Question

The relationship between the acceleration due the gravity (g) and universal gravitational constant (G) may be represented by
(M and R are the mass and the radius of the earth respectively)
A) $G = \dfrac{{gM}}{{\;{R^2}}}$
B) $g = \dfrac{{GM}}{{\;{R^2}}}$
C) $g = \dfrac{G}{{\;{R^2}}}$
D) None of these

Answer 1

Hint
To find the relation between the acceleration due to gravity ($g$) and universal gravitational constant ($G$) we can use Universal law of gravity as reference. According to the universal law of gravity $F = \dfrac{{GMm}}{{\;{R^2}}}$.
And to give the replacement of force from the above equation we can use Newton's second law of the motion- $g = \dfrac{F}{m}$.
Hence using replacement we can easily find the relation between the acceleration due the gravity ($g$) and universal gravitational constant ($G$).

Complete step-by-step answer:
According to the universal law of gravity
$F = \dfrac{{GMm}}{{\;{R^2}}}$ ………… (1)
$F$ = represent the gravitational force between the object,
$G$ = universal gravitational constant.
Here we can replace the force with the acceleration due to the gravity ($g$) by Newton’s second law of Motion
Newton's second law of the motion:
$g = \dfrac{F}{m}$……….. (2)
Substituting the equation (2) in equation (1)
We get,
$g = \dfrac{{GMm}}{{\;{R^2}m}}$
Or in a simple way it can be also written as
$g = \dfrac{{GM}}{{\;{R^2}}}$
Hence the correct answer is given by
g = $\dfrac{{GM}}{{\;{R^2}}}$.
So option (B) is correct.

Note
Sir Isaac Newton developed many theories or laws regarding the motion of the body. The value of G is constant at any point in the universe therefore it is called the universal constant at any point. The value of acceleration due to gravity is $9.8 \dfrac m{s^2}$, its value differs in other planes because it depends on the mass.