Question

Two arms of a $ U $ tube have unequal diameters $ {d_1} = 1.0mm $ and $ {d_2} = 1.0cm $ . If water $ \left( {{\text{Surface}}\,\,{\text{tension}}\,\,7 \times {{10}^{ - 2}}/m} \right) $ is poured into the tube held in the vertical position, the difference of level of water in the $ U $ tube is $ \dfrac{x}{2}cm $ . Find $ x $ Assume the angle of contact to be zero.

Answer 1

Hint: So in this question we have the difference of the level of water is given, now for calculating the $ x $ we will use the formula of change in rise of capillary, and it is given by $ \vartriangle H = \dfrac{{4T}}{{\rho g}}\left[ {\dfrac{1}{{{d_1}}} - \dfrac{1}{{{d_2}}}} \right] $ . And by using this we can solve this question.

Complete step by step solution:
As we know that the rise of water in capillary is $ {H_1} $ and is given by the formula $ \dfrac{{4T}}{{\rho g{d_1}}} $ .
So the change is it will be equal to $ \vartriangle H = {H_1} - {H_2} $
So on substituting the values, we will get the equation as
 $ \Rightarrow \vartriangle H = \dfrac{{4 \times 7 \times {{10}^{ - 2}}}}{{1000 \times 9.8}}\left[ {\dfrac{1}{{{{10}^{ - 3}}}} - \dfrac{1}{{{{10}^{ - 2}}}}} \right] $
And on solving the above equation, we will get the equation as
 $ \Rightarrow \vartriangle H = 2.5 \times {10^{ - 2}}m $
Or it can be written as
 $ \Rightarrow \vartriangle H = 2.5cm $
Since, form the question it is given that
 $ \Rightarrow \dfrac{x}{2} = 2cm $
And on solving it, we get
 $ \Rightarrow x = 5cm $
Therefore, the value of $ x $ will be equal to $ 5cm $ .

Note:
Here in this question while solving it we should not forget to change the unit of the diameter. As the diameter of the unit is given in $ mm $ . So we have to convert them into the $ m $ . Also the change in the level of water should also be converted. So we should take care of the units while solving such types of questions.