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

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.