Question

$a, A$ are areas, $\rho_{m}$ is the density of mercury for a venturimeter the speed of fluid at wide neck is

• A. $v_{1}=\sqrt{\frac{2 \rho_{m} g h}{\rho}}\left[\frac{A}{a}-1\right]^{-\frac{1}{2}}$
• B. $v_{1}=\sqrt{\frac{2 \rho_{m} g h}{\rho}}\left[\frac{a}{A}+1\right]^{-\frac{1}{2}}$
• C. $v_{1}=\sqrt{\frac{2 \rho g h}{\rho_{m}}}\left[\frac{A^{2}}{a^{2}}-1\right]^{-\frac{1}{2}}$
• D. $v_{1}=\sqrt{\frac{2 \rho_{m} g h}{\rho}}\left[\frac{A^{2}}{a^{2}}-1\right]^{-\frac{1}{2}}$

Answer 1

Answer: (D)

Here $A v_{1}=a v_{2}$ or $v_{2}=\left(\frac{A}{a}\right) v_{1}$
Using Bernoulli's theorem,
$P_{1}+\frac{1}{2} \rho v_{1}^{2}=P_{2}+\frac{1}{2} \rho v_{2}^{2}$
$\Rightarrow P_{1}-P_{2}=\frac{1}{2} \rho\left(v_{2}^{2}-v_{1}^{2}\right)$
or $\rho_{m} \cdot g \cdot h=\frac{1}{2} \rho v_{1}^{2}\left[\left(\frac{A}{a}\right)^{2}-1\right]$
Hence, $v_{1}=\sqrt{\frac{2 \rho_{m} \cdot g h}{\rho}}\left[\left(\frac{A}{a}\right)^{2}-1\right]^{-\frac{1}{2}}$