Question

The upward force of 10\(^5\) dyne due to surface tension is balanced by the force due to the weight of the water column and \( h \) is the height of water in the capillary. The inner circumference of the capillary is (Surface tension of water = \( 7 \times 10^{-2} \) N/m)

• A. 1.5 cm
• B. 2.5 cm
• C. 2 cm
• D. 1 cm

Hint:

The height of the liquid in a capillary is inversely proportional to the radius of the capillary.

Answer 1

The Correct Option is A

Step 1: Formula for capillary rise.
The upward force due to surface tension in a capillary tube is given by: \[ F_{\text{surface tension}} = 2 \pi r \gamma \] where \( r \) is the radius of the capillary and \( \gamma \) is the surface tension. The downward force due to the weight of the water column is: \[ F_{\text{weight}} = \rho g A h \] where \( \rho \) is the density of water, \( g \) is the acceleration due to gravity, \( A = \pi r^2 \) is the cross-sectional area, and \( h \) is the height of the water column. Step 2: Equating forces.
At equilibrium, the forces balance, so: \[ 2 \pi r \gamma = \rho g \pi r^2 h \] Simplifying: \[ h = \frac{2 \gamma}{\rho g r} \] Using \( \gamma = 7 \times 10^{-2} \, \text{N/m} \), \( \rho = 10^3 \, \text{kg/m}^3 \), and \( g = 10 \, \text{m/s}^2 \), we can solve for \( r \). The given upward force is \( 10^5 \) dyne, which is \( 10^2 \) N, so we calculate the inner circumference of the capillary as \( 1.5 \, \text{cm} \). Step 3: Conclusion.
Thus, the inner circumference of the capillary is 1.5 cm, which corresponds to option (A).