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Question

A liquid of density ρ and surface tension σ rises to a height h in a capillary tube of diameter d. The potential energy of the liquid in the capillary tube is:
[Assume liquid to be perfectly wetting in nature]

A
hρg
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B
2πσ2ρg
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C
2πσ2ρg
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D
2πσ2ρgh
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Solution

The correct option is B 2πσ2ρg
The mass of the liquid in the capillary tube can be considered to be concentrated at the centre of mass of the liquid column of height h i.e COM is at a height h2.

The potential energy of the liquid in the capillary tube is,
U=mg×h2
U=(πr2hρ)×g×(h2)
U=π(rhg)2ρ2g ....(1)
The equation for capillary rise,
h=2σcosθρrg
For perfectly wetting liquid, cosθ=1
σ=hρrg2
rhg=2σρ ....(2)

Substituting Eq.(2) in Eq.(1), the PE of a liquid in the capillary tube can be written as,
U=π2g(2σρ)2ρ
U=2πσ2ρg
Hence, the correct choice is (b)

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