Two vertical cylindrical vessels A and B of horizontal cross-sectional areas S and 2S are connected at their bottoms with a horizontal tube of cross-sectional area 0.5S. An amount of water is trapped

Question

Two vertical cylindrical vessels A and B of horizontal cross-sectional areas S and 2S are connected at their bottoms with a horizontal tube of cross-sectional area 0.5S. An amount of water is trapped in the vessels under leakproof pistons, one in each cylindrical vessel. The pistons are connected with a light inextensible thread that passes over an ideal pulley as shown in the figure. The pulley is pulled upwards with a constant velocity v . The vessels are rigidly affixed on the horizontal floor. Mark the correct statements.

TestHub · Accepted Answer

Right-side liquid goes down. Continuity: $S v_{1} = 2 S v_{2} \Rightarrow v_{1} = 2 v_{2}$

Constraint (pulley pulled up at v): $\frac{v _{1} - v _{2}}{2} = v \Rightarrow v_{1} - v_{2} = 2 v$

Solving: $v_{2} = 2 v$ , $v_{1} = 4 v$ - (A), (B) correct

Tube flow: $2 S v_{2} = 0.5 S \cdot v_{H} \Rightarrow v_{H} = 8 v$ - (D) correct; flow is towards A (leftwards) - (C) incorrect

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