A man weighs 'W' on the surface of the earth and his weight at a height 'R' from surface of the earth is (`R` is Radius of the earth )

Options

A. W
B. `W/2`
C. `W/4`
D. `W/8`

Answer

To solve the problem of finding the weight of a man at a height equal to the radius of the Earth, we can follow these steps:

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

### Step 2: Determine the height above the Earth's surface
The problem states that the man is at a height \( R \) from the surface of the Earth, where \( R \) is the radius of the Earth. Therefore, the total distance from the center of the Earth to the man is:
\[ h = R + R = 2R \]

### Step 3: Use the formula for gravitational acceleration at height
The gravitational acceleration \( g' \) at a height \( h \) from the surface of the Earth is given by the formula:
\[ g' = \frac{g R^2}{(R + h)^2} \]
Substituting \( h = R \):
\[ g' = \frac{g R^2}{(R + R)^2} = \frac{g R^2}{(2R)^2} = \frac{g R^2}{4R^2} = \frac{g}{4} \]

### Step 4: Calculate the weight at the height \( R \)
The weight of the man at this height \( W' \) can be calculated using the new gravitational acceleration \( g' \):
\[ W' = mg' = m \left(\frac{g}{4}\right) = \frac{mg}{4} \]
Since \( W = mg \), we can substitute:
\[ W' = \frac{W}{4} \]

### Step 5: Conclusion
Thus, the weight of the man at a height \( R \) from the surface of the Earth is:
\[ W' = \frac{W}{4} \]

### Final Answer
The man's weight at a height \( R \) from the surface of the Earth is \( \frac{W}{4} \).

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