What will be the acceleration due to gravity at height `h lt lt R` . Where R is radius of earth and g is acceleration to gravity on the surface earth

Options

A. `(g)/((1 + (h)/(R))^(2))`
B. `g (1 - (2h)/(R))`
C. `(g)/((1 - (h)/(R))^(2))`
D. ` g (1 - (h)/(R))`

Explanation

To find the acceleration due to gravity at a height \( h \) where \( h \ll R \) (with \( R \) being the radius of the Earth and \( g \) being the acceleration due to gravity at the surface), we can follow these steps:

### Step-by-Step Solution:

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

2. **Assume \( h \) is very small compared to \( R \)**:
Since \( h \ll R \), we can use the binomial approximation. The binomial expansion states that for small \( x \):
\[ (1 + x)^n \approx 1 + nx \]
Applying this to our formula, we can rewrite:
\[ (1 + \frac{h}{R})^{-2} \approx 1 - 2\frac{h}{R} \]

3. **Substitute the approximation back into the formula**:
Now substituting this approximation back into the formula for \( g' \):
\[ g' = g \cdot (1 - 2\frac{h}{R}) \]

4. **Final expression for acceleration due to gravity at height \( h \)**:
Thus, the acceleration due to gravity at height \( h \) is:
\[ g' \approx g \left(1 - \frac{2h}{R}\right) \]

### Conclusion:
The final expression for the acceleration due to gravity at a height \( h \) where \( h \ll R \) is:
\[ g' \approx g \left(1 - \frac{2h}{R}\right) \]