If earth has uniform density, and radius 'R'. The value of acceleration due to gravity at distance d above the surface is same d below the surface. If `d=((sqrtx-1)/(2))R`, then find x.

Answer

To solve the problem, we need to find the value of \( x \) given that the distance \( d \) is defined as:

\[ d = \frac{(\sqrt{x} - 1)}{2} R \]

where \( R \) is the radius of the Earth. We know that the acceleration due to gravity at a distance \( d \) above the surface of the Earth is equal to the acceleration due to gravity at a distance \( d \) below the surface of the Earth.

### Step 1: Write the expression for gravitational acceleration above the surface

The acceleration due to gravity at a distance \( d \) above the surface of the Earth is given by:

\[ g_{\text{above}} = \frac{GM}{(R + d)^2} \]

where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

### Step 2: Write the expression for gravitational acceleration below the surface

The acceleration due to gravity at a distance \( d \) below the surface of the Earth can be expressed as:

\[ g_{\text{below}} = \frac{GM'}{(R - d)^2} \]

where \( M' \) is the mass of the Earth that is enclosed within a sphere of radius \( R - d \). Since the Earth has uniform density, we can express \( M' \) as:

\[ M' = \rho \cdot \frac{4}{3} \pi (R - d)^3 \]

where \( \rho \) is the density of the Earth.

### Step 3: Set the two expressions equal to each other

Since \( g_{\text{above}} = g_{\text{below}} \), we have:

\[ \frac{GM}{(R + d)^2} = \frac{G \cdot \rho \cdot \frac{4}{3} \pi (R - d)^3}{(R - d)^2} \]

### Step 4: Simplify the equation

We can cancel \( G \) from both sides:

\[ \frac{M}{(R + d)^2} = \frac{\rho \cdot \frac{4}{3} \pi (R - d)^3}{(R - d)^2} \]

Now, we can express \( M \) in terms of \( \rho \):

\[ M = \rho \cdot \frac{4}{3} \pi R^3 \]

Substituting this into the equation gives:

\[ \frac{\rho \cdot \frac{4}{3} \pi R^3}{(R + d)^2} = \frac{\rho \cdot \frac{4}{3} \pi (R - d)^3}{(R - d)^2} \]

### Step 5: Cancel common terms

We can cancel \( \rho \cdot \frac{4}{3} \pi \) from both sides:

\[ \frac{R^3}{(R + d)^2} = \frac{(R - d)^3}{(R - d)^2} \]

This simplifies to:

\[ \frac{R^3}{(R + d)^2} = (R - d) \]

### Step 6: Cross-multiply and rearrange

Cross-multiplying gives:

\[ R^3 = (R - d)(R + d)^2 \]

Expanding the right side:

\[ R^3 = (R - d)(R^2 + 2Rd + d^2) \]

### Step 7: Substitute \( d \)

Substituting \( d = \frac{(\sqrt{x} - 1)}{2} R \):

\[ R^3 = \left(R - \frac{(\sqrt{x} - 1)}{2} R\right)\left(R^2 + 2R\left(\frac{(\sqrt{x} - 1)}{2} R\right) + \left(\frac{(\sqrt{x} - 1)}{2} R\right)^2\right) \]

### Step 8: Solve for \( x \)

After simplifying the equation, we will find the value of \( x \). The solution will yield:

\[ x = 5 \]

### Final Answer

Thus, the value of \( x \) is:

\[ \boxed{5} \]