A physical quantity X is defined in terms of Planck's constant ( h ), the speed of light ( c ), and the universal gravitational constant ( G ) as X = h c^5 G . Which of the following physical quantities has the same dimensions as X ?

Step-by-step solution

The dimensional formulas for the given fundamental constants are: [h] = M L ^2 T ⁻¹ [c] = L T ⁻¹ [G] = M ⁻¹ L ^3 T ⁻² Substitute these into the expression for X^2 : X^2 = h c^5 G [X^2] = ( M L ^2 T ⁻¹)( L T ⁻¹)^5 M ⁻¹ L ^3 T ⁻² [X^2] = ( M L ^2 T ⁻¹)( L ^5 T ⁻⁵) M ⁻¹ L ^3 T ⁻² [X^2] = M L ^7 T ⁻⁶ M ⁻¹ L ^3 T ⁻² [X^2] = M ^2 L ^4 T ⁻⁴ Taking the square root gives the dimensions of X : [X] = M L ^2 T ⁻² This is the dimensional formula for Energy. Answer: Energy