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Given a needle of mass $m$ modeled by a cylinder of length $l$ and radius $r$ placed on an infinitely large water surface, what is

  1. The maximum depression in the water surface; and
  2. The equation of the shape of the water surface when depressed?

I'm quite sure this is a simple question that already has been modeled theoretically somewhere, but I haven't been able to find a theoretical treatment of the problem with experimentally verifiable quantities.

In addition, can the treatment of the problem be extended to thin films, for example a thin soap film?

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Given the volume of the needle and it's mass density, you can calculate the surface tension force based on a simple free body diagram. You might need to use the concept of a contact angle perhaps and some trigonometry to figure out what the depression is. – drN Mar 1 '13 at 20:21
@drN wouldn't both the contact angle and the amount of needle in contact with water affect things? mathematically I can think of two configurations where the contact angle is identical but the surface area in contact is different, and thus the angle between the surface tension force and the weight of the needle is different. – Vincent Tjeng Mar 2 '13 at 1:57
Good point. I'll ruminate more about this. – drN Mar 2 '13 at 3:51

Here is a link on surface tension that treats the needle example with an equation of balance of forces.

And here is another one .

enter image description here

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Hi there, I think both links provided do not specifically give a value for the amount the needle depresses the water surface by. Would it be correct to say that the needle is essentially "floating" in the water i.e. not depressing the water at all? It seems rather counter-intuitive to me. – Vincent Tjeng Jan 31 '13 at 5:44
In addition, the first link suggests that the force due to surface tension can be exactly anti-parallel to the weight of the needle. Is this in fact physically possible? – Vincent Tjeng Jan 31 '13 at 5:45
it is the vector sum of the ST force on the two sides that HAS to compensate the weight of the needle to float.… . to get a specific value you would need to enter all the constants in the formula in the link . – anna v Jan 31 '13 at 6:02
thank you for your time. are you referring to equation 4.9 in your link? If so, based on my understanding, θ could be calculated by taking the vector sum of the forces due to surface tension and the weight of the needle. However, that still doesn't answer the question of the depression of the surface below the needle or the shape of the surface close to the needle, since the shape of the curve is not determined. – Vincent Tjeng Jan 31 '13 at 7:25

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