If acceleration due to gravity at distance d ` [ < R ] ` from the centre of earth is `beta ` , then its value at distance d above the surface of earth will be [ where R is radius of earth ]

Options

A. `(betaR^(2))/((R+d)^(3))`
B. `(betaR)/(2d)`
C. `(betad)/((R+d)^(3))`
D. `(betaR^(3))/(d(R+d)^(2))`

Answer

To solve the problem, we need to find the acceleration due to gravity at a distance \( d \) above the surface of the Earth, given that the acceleration due to gravity at a distance \( d \) inside the Earth (where \( d < R \), with \( R \) being the radius of the Earth) is \( \beta \).

### Step-by-Step Solution:

1. **Understanding the Acceleration Inside the Earth**:
The acceleration due to gravity \( g \) at a distance \( d \) from the center of the Earth (where \( d < R \)) is given by the formula:
\[ g = \frac{GMd}{R^3} \]
where \( G \) is the universal gravitational constant and \( M \) is the mass of the Earth. According to the problem, at this distance \( d \), the acceleration due to gravity is \( \beta \):
\[ \beta = \frac{GMd}{R^3} \]

2. **Finding the Acceleration Above the Surface of the Earth**:
Now, we need to find the acceleration due to gravity at a distance \( d \) above the surface of the Earth. The distance from the center of the Earth to this point is \( R + d \). The formula for the acceleration due to gravity at this distance is:
\[ g' = \frac{GM}{(R + d)^2} \]

3. **Expressing \( GM \) in Terms of \( \beta \)**:
From the first equation, we can express \( GM \) in terms of \( \beta \):
\[ GM = \frac{\beta R^3}{d} \]

4. **Substituting \( GM \) into the Second Equation**:
Now, substitute \( GM \) into the equation for \( g' \):
\[ g' = \frac{\frac{\beta R^3}{d}}{(R + d)^2} \]

5. **Final Expression for \( g' \)**:
Thus, we can simplify this to:
\[ g' = \frac{\beta R^3}{d (R + d)^2} \]

### Conclusion:
The acceleration due to gravity at a distance \( d \) above the surface of the Earth is:
\[ g' = \frac{\beta R^3}{d (R + d)^2} \]