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 \(d\) from the center of the Earth is \(\beta\).

### Step 1: Understand the relationship between gravity inside and outside the Earth
The acceleration due to gravity at a distance \(d\) from the center of the Earth, where \(d < R\) (the radius of the Earth), is given by:
\[ g' = g \left( \frac{d}{R} \right) \]
where \(g\) is the acceleration due to gravity at the surface of the Earth.

### Step 2: Relate \(\beta\) to \(g\)
From the above equation, we know:
\[ \beta = g \left( \frac{d}{R} \right) \]
From this, we can express \(g\) in terms of \(\beta\):
\[ g = \frac{\beta R}{d} \]

### Step 3: Find the distance above the surface of the Earth
The distance above the surface of the Earth when we are at a distance \(d\) from the center of the Earth is given by:
\[ h = d - R \]
Thus, the distance above the surface of the Earth is \(h\).

### Step 4: Calculate the acceleration due to gravity at height \(h\)
The acceleration due to gravity at a height \(h\) above the Earth's surface is given by:
\[ g_h = \frac{g}{\left(1 + \frac{h}{R}\right)^2} \]
Substituting \(h = d - R\) into the equation:
\[ g_h = \frac{g}{\left(1 + \frac{d - R}{R}\right)^2} = \frac{g}{\left(\frac{d}{R}\right)^2} \]

### Step 5: Substitute \(g\) in terms of \(\beta\)
Now substituting \(g = \frac{\beta R}{d}\):
\[ g_h = \frac{\frac{\beta R}{d}}{\left(\frac{d}{R}\right)^2} = \frac{\beta R}{d} \cdot \frac{R^2}{d^2} = \frac{\beta R^2}{d^3} \]

### Final Result
Thus, the acceleration due to gravity at a distance \(d\) above the surface of the Earth is:
\[ g_h = \frac{\beta R^2}{d^3} \]