Three different liquids are filled in a U-tub | vedclass

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(C) Consider the horizontal level at the bottom of the liquid with density $\rho_3$ in the left arm. The pressure at this level in both arms must be e

Vedclass · Accepted Answer

$\rho_3=2(\rho_2-\rho_1)$

Vedclass · Answer

(C) Consider the horizontal level at the bottom of the liquid with density $\rho_3$ in the left arm. The pressure at this level in both arms must be equal.In the left arm, the pressure is due to the liquid of density $\rho_1$ (height $h$) and the liquid of density $\rho_3$ (height $h/2$).$P_{left} = P_{atm} + \rho_1 gh + \rho_3 g(h/2)$In the right arm, the pressure at the same horizontal level is due to the liquid of density $\rho_2$ (height $h$).$P_{right} = P_{atm} + \rho_2 gh$Equating the pressures: $P_{atm} + \rho_1 gh + \rho_3 g(h/2) = P_{atm} + \rho_2 gh$Dividing by $gh$: $\rho_1 + \frac{\rho_3}{2} = \rho_2$Rearranging for $\rho_3$: $\frac{\rho_3}{2} = \rho_2 - \rho_1 \implies \rho_3 = 2(\rho_2 - \rho_1)$

Three different liquids are filled in a $U$-tube as shown in the figure. Their densities are $\rho_1, \rho_2$ and $\rho_3$ respectively. From the figure, we may conclude that:

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