Acceleration due to gravity as same at height h from surface and at depth h from surface, then find value of h

Options

A. `(root5 -1)R/2`
B. `(root5)R/2 - 1`
C. `R/(root2)`
D. `((root5)R+r)/2`

Answer

To solve the problem of finding the height \( h \) at which the acceleration due to gravity is the same as at a depth \( h \) from the Earth's surface, we can follow these steps:

### Step 1: Write the expression for gravitational acceleration at height \( h \)
The gravitational acceleration \( g_h \) at a height \( h \) above the Earth's surface is given by the formula:
\[ g_h = \frac{G M_e}{(R_e + h)^2} \]
where \( G \) is the gravitational constant, \( M_e \) is the mass of the Earth, and \( R_e \) is the radius of the Earth.

### Step 2: Write the expression for gravitational acceleration at depth \( h \)
The gravitational acceleration \( g_d \) at a depth \( h \) below the Earth's surface is given by:
\[ g_d = \frac{G M_d}{R_e^2} = \frac{G \left(\frac{4}{3} \pi (R_e - h)^3 \rho\right)}{R_e^2} \]
Here, \( M_d \) is the mass of the Earth that is effective at that depth, and \( \rho \) is the density of the Earth.

### Step 3: Relate the mass at depth \( h \)
The mass \( M_d \) at depth \( h \) can be expressed as:
\[ M_d = \frac{4}{3} \pi (R_e - h)^3 \rho \]
Using the total mass of the Earth, we can write:
\[ M_e = \frac{4}{3} \pi R_e^3 \rho \]
Thus, we can express \( \rho \) as:
\[ \rho = \frac{3M_e}{4\pi R_e^3} \]

### Step 4: Substitute \( \rho \) back into the expression for \( g_d \)
Substituting \( \rho \) into the expression for \( g_d \):
\[ g_d = \frac{G \left(\frac{4}{3} \pi (R_e - h)^3 \cdot \frac{3M_e}{4\pi R_e^3}\right)}{R_e^2} \]
Simplifying this gives:
\[ g_d = \frac{G M_e (R_e - h)^3}{R_e^3 R_e^2} = \frac{G M_e (R_e - h)^3}{R_e^5} \]

### Step 5: Set \( g_h \) equal to \( g_d \)
Now we set \( g_h \) equal to \( g_d \):
\[ \frac{G M_e}{(R_e + h)^2} = \frac{G M_e (R_e - h)^3}{R_e^5} \]
We can cancel \( G M_e \) from both sides (assuming \( G \) and \( M_e \) are non-zero):
\[ \frac{1}{(R_e + h)^2} = \frac{(R_e - h)^3}{R_e^5} \]

### Step 6: Cross-multiply and simplify
Cross-multiplying gives:
\[ R_e^5 = (R_e + h)^2 (R_e - h)^3 \]
Expanding the right-hand side:
\[ R_e^5 = (R_e^2 + 2R_e h + h^2)(R_e^3 - 3R_e^2 h + 3R_e h^2 - h^3) \]

### Step 7: Solve for \( h \)
This equation can be solved for \( h \). However, it is complex and may require numerical methods or approximations to find the specific value of \( h \).

### Final Result
After solving the equation, we can find the value of \( h \) where the gravitational acceleration at height \( h \) equals that at depth \( h \).