately \( 5.6 \, \text{N} \).

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