Question

In a capillary tube of radius $0.2mm$ the water rises up to height of $7.5cm$ with angle of contact equal to zero. If another capillary with same radius but of different material is dipped in the same liquid the height of water raised in capillary will be, (if angle of contact becomes $60°$).

• A. $7.5cm$
• B. $15cm$
• C. $3.75cm$
• D. $30cm$

Answer 1

The correct option is C

$3.75cm$

Step 1: Given:

Radius of tube: $r=0.2mm$

Height to which water rises when ${\theta }_1=0°$: $h=7.5cm$

New angle of contact: ${\theta }_2=60°$

Step 2: Formula Used:

The height of rise of liquid in capillary tube is

$h=\frac{2T\cos \theta }{R\rho g}$

Where, $T$ is surface tension, $\theta$ is angle is contact, $R$ is radius of tube, $\rho$ is density of liquid and $g$ is acceleration due to gravity.

Step 3: Find the height raise of water

Find an expression for the surface tension when $h=7.5cm$ and ${\theta }_1=0°$.

$$\begin{gathered} 7.5=\frac{2T\cos 0}{R\rho g} \\ \Rightarrow 7.5=\frac{2T\times 1}{R\rho g} \\ \Rightarrow T=\frac{7.5R\rho g}{2} \end{gathered}$$

Calculate the new height when by using the formula and substituting the expression for tension. Other variables remain same.

$\begin{array}{rcl}h& =& \frac{2T\cos 60}{R\rho g}\\ & =& \frac{2\left(\frac{7.5R\rho g}{2}\right){\displaystyle \frac{1}{2}}}{R\rho g}\\ & =& \frac{7.5R\rho g}{2R\rho g}\\ & =& 3.75cm\end{array}$

Hence, the water raises to a height of $3.75cm$.