In the reported figure of earth, the value of acceleration due to gravity is same at point A and C but it is smaller than that of its value at point B (surface of the earth). The value of $$OA : AB$$ will be $$x : 5$$. The value of $$x$$ is ______

We need to find the value of $$x$$ given that the ratio of the lengths $$OA : AB$$ is expressed as $$x : 5$$.

1. Identify the Given Parameters

From the problem :

• Radius of the Earth ($$R$$) = $$6400\text{ km}$$
• Height of point $$C$$ above the surface ($$h$$) = $$3200\text{ km}$$
• Point $$O$$ represents the center of the Earth.
• Point $$B$$ lies on the surface of the Earth, meaning $$OB = R = 6400\text{ km}$$.
• Point $$A$$ is an internal point at a depth $$d$$ below the surface, or at a distance $$r = OA$$ from the center.

2. Express Acceleration Due to Gravity at Points $$A$$ and $$C$$

• At point $$C$$ (at a height $$h$$ above the surface):
  The formula for acceleration due to gravity at an altitude is:

  $$g_C = \frac{g_0 R^2}{(R + h)^2}$$

  Substitute the given values for $$R$$ and $$h$$:

  $$g_C = \frac{g_0 (6400)^2}{(6400 + 3200)^2} = \frac{g_0 (6400)^2}{(9600)^2} = g_0 \left(\frac{2}{3}\right)^2 = \frac{4}{9}g_0$$

  (where $$g_0$$ is the acceleration due to gravity on the Earth's surface)


• At point $$A$$ (at a distance $$r = OA$$ from the center):
  The formula for acceleration due to gravity inside a solid uniform sphere is:

  $$g_A = \frac{g_0 \cdot r}{R}$$

3. Equate the Gravitational Fields to Find $$OA$$

The problem states that the acceleration due to gravity is identical at points $$A$$ and $$C$$ ($$g_A = g_C$$):

$$\frac{g_0 \cdot r}{R} = \frac{4}{9}g_0$$

Cancel out $$g_0$$ from both sides to solve for $$r$$ (which represents the length $$OA$$):

$$r = OA = \frac{4}{9}R$$

4. Determine the Ratio $$OA : AB$$

Since point $$B$$ lies on the surface, the total distance $$OB = OA + AB = R$$. Let's solve for the segment length $$AB$$:

$$AB = R - OA = R - \frac{4}{9}R = \frac{5}{9}R$$

Now, calculate the ratio of the lengths $$OA : AB$$:

$$\frac{OA}{AB} = \frac{\frac{4}{9}R}{\frac{5}{9}R} = \frac{4}{5}$$

We are given that this ratio is equal to $$x : 5$$:

$$\frac{x}{5} = \frac{4}{5} \implies x = 4$$

Conclusion

The value of $$x$$ is 4.