Problem 73 - A Complete Resource Book in Physics for JEE Main

Question

The change in the value of \(g\) at a height \(h\) above the surface of the earth is the same as that of a depth \(d\) below the surface of earth. When both \(d\) and \(h\) are much smaller than the radius of earth, then which one of the following is correct? (A) \(d=\frac{h}{2}\) (B) \(d=\frac{3 h}{2}\) (C) \(d=2 h\) (D) \(d=h\)

Step-by-Step Solution

Answer

\(d=h\)

Step 1: Calculate the gravitational force above the earth's surface

First, let's calculate the value of gravitational force at a height h above the earth's surface. The expression for gravitational force at a height h above the earth's surface is: \(g_h = g*\frac{R^2}{(R+h)^2}\) where: \(g_h\) - gravitational force at a height h above the earth's surface, \(g\) - gravitational constant, \(R\) - radius of the earth.

Step 2: Calculate the gravitational force below the earth's surface

Now, let's calculate the value of the gravitational force at a depth d below the earth's surface. The expression for gravitational force at a depth d below the earth's surface is: \(g_d = g*(1 - \frac{d}{R})\) where: \(g_d\) - gravitational force at a depth d below the earth's surface.

Step 3: Equate the expressions for gravitational forces

As given in the problem, the change in the value of g is the same at height h and depth d. Therefore, we can equate the expressions for gravitational forces above and below the earth's surface: \(g*\frac{R^2}{(R+h)^2}=g*(1 - \frac{d}{R})\)

Step 4: Simplify the equation and solve for d

To simplify the equation, cancel the g term and cross-multiply: \(\frac{R^2}{(R+h)^2} = 1 - \frac{d}{R}\) Now, let's cross-multiply and rearrange the equation to find d: \(R^2 = R(R + h)^2 - d(R + h)^2\) \(d(R + h)^2 = h^2 R\) As h and d are much smaller than R, we can write: \(d h^2 = h^2 R\) Finally, solving for d, we have: \(d = h\) Therefore, the correct answer is: (D) \(d=h\)