Question

Dimensional formula for the universal gravitational constant G is
A. \[{{M}^{-1}}{{L}^{2}}{{T}^{-2}}\]
B. \[{{M}^{0}}{{L}^{0}}{{T}^{0}}\]
C. \[{{M}^{-1}}{{L}^{3}}{{T}^{-2}}\]
D. \[{{M}^{-1}}{{L}^{3}}{{T}^{-1}}\]

Answer 1

Hint: Dimensional formula of any quantity can be defined as an expression for the unit of a physical quantity in terms of the basic fundamental quantities. The fundamental quantities are mass (M), length (L), and time (T). A dimensional formula is thus expressed in terms of powers of M, L and T. We can solve this question by substituting the dimensions of all the quantities in the formula for gravitational constant.

Complete step-by-step answer:
Gravitational force acting between two objects of masses m1​ and m2​ separated by distance r, \[F=\dfrac{G{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}\]
On transposing the given physical quantities we get,
⟹ \[G=\dfrac{F{{r}^{2}}}{{{m}_{1}}{{m}_{2}}}\]​
Now, The dimensional formula for force is F= \[\left[ ML{{T}^{-2}} \right]\]
The dimensional formula for length is r = \[\left[ L \right]\]
The dimensional formula for mass is m = \[\left[ M \right]\]
Substituting all these dimensions in the given formula we get,
Thus dimensional formula of G is \[G=\dfrac{\left[ ML{{T}^{-2}} \right]{{\left[ L \right]}^{2}}}{{{\left[ M \right]}^{2}}}\]
⟹ \[G={{M}^{-1}}{{L}^{3}}{{T}^{-2}}\]
Hence, the correct option is option C.

Note: The gravitational constant, denoted by the letter G, is an empirical physical constant involved in the calculation of gravitational effects given in Sir Isaac Newton's law of universal gravitation and also in Albert Einstein's general theory of relativity. \[G=6.67408\text{ }\times \text{ }{{10}^{-11}}~{{m}^{3}}~k{{g}^{-1}}~{{s}^{-2}}\]
Gravity or gravitational force can be defined as a natural phenomenon by which all things with mass or energy including planets, stars, galaxies, and even light are brought towards one another by forces of attraction. On Earth, this gravity gives weight to physical objects and matter, and the Moon's gravity causes the ocean tides.