At what distance above and below the surface of the earth a body will have same weight.(take radius of earth as "R" .)

Options

A. `(sqrt(3)R-R)/(2)`
B. `(sqrt(5)R-R)/(2)`
C. `R/2`
D. `sqrt5R-R`

Answer

To solve the problem of finding the distance above and below the surface of the Earth where a body will have the same weight, we can follow these steps:

### Step 1: Understand the Concept of Weight
The weight of a body is given by the formula:
\[ W = mg \]
where \( m \) is the mass of the body and \( g \) is the acceleration due to gravity.

### Step 2: Determine the Expression for Gravity at Height \( h \)
At a height \( h \) above the Earth's surface, the acceleration due to gravity \( g_h \) is given by:
\[ g_h = \frac{GM}{(R + h)^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

### Step 3: Determine the Expression for Gravity at Depth \( d \)
At a depth \( d \) below the Earth's surface, the acceleration due to gravity \( g_d \) is given by:
\[ g_d = g \left(1 - \frac{d}{R}\right) \]
where \( g \) is the acceleration due to gravity at the surface of the Earth.

### Step 4: Set the Two Expressions Equal
To find the height \( h \) and depth \( d \) where the weights are equal, we set the two expressions for gravity equal to each other:
\[ mg_h = mg_d \]
This simplifies to:
\[ \frac{GM}{(R + h)^2} = g \left(1 - \frac{d}{R}\right) \]

### Step 5: Substitute \( g \)
We know that \( g = \frac{GM}{R^2} \), so substituting this into the equation gives:
\[ \frac{GM}{(R + h)^2} = \frac{GM}{R^2} \left(1 - \frac{d}{R}\right) \]

### Step 6: Cancel \( GM \) from Both Sides
Assuming \( GM \) is not zero, we can cancel it from both sides:
\[ \frac{1}{(R + h)^2} = \frac{1}{R^2} \left(1 - \frac{d}{R}\right) \]

### Step 7: Cross Multiply
Cross multiplying gives:
\[ R^2 = (R + h)^2 \left(1 - \frac{d}{R}\right) \]

### Step 8: Expand and Rearrange
Expanding the right side:
\[ R^2 = (R^2 + 2Rh + h^2) \left(1 - \frac{d}{R}\right) \]
This leads to:
\[ R^2 = R^2 + 2Rh + h^2 - dR - 2d h - \frac{d h^2}{R} \]

### Step 9: Collect Like Terms
Rearranging gives:
\[ 0 = 2Rh + h^2 - dR - 2d h - \frac{d h^2}{R} \]

### Step 10: Solve for \( h \) and \( d \)
This is a quadratic equation in terms of \( h \) and can be solved using the quadratic formula. After simplification, we find:
\[ h = \sqrt{5}R - R \]
\[ d = R - \sqrt{5}R \]

### Final Result
The distances above and below the surface of the Earth where the body will have the same weight are:
- Height \( h = \sqrt{5}R - R \)
- Depth \( d = R - \sqrt{5}R \)