At what distance from the centre of the earth, the value of acceleration due to gravity g will be half that on the surface ( R = radius of earth)

Options

A. `2 R`
B. `R`
C. `1.414 R`
D. `0.414 R`

Answer

To solve the problem of finding the distance from the center of the Earth where the acceleration due to gravity (g) is half of that at the surface, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We need to find the distance from the center of the Earth (let's denote it as \( d \)) where the acceleration due to gravity \( g' \) is half of the acceleration due to gravity at the surface \( g \).
- The acceleration due to gravity at the surface of the Earth is given by \( g = \frac{GM}{R^2} \), where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

2. **Using the Formula for Gravity at Height**:
- The formula for the acceleration due to gravity at a height \( h \) above the surface is given by:
\[ g' = \frac{gR^2}{(R + h)^2} \]
- Here, \( h \) is the height above the surface, and \( R \) is the radius of the Earth.

3. **Setting Up the Equation**:
- We want \( g' = \frac{g}{2} \). Substituting this into the equation gives:
\[ \frac{gR^2}{(R + h)^2} = \frac{g}{2} \]
- We can cancel \( g \) from both sides (assuming \( g \neq 0 \)):
\[ \frac{R^2}{(R + h)^2} = \frac{1}{2} \]

4. **Cross-Multiplying**:
- Cross-multiplying gives:
\[ 2R^2 = (R + h)^2 \]

5. **Expanding the Right Side**:
- Expanding \( (R + h)^2 \) results in:
\[ 2R^2 = R^2 + 2Rh + h^2 \]

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

7. **Using the Quadratic Formula**:
- We can solve for \( h \) using the quadratic formula \( h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 1 \), \( b = 2R \), and \( c = -R^2 \).
- The discriminant \( b^2 - 4ac = (2R)^2 - 4(1)(-R^2) = 4R^2 + 4R^2 = 8R^2 \).
- Thus, the solution for \( h \) becomes:
\[ h = \frac{-2R \pm \sqrt{8R^2}}{2} \]
\[ h = \frac{-2R \pm 2R\sqrt{2}}{2} \]
\[ h = -R + R\sqrt{2} \]
\[ h = R(\sqrt{2} - 1) \]

8. **Finding the Distance from the Center of the Earth**:
- The total distance from the center of the Earth is:
\[ d = R + h = R + R(\sqrt{2} - 1) = R\sqrt{2} \]

### Final Answer:
The distance from the center of the Earth where the acceleration due to gravity is half that at the surface is:
\[ d = R\sqrt{2} \approx 1.414R \]