If surface tension ( S ), density ( ), and coefficient of viscosity ( ) are chosen as the fundamental quantities, which of the following combinations has the dimension of length?

Step-by-step solution

Let the dimension of length L be expressed as L = S^x ^y ^z . The dimensional formulas for the given quantities are: [S] = M L^0 T⁻² [ ] = M L⁻³ T^0 [ ] = M L⁻¹ T⁻¹ Substituting these into the expression for length: [ M^0 L^1 T^0 ] = [ M T⁻² ]^x [ M L⁻³ ]^y [ M L⁻¹ T⁻¹ ]^z [ M^0 L^1 T^0 ] = M ^ x+y+z L ^ -3y-z T ^ -2x-z Equating the powers of M, L, and T on both sides, we get a system of linear equations: x + y + z = 0 (Equation 1) -3y - z = 1 (Equation 2) -2x - z = 0 z = -2x (Equation 3) Substitute z = -2x into Eq