Question

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

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
$F={\displaystyle \frac{G\left(m.M\right)}{r^2}}$, where G is called a gravitational constant and r is the distance between them.
So, the formula of universal gravitational constant G is
$\Rightarrow G={\displaystyle \frac{F.r^2}{m.M}}$
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 ${\displaystyle \frac{m}{s^2}}$.
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
$\Rightarrow G={\displaystyle \frac{\left[M^1{L}^1{T}^{-2}\right]\left[L^2\right]}{\left[M^2\right]}}$
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.