Question

If area (A), velocity (V) and density ( $\rho$ ) are taken as fundamental units, what is the dimensional formula for force?

• A. $\left[ AV ^{2} \rho \right]$
• B. $\left[ A ^{2} V \rho\right]$
• C. $\left[ AV \rho^{2}\right]$
• D. $[ AV \rho]$

Answer 1

Answer: (A)

$F = A ^{\alpha} V ^{ \beta } \rho ^{\gamma}$
or $[ MLT ^{-2}] =[ L ^{2}]^{\alpha}[ LT ^{-1}]^{\beta}[ ML ^{-3}]^{\gamma} $ $=\left[ M ^{\gamma} L ^{2 \alpha+\beta-3 \gamma} T ^{-\beta}\right] $
This gives : $\gamma=1, \beta=2, 2 \alpha+\beta-3 \gamma=1$
or $\alpha=1 ; F =\left[ AV ^{2} \rho\right]$