Question

Derive the relationship between '$g$‘ and ’$G$' .

Answer 1

Step 1: Let's assume that,

Mass of the body = $m$

Acceleration due to gravity = $g$

Universal gravitation constant = $G$

Mass of earth = $M_E$

Radius of earth = $R_E$

Step 2: Formula used

$F=\frac{Gxmx{M}_E}{{\left(R_E\right)}^2}$ and

$F=mxg$ ( where $F$ is the force on the body in both cases )

Step 3: Calculation

We know that, $F=\frac{Gxmx{M}_E}{{\left(R_E\right)}^2}$ -----------------------(1)

Also, $F=mxg$ -----------------------(2)

Comparing the values of $F$ from equation (1) and equation (2), we get

$mxg=\frac{Gxmx{M}_E}{{\left(R_E\right)}^2}$

or, $g=\frac{Gx{M}_E}{{\left(R_E\right)}^2}$

Hence, the relation between $g$ and $G$ is $g=\frac{Gx{M}_E}{{\left(R_E\right)}^2}$.