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You are confusing yourself. The statement $P - P_0$ would remain the same is false. Why would it remain the same? There is a certain amount of compressed gas inside the bubble, and there is a force that maintains it compressed. In the first case, this force is just the surface tension. In the second case, it is the surface tension reduced by the ...


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You're calculating the rate of liquid flow through a tube under a specified pressure gradient. The rate of flow changes depending on whether the flow is laminar or turbulent. Laminar flow is described by the Hagen-Poiseuille equation: $$ \Delta P = \frac{8\mu\ell V}{\pi r^4} $$ where $\Delta P$ is the pressure drop, $\mu$ is the viscosity of your saline ...


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Your fractal pattern will fail for reasons already given. However, given a large enough lake, you should be able to stand on a frame which is supported by a very long (perhaps circular) wire. All you need is for the force per meter (assuming uniformly applied) caused by your body weight and the structure itself to be less than the surface tension force ...


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The problem lies in your simplistic assumption that the perimeter is the only thing that matters. The actual force can be no greater that the weight of displaced water (see for example a capillary) and as the force you try to exert, so the amount of water displaced will increase. That doesn't mean you could not use surface tension to "walk on water" - just ...


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Since the force is based on the wetted perimeter, any configuration that would make the perimeter very large in a very small area would be overwhelmed by the surface tension of the water droplets connecting nearby perimeters. So the effective perimeter would be much lower. So you are sunk!


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As if you will see that a capillary tube kept in a beaker filled with water,so the water level rises but if the length of the capillary tube is insufficient than the angle of cos theta will be of 90 it means that its just impossible,and the surface tension force will be just stopped.



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