Petrol (density $= 750 \, kg m^{-3}$) and diesel (density $= 850 \, kg m^{-3}$) enter into two identical venturimeters each with a velocity $10 ms^{-1}$ as shown in the figure. If $\Delta h_1$ is the difference in heights of petrol in the two vertical tubes of venturimeter A and if $\Delta h_2$ is the difference in heights of diesel in the two vertical tubes of the venturimeter B, then $\Delta h_1 : \Delta h_2 =$
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In a venturimeter measuring the flow of a liquid using piezometer tubes (containing the same liquid), the reading $\Delta h$ (head difference) is independent of the liquid's density for a given velocity.
Step 1: Understanding Bernoulli's Principle in Venturimeter:
For a horizontal venturimeter, the pressure difference creates the height difference in the manometer tubes.
Equation: $P_1 - P_2 = \frac{1}{2} \rho v_1^2 \left[ \left(\frac{A_1}{A_2}\right)^2 - 1 \right]$
Step 2: Relating Pressure to Height Difference:
The pressure difference is indicated by the liquid column height difference $\Delta h$ of the fluid flowing through it (since the tubes contain the same liquid as the flow).
\[ P_1 - P_2 = \rho g \Delta h \]
Step 3: Deriving the Expression for $\Delta h$:
Equating the two expressions for pressure difference:
\[ \rho g \Delta h = \frac{1}{2} \rho v_1^2 \left[ \left(\frac{A_1}{A_2}\right)^2 - 1 \right] \]
Notice that the density $\rho$ appears on both sides.
\[ g \Delta h = \frac{1}{2} v_1^2 \left[ \left(\frac{A_1}{A_2}\right)^2 - 1 \right] \]
\[ \Delta h = \frac{v_1^2}{2g} \left[ \left(\frac{A_1}{A_2}\right)^2 - 1 \right] \]
Step 4: Analyzing the Ratio:
Since the venturimeters are identical, the area ratio $A_1/A_2$ is the same. The velocity $v_1$ is given as $10 \, m/s$ for both cases. $g$ is constant.
Therefore, $\Delta h$ depends only on velocity and geometry, not on the density of the liquid.
\[ \Delta h_1 = \Delta h_2 \]
\[ \Delta h_1 : \Delta h_2 = 1 : 1 \]
