Question

A capillary tube of radius 0.1 mm is partly dipped in water (surface tension 70 dyn/cm and glass water contact angle $ \approx 0^\circ $) with $ 30^\circ $ inclined with vertical. The length of water risen in the capillary is ____ cm. (Take $ g = 9.8 $ m/s$^2 $)

• A. \( \frac{82}{5} \)
• B. \( \frac{57}{2} \)
• C. \( \frac{71}{5} \)
• D. \( \frac{68}{5} \)

Hint:

Ensure consistent units throughout the calculation. The vertical height of the liquid column in the capillary is determined by the Jurin's law. When the capillary is inclined, the length of the liquid column along the tube is related to the vertical height through trigonometric relations involving the angle of inclination.

Answer 1

The Correct Option is A

To determine the length of water rise in a capillary, we use the capillary action formula:

\( h = \frac{2T \cos \theta}{r \rho g} \)

Where:

• \( T \) is the surface tension of water, \( T = 70 \) dyn/cm \( = 0.07 \) N/m (since \( 1 \) dyn/cm is \( 0.001 \) N/m)
• \( \theta \) is the contact angle, \( \theta \approx 0^\circ \Rightarrow \cos 0^\circ = 1 \)
• \( r \) is the radius of the capillary, \( r = 0.1 \) mm \( = 0.01 \) cm \( = 0.0001 \) m
• \( \rho \) is the density of water, approximately \( 1000 \) kg/m³
• \( g \) is the acceleration due to gravity, \( 9.8 \) m/s²

Substituting these values into the formula:

\( h = \frac{2 \times 0.07 \times 1}{0.0001 \times 1000 \times 9.8} \)

\( = \frac{0.14}{0.98} \)

\( = 0.142857 \) m \( = 14.2857 \) cm

However, since the capillary tube is inclined at \( 30^\circ \) to the vertical, the actual length of the water column along the tube, \( l \), is given by the relationship:

\( l = \frac{h}{\cos 30^\circ} \)

Where:

• \( \cos 30^\circ = \frac{\sqrt{3}}{2} \approx 0.866 \)

Calculating \( l \):

\( l = \frac{14.2857}{0.866} \)

\( \approx 16.5 \) cm

Thus, the length of water risen in the capillary tube is \( \frac{82}{5} \approx 16.4 \) cm, which matches closely with our calculation due to rounding.

The correct answer is: \( \frac{82}{5} \)