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

To determine the distances above and below the surface of the Earth at which a body will have the same weight, we can break down the problem as follows:

1. Break Down the Problem

1. Identify the gravitational force acting on a body at different distances from the center of the Earth.
2. Establish the equations for gravitational force at a distance $R + h$ (above the Earth's surface) and $R - d$ (below the Earth's surface).

2. Relevant Concepts

1. Gravitational force at a distance $r$ from the center of the Earth is given by: $F = \frac{G M m}{r^2}$ Where:

   • $G$ is the gravitational constant,
   • $M$ is the mass of the Earth,
   • $m$ is the mass of the body,
   • $r$ is the distance from the center of the Earth.

2. For the surface of the Earth, $r = R$.

3. Analysis and Detail

1. Weight above the surface (at distance $h$): $W_{above} = \frac{G M m}{(R + h)^2}$

2. Weight below the surface (at distance $d$):

   • The weight below the surface can be calculated using the formula for gravitational force inside a sphere: $W_{below} = \frac{G M_{\text{inside}} m}{(R - d)^2}$ Where $M_{\text{inside}} = M \left(\frac{R - d}{R}\right)^3$ Thus, the weight below becomes: $W_{below} = \frac{G M \left(\frac{R - d}{R}\right)^3 m}{(R - d)^2} = \frac{G M m}{R^2} \cdot \frac{(R - d)^3}{(R - d)^2 R^3}$ This simplifies to: $W_{below} = \frac{G M m (R - d)}{R^3}$

3. Setting the weights equal: $\frac{G M m}{(R + h)^2} = \frac{G M m (R - d)}{R^3}$ Since $G$, $M$, and $m$ cancel out: $\frac{1}{(R + h)^2} = \frac{R - d}{R^3}$

4. Rearranging the equation: $R^3 = (R - d)(R + h)^2$

4. Verify and Summarize

From the equation derived, we need to simplify it further to find relations for $h$ and $d$. This is a complex equation implying that $h$ and $d$ will not be equal directly, but will have certain properties.

Final Answer

The distances above and below the Earth's surface, where a body experiences the same weight, can be calculated from: $\frac{1}{(R + h)^2} = \frac{R - d}{R^3}$ Where $h$ is the height above the Earth's surface and $d$ is the depth below the surface. Solving this equation can yield precise values for $h$ and $d$. An analytical or numerical solution method is recommended to find specific values as this equation is contextual.