Question

Water rises in a capillary tube to a certain height such that the upward force due to surface tension is balanced by \(63 \times 10^{-4}\,\text{N}\) force due to the weight of the water. The surface tension of water is \(7 \times 10^{-2}\,\text{N m}^{-1}\). The inner diameter of the capillary tube is nearly \((\pi = 22/7)\)

• A. \(6.3 \times 10^{-1}\,\text{m}\)
• B. \(3 \times 10^{-2}\,\text{m}\)
• C. \(7 \times 10^{-2}\,\text{m}\)
• D. \(9 \times 10^{-2}\,\text{m}\)

Hint:

In capillary rise problems, the upward force due to surface tension is given by \(2\pi r T\) when the angle of contact is zero.

Answer 1

The Correct Option is B

Step 1: Write the force due to surface tension.
The upward force due to surface tension acting along the circumference of the capillary tube is:
\[ F = 2\pi r T \] (For water, angle of contact \(\theta \approx 0^\circ\), hence \(\cos\theta = 1\)).

Step 2: Substitute given values.
\[ F = 63 \times 10^{-4}\,\text{N} = 6.3 \times 10^{-3}\,\text{N} \] \[ T = 7 \times 10^{-2}\,\text{N m}^{-1} \]
Step 3: Calculate the radius of the capillary tube.
\[ r = \frac{F}{2\pi T} = \frac{6.3 \times 10^{-3}}{2 \times \frac{22}{7} \times 7 \times 10^{-2}} \] \[ r \approx 1.43 \times 10^{-2}\,\text{m} \]
Step 4: Find the inner diameter.
\[ d = 2r \approx 2.86 \times 10^{-2}\,\text{m} \approx 3 \times 10^{-2}\,\text{m} \]
Step 5: Conclusion.
The inner diameter of the capillary tube is nearly \(3 \times 10^{-2}\,\text{m}\).