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Question

# A uniform capillary tube of inner radius r is dipped vertically into a beaker filled with water. The water rises to a height h in the capillary tube above the water surface in the beaker. The surface tension of water is σ. The angle of contact between water and the wall of the capillary tube is θ. Ignore the mass of water in the meniscus. Which of the following statements is (are) true?

A
For a given material of the capillary tube, h decreases with increase in r
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B
For a given material of the capillary tube, h is independent of σ
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C
If this experiment is performed in a lift going up with a constant acceleration, then h decreases
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D
h is proportional to contact angle θ
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Solution

## The correct option is C If this experiment is performed in a lift going up with a constant acceleration, then h decreasesrelation between surface tension to water density is given by, 2σR=ρgh ⇒h=2σRρg .....(1) where, R= Radius of meniscus r= Radius of capillary h=height of tube Relation between radius of capillary and contact angle θ R=rcosθ .....(2) θ = contact angle putting eq. (2) value into eq. (1) h=2σcosθrρg .....(3) from the above expression a relationship between h and r is, h∝1r so as h decreases r increase simultaneously. Hence, option A is correct. For option B, ⇒h=2σRρg h∝σ So, for a given material of the capillary tube, h is dependent on σ. Hence option B is wrong. For option C, If lift is going upwards with the constant acceleration then the acceleration due to the gravity will be glift=g+a So, from equation 3, h=2σcosθrρ(g+a) where, h∝1(g+a) hence, option C is correct. For option D, from eq. (3) h=2σcosθrρg where, h∝cosθ not h∝θ Hence, option D is wrong.

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