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Question: Dimensional formula of universal gravitational constant G is- \( \left( a \right){M^{ - 1}}{L^...

Dimensional formula of universal gravitational constant G is-
(a)M−1L3T−2 (b)M−1L2T−2 (c)M−2L3T−2 (d)M−2L2T−2  \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}} \\\

Explanation

Solution

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=G(m.M)r2F = \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
⇒G=F.r2m.M\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 M1M^1.
And we know the S.I unit of acceleration (a) is ms2\dfrac{m}{s^{2}}.
The dimension of meter is L1L^1 and the dimension of second (s) is T1T^1.
So the dimension of acceleration is L1T−2L^1 T ^{-2}.
Therefore, the dimension of force (F) is the product of mass and acceleration i.e M1M^1 L1T−2L^1 T ^{-2} =M1L1T−2{M^1}{L^1}{T^{ - 2}}.
And we all know distance is measured in meters so the dimension of r is L1L^1.
Therefore, the dimension of G is
⇒G=[M1L1T−2][L2][M2]\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=[M−1L3T−2]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=[M−1L3T−2]G = \left[ {{M^{ - 1}}{L^3}{T^{ - 2}}} \right] we can say that universal gravitational constant has negative dimensions of mass.