If `M` is the mass of the earth and `R` its radius, the radio of the gravitational acceleration and the gravitational constant is

Options

A. `(a)(R^(2))/(M)`
B. `(b)(M)/(R^(2))`
C. `(c)MR^(2)`
D. `(d)(M)/(R)`

Explanation

To find the ratio of gravitational acceleration (g) to the gravitational constant (G), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Gravitational Force Equation**:
The gravitational force (F) acting on a mass (m) at the surface of the Earth can be expressed using Newton's law of gravitation:
\[ F = \frac{G M m}{R^2} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( R \) is the radius of the Earth.

2. **Relate Force to Gravitational Acceleration**:
The gravitational force can also be expressed in terms of mass and acceleration due to gravity (g):
\[ F = m g \]

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

4. **Cancel Mass (m)**:
Assuming \( m \neq 0 \), we can divide both sides of the equation by \( m \):
\[ g = \frac{G M}{R^2} \]

5. **Rearrange to Find the Ratio**:
We are interested in the ratio of \( g \) to \( G \):
\[ \frac{g}{G} = \frac{G M}{R^2 G} \]
This simplifies to:
\[ \frac{g}{G} = \frac{M}{R^2} \]

6. **Find the Required Ratio**:
To express the ratio of \( G \) to \( g \), we take the reciprocal:
\[ \frac{G}{g} = \frac{R^2}{M} \]

### Final Result:
The ratio of the gravitational constant \( G \) to the gravitational acceleration \( g \) is:
\[ \frac{G}{g} = \frac{R^2}{M} \]