A U-tube is partially filled with mercury. Some water is poured in its one arm and a liquid in the other arm so that the level of mercury in both arms is same. What is the ratio of lengths of water and liquid in the two arms of the U-tube?
Density of water $= 1000 k g m^{- 3}$ and density of liquid $= 900 k g m^{- 3}$ .

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Solution

Since the level of mercury in the two arms of the U-tube is same, therefore,
Pressure exerted by water columnin one arm =pressure exerted by liquid column in other arm
i.e., $h_{1} d_{1} g = h_{2} d_{2} g$
$\Rightarrow \frac{h_{1}}{h_{2}} = \frac{d_{2}}{d_{1}} = \frac{900 k g m^{- 3}}{1000 k g m^{- 3}} = \frac{9}{10}$ .
Hence, the ratio of lengths of water and the liquid is $9 : 10$ .

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Since the level of mercury in the two arms of the U-tube is same, therefore, Pressure exerted by water columnin one arm =pressure exerted by liquid column in other armi.e., h1d1g=h2d2g⇒h1h2=d2d1=900kgm−31000kgm−3=910.Hence, the ratio of lengths of water and the liquid is 9:10.