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A glass U-tube is shown in its "Initial Condition," 1). The tube is filled with mercury to the level "0 - - 0", its legs are open at their tops to the atmosphere. The cross-sectional areas, left and right, are AL and AR. Beside the U-tube is a cup containing mw grams of water.
The event is that the water is poured into the right leg whereupon the fluids, under gravity, attain a second state of equilibrium. We expect the level of mercury in the left leg to rise above "0 - - 0." Represent the densities of water and mercury as: ρw and ρm. Obtain an algebraic expression for the final height of mercury in the left leg, hm.
Solution: A needed first step is to sketch an "anticipated" final condition of the system and to apply some dimensions to it. It is not necessary that you draw the "proper" sketch but simply that you get something to start with. For instance see Figure 2). To proceed we look at the system/event in its final condition in terms of its properties: geometry, mass and momentum.
Hence there are three equations with three unknown quantities: L, hw and hm. We leave it here.