JEE Advanced 2014 · Paper 2 · Physics Q10

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PhysicsSingle correct+3 / -1Question 10 of 60

A glass capillary tube is of the shape of a truncated cone with an apex angle $\alpha$ so that its two ends have cross sections of different radii. When dipped in water vertically, water rises in it to a height $h$ , where the radius of its cross section is $b$ . If the surface tension of water is $S$ , its density is $\rho$ , and its contact angle with glass is $\theta$ , the value of $h$ will be ( $g$ is the acceleration due to gravity)

A truncated-cone glass capillary, wider at the bottom and narrowing upward, dipped vertically in water; the water column inside rises to height h above the outside water level.

\dfrac{2S}{b\rho g}\cos(\theta-\alpha)

\dfrac{2S}{b\rho g}\cos(\theta+\alpha)

\dfrac{2S}{b\rho g}\cos(\theta-\alpha/2)

\dfrac{2S}{b\rho g}\cos(\theta+\alpha/2)

Answer and worked solution

Official answer: D

Angle of the surface tension. The wall of the cone is tilted by $\alpha/2$ from the vertical, and the tube narrows upward. The liquid surface meets the wall at the contact angle $\theta$ , so the pull of surface tension makes an angle $\theta+\alpha/2$ with the vertical.

Balance at the meniscus (radius $b$ ), with the column treated as of radius $b$ :
$2\pi bS\cos\left(\theta+\frac\alpha2\right)=\pi b^2h\rho g\implies h=\frac{2S}{b\rho g}\cos\left(\theta+\frac\alpha2\right).$

(D)

Topics:Capillarity Surface tension

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JEE Advanced 2014 · Paper 2, Physics question 10