If Force ( F ), Velocity ( v ), and Density ( ) are taken as the fundamental quantities, then what will be the dimensional formula for the Universal Gravitational Constant ( G )?

Step-by-step solution

Let the dimensional formula of the universal gravitational constant be given by: [G] = [F]^a [v]^b [ ]^c Writing the standard dimensional formulas for each quantity: [G] = [ M ⁻¹ L ^3 T ⁻²] [F] = [ M L T ⁻²] [v] = [ L T ⁻¹] [ ] = [ M L ⁻³] Substituting these into the equation: [ M ⁻¹ L ^3 T ⁻²] = [ M L T ⁻²]^a [ L T ⁻¹]^b [ M L ⁻³]^c [ M ⁻¹ L ^3 T ⁻²] = [ M ^ a+c L ^ a+b-3c T ^ -2a-b ] Equating the powers of M , L , and T on both sides: a + c = -1 (Equation 1) a + b - 3c = 3 (Equation 2) -2a - b = -2 (Equation 3) F