# Measurement of Surface Tension by Capillary Rise Method | Physics Grade 11 Notes

### Fluid Mechanics

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Measurement of Surface Tension by Capillary Rise Method
Consider a capillary tube of radius r open at both ends and dipped into a liquid which has a concave meniscus as shown in the figure. Let θ be the contact, h be the height of liquid rises, ϱ be the density of the liquid, and T be the surface tension of the liquid. The surface tension force causes the liquid to exert a downward directed force T on the walls of the tube. This force T acts along the tangent at the point of contact A. From newton’s third law of motion, the tube exerts equal and opposite reaction R. the reaction (R = T) can be resolved into two rectangular components so that Tsinθ and Tcosθ are obtained along horizontal and vertical respectively. The horizontal component cancels to each other whereas vertical components are added which pulls the liquid upward. The component T cosθ acts along the whole circumference of the meniscus.

.: Total upward force = Tcosθ * 2πr

Volume of liquid in the tube above the free surface liquid is given by
V = volume of cylinder of height h and radius r + Volume of cylinder of height r and radius r – volume of hemisphere of radius r
= πr2h + πr2 r + - ½  (4/3 πr3)
= πr2h + πr3 – 2/3 πr3
= πr2h + 1/3 πr3
V = πr2 (h + r/3)

Weight of liquid column = Mass of liquid column * g
= ϱ * V * g
= ϱπr2 (h + r/3) g

For equilibrium,
Total upward force = Weight of the liquid column
Tcosθ * 2πr = ϱπr2 (h + r/3)g
or, T = ϱ r (h + r/3) g/2cosθ

When the tube is very fine bore, r/3 may be neglected compared to h, so surface tension T becomes
T = ϱrhg / 2cosθ
H = 2T cosθ / ϱ r g

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