Question

A glass capillary tube with a radius \( r = 0.02 \, {cm} \) is immersed into water to a depth of \( d = 2 \, {cm} \). To blow an air bubble out of the lower end of the tube, the pressure required is: 
Given:
Surface tension \( T = 7 \times 10^{-2} \, {N/m}^{-1} \)
Density of water \( \rho = 10^3 \, {kg/m}^{-3} \)
Acceleration due to gravity \( g = 10 \, {m/s}^{-2} \)

• A. 480 Nm\(^{-2}\)
• B. 900 Nm\(^{-2}\)
• C. 200 Nm\(^{-2}\)
• D. 700 Nm\(^{-2}\)

Hint:

Accurate unit conversions and understanding of fluid statics principles are crucial for solving problems in fluid mechanics involving capillary actions and bubble formation.

Answer 1

The Correct Option is B

Step 1: Convert the units of radius to meters. \[ r = 0.02 \, {cm} = 0.0002 \, {m} \] Step 2: Calculate the hydrostatic pressure at the depth of the tube. \[ P_{{hydrostatic}} = \rho \times g \times d \] \[ P_{{hydrostatic}} = 1000 \, {kg/m}^{-3} \times 10 \, {m/s}^{-2} \times 0.02 \, {m} = 200 \, {Nm}^{-2} \] Step 3: Calculate the additional pressure required to overcome surface tension at the tube's exit. \[ P_{{surface}} = \frac{2T}{r} \] \[ P_{{surface}} = \frac{2 \times 0.07 \, {N/m}^{-1}}{0.0002 \, {m}} = 700 \, {Nm}^{-2} \] Step 4: Calculate the total pressure required. \[ P_{{total}} = P_{{hydrostatic}} + P_{{surface}} \] \[ P_{{total}} = 200 \, {Nm}^{-2} + 700 \, {Nm}^{-2} = 900 \, {Nm}^{-2} \]