Question

What will be the formula of mass of the earth in terms of g, R and G?
 A. ${{g}^{2}}(\dfrac{R}{G})$
 B. $G(\dfrac{{{R}^{2}}}{g})$
 C. $G(\dfrac{{{R}^{2}}}{g})$
 D. $g(\dfrac{{{R}^{2}}}{G})$

Answer 1

Hint: The formula of mass of the earth in terms of g, R and G is obtained by newton’s law of gravitation. Newton’s law of gravitation is similar to Coulomb's law of electrical forces and the magnitude of the force on each object is the same, consistent with Newton's third law.Force which has both direction and magnitude(vector quantity).

Complete step-by-step solution:
The earth mass is a standard unit of mass that is used to indicate the masses of other planets. One solar mass is approximately equal to 333,000 earth masses. The mass of the moon is about 1.2% of that of the earth.
Mass of earth and moon is approximately equal to $6.0456\times {{10}^{-3}}kg$
Newton’s law of gravitation
It stated that any particle attracts any other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to square of the distance between their centers
$F=G\dfrac{{{m}_{1}}{{m}_{2}}}{{{R}^{2}}}$ $\cdots \cdots (1)$
F=force
G=gravitational constant
${{m}_{1}}$ = mass of object 1
${{m}_{2}}$ =mass of object 2
R=distance between centers of the masses
The fundamental force equation is given as
$F=ma$ $\cdots \cdots (2)$
m=mass
a =acceleration
We know that the acceleration due to gravity is equal to $9.8m{{s}^{-2}}$
The gravitational constant (G) is $6.673\times {{10}^{-11}}N{{m}^{2}}/k{{g}^{2}}$
The radius of the earth is $6.37\times {{10}^{6}}m$
Equating equation $(1)$ and $(2)$
We get
$F=G\dfrac{{{m}_{1}}{{m}_{2}}}{{{R}^{2}}}$ $=ma$
Mass cancels out .when we rearrange the equation and plug all the numbers in
\[G\dfrac{m}{{{R}^{2}}}=g\]
$m=g(\dfrac{{{R}^{2}}}{G})$
So option is D correct
The earth gains mass every day approximate mass of earth is
 $m=5.96\times {{10}^{24}}kg$

Note: Students the formula of mass of earth is obtained by Newton's law of gravitation and also by making use of fundamental force equations. Final mass formula as to be in terms g R and G and gravity is the weakest force of all other fundamental forces and gravitation is the proposed carrier particle for gravity.