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JEE Main 20149 AprilOnline
Exam
JEE Main
Year
2014
Date
9 April
Mode
Online
Subject
Physics
Chapter
Mechanical Properties of Fluids

JEE Main 2014 · 9 April · Online · Physics · Mechanical Properties of Fluids

A capillary tube is immersed vertically in water and the height of...

A capillary tube is immersed vertically in water and the height of the water column is xx. When this arrangement is taken into a mine of depth d, the height of the water column is yy. If R is the radius of earth, the ratio xy\dfrac{x }{y } is :

Options

  1. A(R−dR+d)\left(\dfrac{\text{R} - \text{d}}{\text{R} + \text{d}}\right)
  2. B(R+dR−d)\left(\dfrac{\text{R} + \text{d}}{\text{R} - \text{d}}\right)
  3. C(1−2dR)\left(1 - \dfrac{2 \text{d}}{\text{R}}\right)
  4. D(1−dR)\left(1 - \dfrac{\text{d}}{\text{R}}\right)

Correct answer

D. (1−dR)\left(1 - \dfrac{\text{d}}{\text{R}}\right)

Step-by-step solution

We know that

h=2T cosθρgr\text{h} = \dfrac{2 \text{T cos} \theta}{ρ \text{gr}}

∴x=2Tcosθρgr∴ \text{x} = \dfrac{2 \text{T} cos \theta}{ρ \text{gr}}

Now, value of geff at depth 'd' is

geff=g(1−dR){\text{g}}_{\text{eff}} = \text{g} \left(1 - \dfrac{\text{d}}{\text{R}}\right)

∴y=2Tcosθρgeff⋅r∴ \text{y} = \dfrac{2 \text{T} cos \theta}{ρ {\text{g}}_{\text{eff}} \cdot \text{r}}

∴xy=geffg=1−dR∴ \dfrac{\text{x}}{\text{y}} = \dfrac{{\text{g}}_{\text{eff}}}{\text{g}} = 1 - \dfrac{\text{d}}{\text{R}}
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