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_E) in terms of gravitational acceleration (g), the radius of the Earth (R), and the gravitational constant (G), we can follow these steps:

### Step 1: Understand the relationship between gravitational force and gravitational acceleration
The gravitational force (F) acting on a mass (m) at the surface of the Earth is given by Newton's law of gravitation:
\[ F = \frac{G \cdot M_E \cdot m}{R^2} \]
where:
- \( G \) is the gravitational constant,
- \( M_E \) is the mass of the Earth,
- \( R \) is the radius of the Earth,
- \( m \) is the mass of the object experiencing the gravitational force.

### Step 2: Relate gravitational force to gravitational acceleration
The gravitational force can also be expressed in terms of gravitational acceleration (g):
\[ F = m \cdot g \]
where \( g \) is the acceleration due to gravity at the surface of the Earth.

### Step 3: Set the two expressions for force equal to each other
Since both expressions represent the same force acting on the mass \( m \), we can set them equal:
\[ m \cdot g = \frac{G \cdot M_E \cdot m}{R^2} \]

### Step 4: Simplify the equation
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ g = \frac{G \cdot M_E}{R^2} \]

### Step 5: Solve for the mass of the Earth (M_E)
To find \( M_E \), we can rearrange the equation:
\[ M_E = \frac{g \cdot R^2}{G} \]

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

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