Let's analyze the dimensions of each option step by step. ### **Given Dimensions** - \( V \): velocity (\( LT^{-1} \)) - \( L \): length (\( L \)) - \( \rho \): density (\( ML^{-3} \)) - \( p \): pressure (\( ML^{-1}T^{-2} \)) - \( \mu \): dynamic viscosity (\( ML^{-1}T^{-1} \)) We need to check which option is **NOT dimensionless**. --- ## **Option A:** \( \frac{\rho L V}{\mu} \) Let's write dimensions for numerator and denominator: - \( \rho \): \( ML^{-3} \) - \( L \): \( L \) - \( V \): \( LT^{-1} \) - \( \mu \): \( ML^{-1}T^{-1} \) So, \[ \rho L V = (ML^{-3}) \cdot (L) \cdot (LT^{-1}) = M L^{-3+1+1} T^{-1} = M L^{-1} T^{-1} \] \[ \frac{\rho L V}{\mu} = \frac{M L^{-1} T^{-1}}{M L^{-1} T^{-1}} = 1 \] **Option A is dimensionless.** --- ## **Option B:** \( \frac{p}{\rho V^2} \) - \( p \): \( ML^{-1}T^{-2} \) - \( \rho \): \( ML^{-3} \) - \( V^2 \): \( (LT^{-1})^2 = L^2 T^{-2} \) \[ \rho V^2 = (ML^{-3}) \cdot (L^2 T^{-2}) = M L^{-3+2} T^{-2} = M L^{-1} T^{-2} \] \[ \frac{p}{\rho V^2} = \frac{M L^{-1} T^{-2}}{M L^{-1} T^{-2}} = 1 \] **Option B is dimensionless.** --- ## **Option C:** \( \frac{\rho L}{V^2} \) - \( \rho \): \( ML^{-3} \) - \( L \): \( L \) - \( V^2 \): \( (LT^{-1})^2 = L^2 T^{-2} \) \[ \rho L = (ML^{-3}) \cdot (L) = M L^{-2} \] \[ \frac{\rho L}{V^2} = \frac{M L^{-2}}{L^2 T^{-2}} = M L^{-2} \cdot L^{-2} T^2 = M L^{-4} T^2 \] **Option C is NOT dimensionless.** --- ## **Final Answer** \[ \boxed{C} \] **Option (C) is NOT dimensionless.**