What will be the formula of mass of the earth in terms of `g` , `R` and `G` ?

Options

A. `g^(2)(R//G)`
B. `G(R^(2)//g)`
C. `G(R//g)`
D. `g(R^(2)//G)`

Answer

To derive the formula for the mass of the Earth (M) in terms of gravitational acceleration (g), the radius of the Earth (R), and the universal gravitational constant (G), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Gravitational Force**:
The gravitational force (F) acting on a mass (m) placed on the surface of the Earth can be expressed as:
\[ F = m \cdot g \]
where \( g \) is the acceleration due to gravity at the surface of the Earth.

2. **Apply Newton's Law of Gravitation**:
According to Newton's law of gravitation, the gravitational force between two masses (the Earth and the object) is given by:
\[ F = \frac{G \cdot M \cdot m}{R^2} \]
where \( M \) is the mass of the Earth, \( R \) is the radius of the Earth, and \( G \) is the universal gravitational constant.

3. **Set the Two Expressions for Force Equal**:
Since both expressions represent the same force acting on the mass \( m \), we can set them equal to each other:
\[ m \cdot g = \frac{G \cdot M \cdot m}{R^2} \]

4. **Cancel the Mass (m)**:
We can cancel \( m \) from both sides of the equation (assuming \( m \neq 0 \)):
\[ g = \frac{G \cdot M}{R^2} \]

5. **Rearrange the Equation to Solve for M**:
To find the mass of the Earth \( M \), we can rearrange the equation:
\[ M = \frac{g \cdot R^2}{G} \]

### Final Formula:
Thus, the formula for the mass of the Earth in terms of \( g \), \( R \), and \( G \) is:
\[ M = \frac{g \cdot R^2}{G} \]