Question

Derive relation between $g$ and $G$

Answer 1

Newton's law of gravitation states that the force between two unknown masses is directly proportional to the force acting between them. The force is inversely proportional to the square of distance between the masses. Also these masses experience acceleration due to gravity as well.

Complete step by step solution:
The acceleration of a body in free fall due to the massive body's gravity is g. The attraction force between two objects with a unit mass divided in some portion of this universe by a unit distance is G.

The gravity of any large body is g. The inertia on an object. A universal gravitational constant denoting G is the attraction force between any two masses divided by unit size. There is no proportional relationship between G and g. That implies that they are distinct bodies.

Let’s consider two bodies of masses $M$ and $m$ kept at distance $r$ from each other, now, according to Newton’s law of gravitation, we know that,
$F = \dfrac{{GMm}}{{{r^2}}}$
$F$ is the force between the two bodies
$G$ is the gravitational constant
$M$ is mass for first body
$m$ is mass of second body
$r$ is distance between two bodies
Let us consider that the first body is earth with mass $M$, $r$ radius. Now the force acting along the body will be
$F = mg$
$g$ is acceleration due to gravity
From the above two equations, we can write that,
$\dfrac {{GMm}} {{{r^2}}} = mg$
$\Rightarrow g = \dfrac{{GM}}{{{r^2}}}$

Hence we have a relation between $g$ and $G$.

Note: Although the relationship between g and G in physics can be expressed in a shape. Because of the gravity and the universal gravity, there is no relation between the acceleration and the G value. For some point in this world, the value of G is constant. G and g are not mutually based.