Question

SI unit of G in Newton’s law of gravitation is
A. $\dfrac{{{N^2}{m^2}}}{{kg}}$
B. $\dfrac{{N{m^2}}}{{kg}}$
C. $\dfrac{{Nm}}{{kg}}$
D. $\dfrac{{N{m^2}}}{{k{g^2}}}$

Answer 1

Hint: Units and Dimensions play key roles in physics. Various quantities have various units. SI units of the main fundamental quantities are kilogram for mass while meter for length second for time kelvin for temperature while ampere for the current. By using these units we will solve this question.

Formula used:
$F = \dfrac{{GMm}}{{{r^2}}}$

Complete step by step answer:
Gravitational force will be acting between any two masses at a particular distance. Since the magnitude of that force is very small nobody feels it. There are some similarities between the gravitational force and electrostatic force. gravitational force acts between the two masses whereas electrostatic force acts between the two charges. Both follow the inverse square law which means that the force will be inversely proportional to the square of distance between them.
So Newton discovered this gravitational force and due to this everybody on the earth is acted upon by the force due to the earth and this is what we call as the weight of the body.
We have the expression for the force between earth of mass ‘M’ and body of mass ‘m’ which are separated at the distance ‘r’ from the center of the earth
That expression is
$F = \dfrac{{GMm}}{{{r^2}}}$
Where ‘G’ is the gravitational constant.
So
$F = \dfrac{{GMm}}{{{r^2}}}$
$ \Rightarrow G = \dfrac{{F{r^2}}}{{Mm}}$
SI Units of force is newton(N) and mass is kilogram(kg) and distance(radius) is meter(m). by substituting the above formula with units, we get
$G = \dfrac{{F{r^2}}}{{Mm}}$
$\eqalign{
  & \Rightarrow G = \dfrac{{N{m^2}}}{{(kg)(kg)}} \cr
  & \therefore G = \dfrac{{N{m^2}}}{{k{g^2}}} \cr} $
Hence option D is correct.

Note:
There are some quantities which have units while don't have dimensions. Few instances for that are plane angle which has unit or radian but no dimensions and solid angle which has a unit of steradian and no dimension and angular displacement also has unit of radian but no dimension.