A body weighs W newton at the surface of the earth. Its weight at a height equal to half the radius of the earth, will be

Options

A. `(W)/(2)`
B. `(2W)/(3)`
C. `(4W)/(9)`
D. `(8W)/(27)`

Answer

To find the weight of a body at a height equal to half the radius of the Earth, we can follow these steps:

### Step 1: Understand the weight at the surface of the Earth
The weight of the body at the surface of the Earth is given as \( W \) newtons. The weight can be expressed as:
\[ W = mg \]
where \( m \) is the mass of the body and \( g \) is the acceleration due to gravity at the surface of the Earth.

### Step 2: Define the height
The height \( h \) at which we want to find the weight is equal to half the radius of the Earth. If we denote the radius of the Earth as \( R \), then:
\[ h = \frac{R}{2} \]

### Step 3: Calculate the acceleration due to gravity at height \( h \)
The formula for the acceleration due to gravity at a height \( h \) above the surface of the Earth is given by:
\[ g_h = \frac{g \cdot R^2}{(R + h)^2} \]
Substituting \( h = \frac{R}{2} \):
\[ g_h = \frac{g \cdot R^2}{\left(R + \frac{R}{2}\right)^2} = \frac{g \cdot R^2}{\left(\frac{3R}{2}\right)^2} \]
\[ g_h = \frac{g \cdot R^2}{\frac{9R^2}{4}} = \frac{4g}{9} \]

### Step 4: Calculate the weight at height \( h \)
Now, we can find the weight of the body at height \( h \) using the new value of gravity:
\[ W_h = mg_h \]
Substituting \( g_h \):
\[ W_h = m \left(\frac{4g}{9}\right) \]
Since \( W = mg \), we can express \( m \) in terms of \( W \):
\[ m = \frac{W}{g} \]
Thus:
\[ W_h = \frac{W}{g} \cdot \frac{4g}{9} = \frac{4W}{9} \]

### Conclusion
The weight of the body at a height equal to half the radius of the Earth is:
\[ W_h = \frac{4W}{9} \]

### Final Answer
The weight at a height equal to half the radius of the Earth will be \( \frac{4W}{9} \).

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