Question

What is the dimensional formula of universal gravitational constant ‘G’? The gravitational force of attraction between two objects of masses ${{\text{m}}_1}$ and ${{\text{m}}_2}$ separated by a distance d is given by ${\text{F}} = \dfrac{{{\text{G}}{{\text{m}}_1}{{\text{m}}_2}}}{{{{\text{d}}^2}}}$ where ‘G’ is the universal gravitational constant.
$
  {\text{A}}{\text{. [ML}}{{\text{T}}^{ - 2}}{\text{]}} \\
  {\text{B}}{\text{. [}}{{\text{M}}^{ - 1}}{{\text{L}}^2}{{\text{T}}^2}{\text{]}} \\
  {\text{C}}{\text{. [}}{{\text{M}}^{ - 1}}{{\text{L}}^3}{{\text{T}}^{ - 2}}{\text{]}} \\
  {\text{D}}{\text{. [}}{{\text{M}}^2}{{\text{L}}^2}{{\text{T}}^{ - 2}}{\text{]}} \\
 $

Answer 1

Hint: Here, we will proceed by writing down the formula corresponding to the gravitational law and then we will apply the dimensional analysis. Then, dimensions of all the quantities are written in terms of basic dimensions.

Complete answer:
According to the law of gravitation, we can say that the gravitational force between two objects of masses ${{\text{m}}_1}$ and ${{\text{m}}_2}$ separated by a distance d is given by

${\text{F}} = \dfrac{{{\text{G}}{{\text{m}}_1}{{\text{m}}_2}}}{{{{\text{d}}^2}}}{\text{ }} \to {\text{(1)}}$ where G denotes the universal gravitational constant
As we know that the dimensional formula of mass m is [M], the dimensional formula of distance d is [L], the dimensional formula of force F is ${\text{[ML}}{{\text{T}}^{ - 2}}{\text{]}}$
Rearranging the equation (1), we get

${\text{G}} = \dfrac{{{\text{F}}{{\text{d}}^2}}}{{{{\text{m}}_1}{{\text{m}}_2}}}{\text{ }} \to {\text{(2)}}$
Applying dimensional analysis on the above equation, we get
Dimensional formula of G = $\dfrac{{[{\text{Dimensional formula of F}}]{{[{\text{Dimensional formula of d}}]}^2}}}{{[{\text{Dimensional formula of }}{{\text{m}}_1}][{\text{Dimensional formula of }}{{\text{m}}_2}]}}$
$ \Rightarrow $ Dimensional formula of G = \[\dfrac{{{\text{[ML}}{{\text{T}}^{ - 2}}{\text{]}}{{[{\text{L}}]}^2}}}{{[{\text{M}}][{\text{M}}]}} = \dfrac{{{\text{[ML}}{{\text{T}}^{ - 2}}{{\text{L}}^2}{\text{]}}}}{{{{[{\text{M}}]}^2}}} = {\text{[M}}{{\text{L}}^3}{{\text{T}}^{ - 2}}{{\text{M}}^{ - 2}}{\text{]}} = {\text{[}}{{\text{M}}^{ - 1}}{{\text{L}}^3}{{\text{T}}^{ - 2}}{\text{]}}\]
Therefore, the dimensional formula of universal gravitational constant is \[{\text{[}}{{\text{M}}^{ - 1}}{{\text{L}}^3}{{\text{T}}^{ - 2}}{\text{]}}\].

So, the correct answer is “Option C”.

Note:
We have taken the dimensional formula for force F as ${\text{[ML}}{{\text{T}}^{ - 2}}{\text{]}}$. This is because force is simply the product of mass and acceleration. The dimensional formula for mass is [M] and that for acceleration (rate of change of the speed with respect to time) is ${\text{[L}}{{\text{T}}^{ - 2}}{\text{]}}$. As, F = ma $ \Rightarrow {\text{ F}} = [{\text{M}}]{\text{[L}}{{\text{T}}^{ - 2}}{\text{]}} = [{\text{ML}}{{\text{T}}^{ - 2}}]$.