Question

If \[G\] is universal gravitational constant and \[g\] is acceleration due to gravity, the dimensions of \[G\] will be
A.\[\left[ {{M^1}{L^2}T} \right]\]
B.\[\left[ {{M^{ - 1}}{L^3}{T^{ - 2}}} \right]\]
C.\[\left[ {{M^2}LT} \right]\]
D.\[\left[ {{M^1}{L^3}{T^{ - 2}}} \right]\]

Answer 1

Hint:Use the relation between the acceleration due to gravity on the surface of the planet, mass of the planet and radius of the planet. Determine the dimensions of all the physical quantities from their units and substitute them in the relation to obtain the dimensions of universal gravitational constant.

Formula used:
The expression for the acceleration due to gravity \[g\] is
\[g = \dfrac{{GM}}{{{R^2}}}\] …… (1)
Here, \[G\] is the universal gravitational constant, \[M\] is the mass of the planet and \[R\] is the radius of the planet.

Complete step by step answer:
The dimensions of a physical quantity are the powers raised to the fundamental physical quantities.The dimensions of a physical quantity can be used to determine the unit of that physical quantity.We can determine the dimensions of the universal gravitational constant \[G\] by using equation (1) as we know the units of all other physical quantities involved in the formula.Rearrange equation (1) for the universal gravitational constant \[G\].
\[G = \dfrac{{g{R^2}}}{M}\] …… (2)
The term \[g\] in the above expression is the acceleration due to gravity and has the same unit as that of the acceleration. The SI unit of acceleration is \[{\text{m}} \cdot {{\text{s}}^{{\text{ - 2}}}}\]. Hence, the dimensional formula for acceleration due to gravity is \[\left[ {{M^0}{L^1}{T^{ - 2}}} \right]\].
The term \[R\] in the above expression is the radius of the planet and has the same unit as that of the length. The SI unit of radius is \[{\text{m}}\]. Hence, the dimensional formula for radius is \[\left[ {{M^0}{L^1}{T^0}} \right]\].
The term \[M\] in the above expression is the mass of the planet and has the same unit as that of the normal mass. The SI unit of radius is \[{\text{kg}}\]. Hence, the dimensional formula for mass is \[\left[ {{M^1}{L^0}{T^0}} \right]\].
We can now determine the dimensional formula for the universal gravitational constant by substituting the dimensions of the respective physical quantities and solving it.
\[G = \dfrac{{\left[ {{M^0}{L^1}{T^{ - 2}}} \right]{{\left[ {{M^0}{L^1}{T^0}} \right]}^2}}}{{\left[ {{M^1}{L^0}{T^0}} \right]}}\]
\[ \Rightarrow G = \dfrac{{\left[ {{M^0}{L^1}{T^{ - 2}}} \right]\left[ {{M^0}{L^2}{T^0}} \right]}}{{\left[ {{M^1}{L^0}{T^0}} \right]}}\]
\[ \Rightarrow G = \dfrac{{\left[ {{M^0}{L^3}{T^{ - 2}}} \right]}}{{\left[ {{M^1}{L^0}{T^0}} \right]}}\]
\[ \therefore G = \left[ {{M^{ - 1}}{L^3}{T^{ - 2}}} \right]\]
Therefore, the dimensional formula for universal gravitational constant is \[\left[ {{M^{ - 1}}{L^3}{T^{ - 2}}} \right]\].

Hence, the correct option is B.

Note: One can also use the equation for Newton's gravitational law of attraction to determine the dimensions of the universal gravitational constant. The students should also keep in mind that \[M\] in the dimensional formula represents the mass and \[L\] represents length. The students may get confused about it as the unit of length meter is also represented as m.