To determine the ratio of the gravitational acceleration (g) to the gravitational constant (G), we need to start by understanding the relationship between them.
1. Gravitational Acceleration (g):
The formula for the gravitational acceleration at the surface of a planet (in this case, the Earth) is given by:
[tex]\[
g = \frac{G \cdot M}{R^2}
\][/tex]
where:
- [tex]\( G \)[/tex] is the universal gravitational constant.
- [tex]\( M \)[/tex] is the mass of the Earth.
- [tex]\( R \)[/tex] is the radius of the Earth.
2. Ratio of [tex]\( g \)[/tex] to [tex]\( G \)[/tex]:
We are required to find the ratio [tex]\( \frac{g}{G} \)[/tex].
Substitute the formula for [tex]\( g \)[/tex] into this ratio:
[tex]\[
\frac{g}{G} = \frac{\frac{G \cdot M}{R^2}}{G}
\][/tex]
3. Simplify the Expression:
Simplify the fraction by canceling out [tex]\( G \)[/tex]:
[tex]\[
\frac{g}{G} = \frac{G \cdot M}{R^2 \cdot G} = \frac{M}{R^2}
\][/tex]
Therefore, the ratio of the gravitational acceleration (g) to the gravitational constant (G) is [tex]\( \frac{M}{R^2} \)[/tex].
So, the correct answer is:
(b) [tex]\(\frac{M}{R^2}\)[/tex]
