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Suppose you take a loop of string and place it on the surface of a liquid, the string should become taut due to surface tension. How would you be able to calculate the tension in the loop of string?

My main difficulty in solving this problem is the direction in which surface tension force act on the thread - perpendicular or tangential at any point on the thread?

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  • $\begingroup$ Interesting: why would the external tension (due to surface tension on the outside of the loop) differ from the compressive tension (due to surf. tension on the inside of the loop)? $\endgroup$ – Carl Witthoft Jan 30 '14 at 15:58
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    $\begingroup$ "should" Aristotle vs. Galileo! Do the experiment. Take a length of buoyant hydrophilic string, make a loop, and float it on a clean water surface. Does it collapse upon itself for depressing the surface where it touches? Do the same for a hydrophobic string (the example above, but waxed, e.g., dental floss). Repeat after adding a drop of detergent. Get back to us with the results. $\endgroup$ – Uncle Al Jan 30 '14 at 16:36
  • $\begingroup$ The surface tension will be the same inside and outside the loop, so the loop won't stretch or indeed move in any way. You could put the loop on a soap film then burst the film inside the loop. That would stretch it. $\endgroup$ – John Rennie Jan 30 '14 at 17:42
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Even though you may not have seen this before, but the string will not necessarily always behave the same.

If it is just the string and the liquid then you will get an effect called the Cheerios effect (Am. J. Phys. 73, 817 (2005)) that causes the string to clot. This is caused by the string locally deforming the surface due to a balance of buoyancy and wetting adhesion, which causes the surface energy to go up. Minimization of the surface energy drives the string to clot together such that it deforms the surface as little as possible.

Note that this clotting will only occur if the energy gain for by clotting is not balanced by an energetic contribution for the bending of the string i.e. if it would be a rubber like material it will probably not (fully) clot, but with a normal piece of cotton string I think it will.

A nice circular loop can form if Marangoni effects play as suggested by Mark Rovetta. In this case adding a detergent or a bit of oil inside the string will cause the surface tension to lower such that it is energetically favorable to create more surface area with this low surface tension, causing the liquid with oil/detergent to flow outwards and open up the string into a loop.

I have explained this in terms of energies (because that is so much easier when thinking about surface tension effects), but to answer the part of your question on the direction of the force: it acts perpendicular to thread because that is the direction of the energy gradient when clotting or expanding. The tension in the thread itself is indeed parallel to the thread as explanation by Mark.

Footnote: I just found out that this question is strongly related to Why does a cork float to the side of a glass?, in particular the answer by Pulsar is useful in this context.

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Because it is string, virtually one dimensional, all the force it can support is parallel its length - the tension T.

If you put a drop of oil on the water, inside the loop, it spreads because the energy of the oil-water interface is less than that oil-air interface.

The loop stops expanding (equilibrium) when an incremental increase in oil-water area releases just enough energy to do the work to incrementally stretch the string in critical tension Tc (hint: use the elastic modulus of the string to solve.)

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  • $\begingroup$ Sorry, sir. My bad $\endgroup$ – user34304 Jan 31 '14 at 13:58

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