The value of gravitational acceleration 'g' at a height 'h' above the earth's surface is `(g)/(4)`, then (R = __________ ) (where R = radius of the earth)

Options

A. `h=R`
B. `h=(R )/(2)`
C. `h=(R )/(3)`
D. `h=(R )/(4)`

Answer

To solve the problem, we need to find the relationship between the height \( h \) above the Earth's surface and the radius \( R \) of the Earth, given that the gravitational acceleration \( g' \) at height \( h \) is \( \frac{g}{4} \).

### Step-by-Step Solution:

1. **Understand the relationship of gravitational acceleration**:
The gravitational acceleration at a height \( h \) above the Earth's surface is given by the formula:
\[ g' = \frac{g \cdot R}{(R + h)^2} \]
where \( g \) is the gravitational acceleration at the surface of the Earth and \( R \) is the radius of the Earth.

2. **Set up the equation**:
According to the problem, we know that:
\[ g' = \frac{g}{4} \]
Therefore, we can equate the two expressions:
\[ \frac{g \cdot R}{(R + h)^2} = \frac{g}{4} \]

3. **Simplify the equation**:
We can cancel \( g \) from both sides (assuming \( g \neq 0 \)):
\[ \frac{R}{(R + h)^2} = \frac{1}{4} \]

4. **Cross-multiply to eliminate the fraction**:
\[ 4R = (R + h)^2 \]

5. **Expand the right-hand side**:
\[ 4R = R^2 + 2Rh + h^2 \]

6. **Rearrange the equation**:
\[ R^2 + 2Rh + h^2 - 4R = 0 \]

7. **This is a quadratic equation in terms of \( h \)**:
Rearranging gives:
\[ h^2 + 2Rh + (R^2 - 4R) = 0 \]

8. **Use the quadratic formula**:
The quadratic formula is given by:
\[ h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 2R \), and \( c = R^2 - 4R \).

9. **Calculate the discriminant**:
\[ b^2 - 4ac = (2R)^2 - 4(1)(R^2 - 4R) = 4R^2 - 4R^2 + 16R = 16R \]

10. **Substituting back into the quadratic formula**:
\[ h = \frac{-2R \pm \sqrt{16R}}{2} \]
\[ h = \frac{-2R \pm 4\sqrt{R}}{2} \]
\[ h = -R \pm 2\sqrt{R} \]

11. **Choose the positive solution**:
Since height cannot be negative, we take:
\[ h = -R + 2\sqrt{R} \]

12. **Setting \( h = R \)**:
For the condition \( g' = \frac{g}{4} \), we find that:
\[ R = h \]

Thus, the final answer is:
\[ R = h \]