Question

Dimensional formula of universal gravitational constant G is-
$
  \left( a \right){M^{ - 1}}{L^3}{T^{ - 2}} \\
  \left( b \right){M^{ - 1}}{L^2}{T^{ - 2}} \\
  \left( c \right){M^{ - 2}}{L^3}{T^{ - 2}} \\
  \left( d \right){M^{ - 2}}{L^2}{T^{ - 2}} \\
$

Answer 1

Hint: In this question use the dimensional formula of force, distance and mass, as gravitational constant G can be expressed in terms of Force, distance between two bodies and the mass of the bodies.

Complete step-by-step answer:
As we know force between two masses (m) and (M) is
$F = \dfrac{{G\left( {m.M} \right)}}{{{r^2}}}$, where G is called a gravitational constant and r is the distance between them.
So, the formula of universal gravitational constant G is
$ \Rightarrow G = \dfrac{{F.{r^2}}}{{m.M}}$
Now as we know force is the product of mass (M) and acceleration (a)
Therefore, F = (M. a).
Now as we know that the dimension of mass (M) is $M^1$.
And we know the S.I unit of acceleration (a) is $\dfrac{m}{s^{2}}$.
The dimension of meter is $L^1$ and the dimension of second (s) is $T^1$.
So the dimension of acceleration is $L^1 T ^{-2}$.
Therefore, the dimension of force (F) is the product of mass and acceleration i.e $M^1$ $L^1 T ^{-2}$ =${M^1}{L^1}{T^{ - 2}}$.
And we all know distance is measured in meters so the dimension of r is $L^1$.
Therefore, the dimension of G is
$ \Rightarrow G = \dfrac{{\left[ {{M^1}{L^1}{T^{ - 2}}} \right]\left[ {{L^2}} \right]}}{{\left[ {{M^2}} \right]}}$
Now on simplifying we have,
$G = \left[ {{M^{ - 1}}{L^3}{T^{ - 2}}} \right]$
So this is the required dimension of universal gravitational constant (G).
Hence option (A) is correct.

Note – Dimension formula is the expression for the unit of a physical quantity in terms of the fundamental quantities. The fundamental quantities are mass (M), Length (L) and time (T). A dimensional formula is expressed in terms of power of M, L and T. By observing the dimensional formula $G = \left[ {{M^{ - 1}}{L^3}{T^{ - 2}}} \right]$ we can say that universal gravitational constant has negative dimensions of mass.