Question

If \( M \) is the mass of water that rises in a capillary tube of radius \( r \), then the mass of water which will rise in a capillary tube of radius \( 2r \) is:

• A. \( M \)
• B. \( \frac{M}{2} \)
• C. \( 4M \)
• D. \( 2M \)

Hint:

The mass of water in a capillary tube depends linearly on the radius of the tube. Doubling the radius of the capillary will double the mass of the water it holds.

Answer 1

The Correct Option is D

The height \( h \) of water in a capillary tube is determined by the capillary rise formula: \[h = \frac{2T \cos \theta}{r \rho g},\] where \( T \) denotes surface tension, \( \theta \) is the angle of contact, \( r \) is the tube radius, \( \rho \) is the liquid density, and \( g \) is the acceleration due to gravity. The volume of water in the capillary is calculated as: \[V = \pi r^2 h = \pi r^2 \cdot \frac{2T \cos \theta}{r \rho g}.\] Upon simplification: \[V \propto r.\] The mass of water in the capillary is: \[m = \rho V \propto r.\] For a capillary with a radius of \( 2r \), the updated mass is: \[m_2 = 2m_1,\] indicating that the water mass is directly proportional to the capillary tube's radius. Therefore, the new mass is: \[2M.\] Final Answer: \[\boxed{2M}.\]