Engineering
Physics
Capillary and Pressure Difference due to Surface Tension
Question

A cylindrical capillary tube of 0.2 mm radius is made by joining two capillaries T1 and T2 of different materials having water contact angles of 0° and 60°, respectively. The capillary tube is dipped vertically in water in two different configurations, case I and II as shown in figure. Which of the following option(s) is(are) correct ?

(Surface tension of water = 0.075 N/m, density of water = 1000 kg/m3, take g = 10 m/s2)

For case I, if the capillary joint is 5 cm above the water surface, the height of water column raised in the tube will be more than 8.75 cm. (Neglect the weight of the water in the meniscus)

The correction in the height of water column raised in the tube, due to weight of water contained in the meniscus, will be different for both cases.

For case I, if the joint is kept at 8 cm above the water surface, the height of water column in the tube will be 7.5 cm. (Neglect the weight of the water in the meniscus)

For case II, if the capillary joint is 5 cm above the water surface, the height of water column raised in the tube will be 3.75 cm. (Neglect the weight of the water in the meniscus)

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Solution

h=2TcosθρgR   ;   h1=2×0.075×cos0°1000×10×0.2×103

⇒ h1 = 75 mm (in T1) [If we assume entire tube of T1]

⇒ h2=2×0.075×cos60°1000×10×0.2×103  = 37.5 mm (in T2) [If we assume entire tube of T2]

Option (A) : Since contact angles are different so correction in the height of water column raised in the tube will be different in both the cases.

Option (B) : If joint is 5 cm is above water surface, then lets say water crosses the joint by height h, then:

 P02Tr+ρgh+ρg×5×102                      

    = P0

cosθ=Rr·r=Rcosθ

 

ρg(h+5×102)=2TcosθR

 

h=2×0.075×cos60°0.2×103×1000×10=5×102

⇒ h = –ve, not possible, so liquid will not cross the interface, but angle of contact at the interface will change, to balance the pressure,

So option (B) is wrong.

Option (C) : If interface is 8 cm above water then water will not even reach the interface, and water will rise till 7.5 cm only in T1, so option (C) is right.

Option (D) : If interface is 5 cm above the water in vessel, then water in capillary will not even reach the interface. Water will reach only till 3.75 cm, so option (D) is right.