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, we will follow these steps:

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

### Step 2: Use the formula for gravitational force.
The weight of the body can be expressed using the formula for gravitational force:
\[ W = \frac{G M m}{R^2} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the body,
- \( R \) is the radius of the Earth.

### Step 3: Determine the new height.
The body is raised to a height \( h = \frac{R}{2} \) (half the radius of the Earth).

### Step 4: Apply the formula for weight at height \( h \).
When the body is at height \( h \), the weight can be calculated using the modified formula:
\[ W' = \frac{G M m}{(R + h)^2} \]
Substituting \( h = \frac{R}{2} \):
\[ W' = \frac{G M m}{\left(R + \frac{R}{2}\right)^2} = \frac{G M m}{\left(\frac{3R}{2}\right)^2} = \frac{G M m}{\frac{9R^2}{4}} = \frac{4 G M m}{9 R^2} \]

### Step 5: Relate the new weight to the original weight.
Since we know the original weight \( W = \frac{G M m}{R^2} \), we can express the new weight \( W' \) in terms of the original weight:
\[ W' = \frac{4}{9} W \]

### Step 6: Substitute the known weight.
Now, substituting the known weight:
\[ W' = \frac{4}{9} \times 12.6 \, \text{N} \]

### Step 7: Calculate the new weight.
Calculating this gives:
\[ W' = \frac{4 \times 12.6}{9} = \frac{50.4}{9} \approx 5.6 \, \text{N} \]

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

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