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Question
Test if the following equation is dimensionally correct:
\[v = \sqrt{\frac{P}{\rho}},\]
where v = velocity, ρ = density, P = pressure
Solution
\[\nu = \sqrt{\left( \frac{P}{\rho} \right)}\]
Velocity, [ν] = [LT−1]
Pressure,
\[P = \frac{\left[ F \right]}{\left[ A \right]} = \left[ {ML}^{- 1} T^{- 2} \right]\]
Density,
\[\left[ \rho \right] = \frac{\left[ M \right]}{\left[ V \right]} = \left[ {ML}^{- 3} T^0 \right]\]
Now,
\[\sqrt{\frac{P}{\rho}} = \left[ \frac{\left[ {ML}^{- 1} T^{- 2} \right]}{\left[ {ML}^{- 3} \right]} \right]^\frac{1}{2} = \left[ L^2 T^{- 2} \right]^{1/2} = \left[ {LT}^{- 1} \right]\]
Since the dimensions of both sides of the equation are the same, the equation is dimensionally correct.
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