Question

The lower end of a capillary tube is dipped into water and it is seen that water rises through \(7.5\,\text{cm}\) in the capillary. Given, surface tension of water is \(7.5 \times 10^{-2}\,\text{N m}^{-1}\) and angle of contact is zero. Find the diameter (in mm) of the capillary tube. (Given \(g = 10\,\text{m s}^{-2}\))

Hint:

Use \(h = \frac{2T}{\rho g r}\) for water (since \(\theta = 0^\circ\)).
Always convert cm to m carefully — most mistakes happen here!

Answer 1

Correct Answer: 0.4

Concept: Capillary rise is given by: \[ h = \frac{2T \cos \theta}{\rho g r} \]

Step 1: Given data \[ h = 7.5\,\text{cm} = 0.075\,\text{m}, \quad T = 7.5 \times 10^{-2}\,\text{N/m} \] \[ \theta = 0^\circ \Rightarrow \cos \theta = 1, \quad \rho = 1000\,\text{kg/m}^3, \quad g = 10\,\text{m/s}^2 \]

Step 2: Substitute values \[ 0.075 = \frac{2 \times 7.5 \times 10^{-2}}{1000 \times 10 \times r} \] \[ 0.075 = \frac{0.15}{10000\,r} \]

Step 3: Solve for \(r\) \[ 0.075 \times 10000\,r = 0.15 \] \[ 750\,r = 0.15 \] \[ r = \frac{0.15}{750} = 2 \times 10^{-4}\,\text{m} = 0.2\,\text{mm} \]

Step 4: Diameter \[ d = 2r = 0.4\,\text{mm} \] Final: \[ {0.4\,\text{mm}} \]