What will be the formula of the mass in terms of g,R and G ? (R = radius of the earth)

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 the acceleration due to gravity (g), the radius of the Earth (R), and the gravitational constant (G), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Forces**:
When an object of mass \( m \) is at the surface of the Earth, it experiences a gravitational force acting downwards, which can be expressed as:
\[ F = mg \]
where \( g \) is the acceleration due to gravity at the surface of the Earth.

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

3. **Setting the Forces Equal**:
Since both expressions represent the same gravitational force acting on the object, we can set them equal to each other:
\[ mg = \frac{GMm}{R^2} \]

4. **Canceling the Mass of the Object**:
We can cancel the mass \( m \) from both sides of the equation (assuming \( m \neq 0 \)):
\[ g = \frac{GM}{R^2} \]

5. **Rearranging the Equation**:
To find the mass of the Earth \( M \), we can rearrange the equation:
\[ M = \frac{gR^2}{G} \]

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