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 given by the capillary rise formula: \[ h = \frac{2T \cos \theta}{r \rho g}, \] where \( T \) is the surface tension, \( \theta \) is the angle of contact, \( r \) is the radius of the tube, \( \rho \) is the density of the liquid, and \( g \) is the acceleration due to gravity. The volume of water in the capillary is: \[ V = \pi r^2 h = \pi r^2 \cdot \frac{2T \cos \theta}{r \rho g}. \] Simplifying: \[ V \propto r. \] The mass of water in the capillary is: \[ m = \rho V \propto r. \] For a capillary with radius \( 2r \), the new mass is: \[ m_2 = 2m_1, \] which shows that the mass of water is directly proportional to the radius of the capillary tube. Thus, the new mass is: \[ 2M. \] Final Answer: \[ \boxed{2M}. \]