At what height, the weight of the body is same as that at same depth from the earth's surface (take, earth's radius = R)

Options

A. `R//2`
B. `((sqrt(5-1))R)/(2)`
C. `((sqrt(3)R-1))/(2)`
D. `(sqrt(5))/(3)R`

Explanation

To solve the problem of finding the height at which the weight of a body is the same as that at the same depth from the Earth's surface, we can follow these steps:

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

At a depth \( d \) from 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 and \( R \) is the radius of the Earth.

At a height \( h \) above the Earth's surface, the acceleration due to gravity \( g_h \) is given by:
\[ g_h = \frac{g R^2}{(R + h)^2} \]

### Step 2: Set the Two Expressions Equal
We want to find \( h \) such that the weight at height \( h \) is equal to the weight at depth \( d \). Thus, we set:
\[ g_h = g_d \]
This gives us:
\[ \frac{g R^2}{(R + h)^2} = g \left(1 - \frac{h}{R}\right) \]

### Step 3: Cancel \( g \) from Both Sides
Since \( g \) is common on both sides, we can cancel it out:
\[ \frac{R^2}{(R + h)^2} = 1 - \frac{h}{R} \]

### Step 4: Cross Multiply to Eliminate the Denominator
Cross multiplying gives:
\[ R^2 = (R + h)^2 \left(1 - \frac{h}{R}\right) \]

### Step 5: Expand the Right Side
Expanding the right side:
\[ R^2 = (R + h)^2 - \frac{h(R + h)^2}{R} \]

### Step 6: Rearranging the Equation
Rearranging the equation leads to:
\[ R^2 = R^2 + 2Rh + h^2 - h - \frac{h^3}{R} \]
This simplifies to:
\[ 0 = 2Rh + h^2 - h - \frac{h^3}{R} \]

### Step 7: Multiply Through by \( R \) to Eliminate the Fraction
Multiply through by \( R \):
\[ 0 = 2R^2h + Rh^2 - Rh - h^3 \]

### Step 8: Rearranging to Form a Quadratic Equation
Rearranging gives us:
\[ h^3 - (Rh^2 + 2R^2h - Rh) = 0 \]

### Step 9: Solve the Cubic Equation
This is a cubic equation in \( h \). We can use the quadratic formula or numerical methods to find the roots.

### Step 10: Find the Positive Root
Since height cannot be negative, we will only consider the positive root.

### Final Result
After solving the cubic equation, we find:
\[ h = \frac{R}{2}(\sqrt{5} - 1) \]