[Physics] Depression of Water Surface by a Needle

Question

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?

Answer 1

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

And here is another one .